Research ArticleOn the Optimal Auxiliary Linear Operator for the SpectralHomotopy Analysis Method Solution of Nonlinear OrdinaryDifferential Equations
S S Motsa
School of Mathematics Statistics and Computer Science University of KwaZulu-Natal Private Bag X01Pietermaritzburg Scottsville 3209 South Africa
Correspondence should be addressed to S S Motsa sandilemotsagmailcom
Received 9 July 2014 Accepted 31 July 2014 Published 14 August 2014
Academic Editor Mohammad Mehdi Rashidi
Copyright copy 2014 S S Motsa This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The purpose of this study is to identify the auxiliary linear operator that gives the best convergence and accuracy in theimplementation of the spectral homotopy analysis method (SHAM) in the solution of nonlinear ordinary differential equationsThe auxiliary linear operator is an essential element of the homotopy analysis method (HAM) algorithm that strongly influencesthe convergence of the method In this work we introduce new procedures of defining the auxiliary linear operators and comparesolutions generated using the new linear operators with solutions obtained using well-known linear operators The applicabilityand validity of the proposed linear operators is tested on four highly nonlinear ordinary differential equations with fluid mechanicsapplications that have recently been reported in the literature The results from the study reveal that the new linear operators givebetter results than the previously used linear operators The identification of the optimal linear operator will direct future researchon further applications of HAM-based methods in solving complicated nonlinear differential equations
1 Introduction
Thehomotopy analysismethod (HAM) is an analyticmethodthat has been widely used for solving highly nonlinearequations with applications in computational and appliedmathematics economics and finance engineering andmanyother areas of fundamental science A systematic expositionof features of the HAM and its application can be found inrecent books by Liao [1ndash3] and Vajravelu and Van Gorder[4] A distinctive characteristic of the HAM that sets itapart from all other analytical methods is the presence ofa convergence-controlling parameter and the flexibility toselect auxiliary functions and linear operators in order toguarantee convergence and improve accuracy of the approx-imate solutions This makes the HAM suitable for solvingmany highly nonlinear problems including those that do notcontain small or large embedded parameters
A discrete numerical approach that uses the principles ofthe traditional HAM and combines them with the Cheby-shev spectral collocation method was introduced by Motsa
et al [5 6] and called spectral homotopy analysis method(SHAM) In the traditional HAM approach the options ofinitial approximations and auxiliary linear operators andfunctions are limited by the requirement that the obtainedapproximate solution must be a continuous analytical seriessolution In the SHAMapplication this restriction is removedand consequently the SHAM can accommodate infiniteoptions of initial approximations auxiliary linear operatorsand convergence-controlling auxiliary functions This makesthe SHAM more robust and capable of solving a widerrange of complicated nonlinear equations than its analyticalcounterpart
The effect of different auxiliary linear operators on theaccuracy and convergence of the HAM has been studiedby different authors in the past Van Gorder and Vajravelu[7] presented different methods for selecting auxiliary linearoperators and gave the necessary and sufficient conditionsfor the convergence of the HAM series solutions The aim ofthe illustrative study of [7] was to demonstrate how differentlinear operators can be obtained A comparative analysis of
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 697845 15 pageshttpdxdoiorg1011552014697845
2 Mathematical Problems in Engineering
the accuracy and convergence of the approximate solutionsobtained using their proposed linear operators was not con-sidered Shan and Chaolu [8] proposed a linear operator con-structed by forming a differential equation using the guessedinitial solution that satisfies the boundary conditions Themethod proposed in [8] was demonstrated using theThomasFermi equation and a nonlinear heat transfer equation VanGorder [9] discussed methods for selecting auxiliary linearoperators in solving the Painleve equations Lane-Emdenequation and a third order equation that models a nonlinearstretching sheet in a fluid flow problem It was noted thatfor some auxiliary linear operators the rate of convergencemay be slow and for others the rate of convergence wasrapid All the studies reviewed so far however suffer fromthe fact that the linear operators have to be carefully selectedin such a way that the solution method gives analyticalsolutions made up of simple functions such as polynomialsdecaying exponential sines cosines rational functions andother elementary functions or products of such functionswhich are fairly easy to integrate A particular method ofselecting a linear operator such as the method of completedifferential matching used in [7 9] can only be used in veryfew nonlinear equations if the output solution is expectedto be a simple series function Despite being suggested in[7] as a choice of linear operator that potentially can givebetter convergence and accuracy the method of completedifferential matching has not been widely used in the HAMliterature because it often leads to higher order decomposedequations that cannot be integrated analytically
In seeking to identify auxiliary parameters and operatorsthat give the best accuracy and convergence of the HAMalgorithm a more suitable study would seek to identify theoptimal parameters from the infinitely many possibilities andnot to search from the small set of options that guaranteeexact solutions of the decomposed equations The aim of thispaper is to determine the auxiliary linear operator and initialapproximation that give the optimal accuracy and conver-gence in the HAM solution of highly nonlinear boundaryvalue problems defined in bounded domainsThe study givesa systematic way of selecting initial approximations auxiliarylinear operators and convergence controlling parameterNew methods of defining linear operators in the contextof the discrete spectral method based version of the HAM(SHAM) are proposed The accuracy and convergence of theSHAM algorithms derived using the proposed new linearoperators are validated on the well-known nonlinear differ-ential equations that have been previously solved using theHAMSHAM algorithms with the standard linear operatorsIn particular we consider the Darcy-Brinkman-Forchheimermodel [5 10] Jeffery-Hamel equation [6 11ndash13] the laminarviscous flow in a semiporous channel subject to a transversemagnetic field [14ndash16] and the model of two-dimensionalviscous flow in a rectangular domain bounded by twomovingporous walls [17ndash20]
The remainder of the paper is organized as followsIn Section 2 we give a brief description of the SHAM forgeneral nonlinear ordinary differential equations Section 3introduces the auxiliary linear operators that are used inthe SHAM algorithm Section 4 describes the application
of the SHAM with the proposed linear operators in theproblems that are selected for numerical experimentationThe numerical simulations and results are presented inSection 5 Finally we conclude and describe future work inSection 6
2 Description of the Spectral HomotopyAnalysis Method in a General Setting
In this section we give the basic description of the homotopyanalysis method for the solution of a general nonlineardifferential equation of order 119901 (where 119901 is a positive integer)given by
where 120572119895(119909) and Φ(119909) are known functions of the indepen-dent variable 119909 and 119910(119909) is an unknown functionThe primesin (1) denote differentiation with respect to 119909 and 119910(0) equiv 119910(119909)The function 119873 represents the nonlinear component of thegoverning equation For illustrative purposes we assume that(1) is to be solved in the domain 119909 isin [119886 119887] subject to theseparated boundary conditions
where 119861119886 and 119861119887 are linear operatorsIn the framework of the homotopy analysis method
(HAM) [1ndash4] we define the following zeroth order deforma-tion equations
(1 minus 119902)L [119884 (119909 119902) minus 1199100 (119909)] = 119902ℏ N [119884 (119909 119902)] minus Φ (119909)
(3)
where 119902 isin [0 1] denotes an embedding parameter 119884(119909 119902) isa kind of continuous mapping function of 119910(119909) and ℏ is theconvergence-controlling parameter The nonlinear operatorN is defined from the governing equation (1) as
By differentiating the zeroth order equations (3) 119898 timeswith respect to 119902 setting 119902 = 0 and finally dividing theresulting equations by119898 we obtain the following119898th orderdeformation equations
L [119910119898 (119909) minus (120594119898 + ℏ) 119910119898minus1 (119909)] = ℏ119877119898minus1 [1199100 1199101 119910119898minus1]
From the solutions of (6) the approximate solution for119910(119909) is determined as the series solution
119910 (119909) =
+infin
sum
119896=0
119910119896 (119909) (8)
AHAMsolution is said to be of order119870 if the above seriesis truncated at 119896 = 119870 that is if
119910 (119909) =
119870
sum
119898=0
119910119898 (119909) (9)
For nontrivial linear operators the higher order equa-tions (6) cannot be integrated using analytical means Con-sequently numerical approaches are employed When theChebyshev spectral collocation method is used to solve (6)the method is called spectral homotopy analysis method[3 5 6]
In using the SHAM the initial guess is obtained simplyas a solution of the linear part of the governing equation (1)subject to the underlying boundary conditions (2)That is wesolve
In essence collocation approximates the solution 119910(119909)using an interpolating polynomial of degree119872which satisfiesthe boundary conditions and the differential equations atall points called the collocation points 119909119895 where 119895 =
0 1 119872 In the Chebyshev spectral collocation methodthe collocation points are chosen to be the extrema ofChebyshev polynomials of degree 119879119872 on the interval minus1 le120578 le 1 defined as
120578119895 = cos(120587119895
119872) 119895 = 0 1 119872 (11)
We use the transformation 119909 = (119887 minus 119886)(120578 + 1)2 to map theinterval [119886 119887] to [minus1 1] The so-called differentiation matrix119863 is used to approximate the derivatives of the unknownvariables 119910(119909) at the collocation points as the matrix vectorproduct
Thematrix119863 is of size (119872+1)times(119872+1) and its entries aredefined in [21 22] In the SHAM algorithm the continuousderivatives of the higher order deformation equations arereplaced by the discrete Chebyshev differentiation matricesAs a result the higher order deformation equations reduceto matrix equations and are solved using standard techniquesfor solving linear systems of equations
3 Definition and Selection of LinearAuxiliary Operators
In this section we describe the different approaches used todefine the auxiliary linear operators in the SHAM algorithmIn order to highlight the subtle differences between thevariety of linear operators it is convenient to express thenonlinear operator of the governing equation (1) using a sumformula of the derivative products as follows
where 120573119903119895(119909) (119895 = 0 1 119901) are coefficients of thenonlinear terms containing 119910119903 as the highest derivative Inthis way (1) becomes
The selection of auxiliary linear operators in the appli-cation of the homotopy analysis method was described byVan-Gorder and Vajravelu [7] in a fairly general setting Inparticular three methods namely the method of highest
4 Mathematical Problems in Engineering
order differential matching linear partition matching andcomplete differential matching were defined in [7] Below wegive a definition of the linear operator selection of [7] andpresent other options that can be implemented under theSHAM algorithm for any nonlinear differential equation thatcan be represented by (16)
(i) Method of the Highest Order Differential Matching Thisapproach defines the auxiliary linear operator using only thehighest order derivative This approach is the most widelyused approach in the HAM solution of nonlinear differentialequations defined in finite domains With reference to (16)the linear operator is selected as
L1 [119910] =119889119901119910
119889119909119901 (17)
(ii) Method of Linear Partition Matching This approachdefines the auxiliary linear operator to be the collection ofall linear terms in the governing equation With reference to(16) we set
L2 [119910] =
119901
sum
119895=0
120572119895 (119909)119889119895119910
119889119909119895 (18)
(iii) Method of Complete Differential Matching In thisapproach the collection of all linear operators and somenonlinear factors of the governing nonlinear equations areused to define the linear operator The aim is to ensurethat all the terms of the governing nonlinear differentialequation contribute to the auxiliary linear operator selectedThe following rules were defined in [7]
(a) in the case where we have a termwhich is the productof derivatives we take the higher order derivative inthe term
(b) if the term has a product of derivatives with functionsof the unknown function we again take the highestorder derivative in the term
(c) in the case where we have a nonlinear expression injust the unknown function we take the function itself
Thus applying the above rules to (16) we set
L3 [119910] =
119901
sum
119895=0
120572119895 (119909)119889119895119910
119889119909119895+
119901
sum
119903=0
119889119903119910
119889119909119903
119903
sum
119895=0
120573119903119895 (19)
(iv) Linear Partition Mapping after Transformation Thisapproach uses linear partition mapping after the transforma-tion 119910(119909) = 119911(119909)+1199100(119909) where the function 1199100(119909) is carefullychosen to satisfy the underlying boundary conditions Inearlier versions of the SHAM[5 6] it was suggested that1199100(119909)be defined as a solution of (10) which is solved subject to
the given boundary conditions Substituting in the governingequation (1) to express the equation in terms of 119911(119909) gives
119901
sum
119895=0
120572119895 (119909) 119911(119895)(119909) +
120597119873
120597119911(1199100 119910
1015840
0 119910
(119901)
0) 119911
+120597119873
1205971199111015840(1199100 119910
1015840
0 119910
(119901)
0) 1199111015840
+ sdot sdot sdot +120597119873
120597119911(119901)(1199100 119910
1015840
0 119910
(119901)
0) 119911(119901)+Nonlinear Terms
= Φ (119909) minus
119901
sum
119895=0
120572119895 (119909) 119910(119895)
0(119909) minus 119873 (1199100 119910
1015840
0 119910
(119901)
0)
(20)
Thus the linear operator is set to be
L4 [119911] =
119901
sum
119895=0
120572119895 (119909) 119911(119895)(119909) +
120597119873
120597119911(1199100 119910
1015840
0 119910
(119901)
0) 119911
+ sdot sdot sdot +120597119873
120597119911(119901)(1199100 119910
1015840
0 119910
(119901)
0) 119911(119901)
(21)
(v) Modified Complete Differential Matching Method Thisis a new approach that is proposed as a modification ofthe method of complete differential matching of [7] In theoriginal approach [7] from the group of nonlinear termsof the governing equation only the highest derivative isselected In the proposed approach we want to ensure thatall the terms of the nonlinear groups that make up the termsof the differential equation contribute to the linear operatorWe define the following rules which are a modification of therules set in [7]
(a) In the case where we have a termwhich is the productof derivatives we take the higher order derivativein the term and approximate each function in theremaining derivative product by 1199100(119909) For exampleif the nonlinear product is 1199101015840(119909)11991010158401015840(119909)119910101584010158401015840(119909) we set1199101015840
0(119909)11991010158401015840
0(119909)119910101584010158401015840(119909) where 1199100(119909) is the solution of (10)
(b) If the term has a product of derivatives with func-tions of the unknown function we again take thehighest order derivative in the term and approximateeach function in the remaining derivative productby 1199100(119909) For example if the nonlinear product is1199102(119909)119910101584010158401015840(119909) we set 1199102
0(119909)119910101584010158401015840(119909) where 1199100(119909) is the
solution of (10)(c) In the case where we have a nonlinear expression in
just the unknown function we take the function itselfand approximate the rest of the functions by 1199100(119909)For example if the nonlinear product is 1199102(119909) we set1199100(119909)119910(119909) where 1199100(119909) is the solution of (10)
4 Numerical Experiments
In this section we illustrate how using a different linearoperator can significantly improve convergence and accuracy
Mathematical Problems in Engineering 5
of the SHAM For illustration purposes we consider examplesof nonlinear differential equations that have previously beensolved in the published literature using theHAMSHAMwithstandard linear operators It is worth mentioning that in thepreviousHAM-based investigations the linear operators werechosen in such a way that
(i) the obtained approximate solution is a continuousanalytical expression
(ii) the HAM series solution conforms to a predefinedrule of solution expression
The spectral method based approach of [5 6] was aimedat removing the above restrictions by considering linearoperators which are defined from part of the governingdifferential equation In this case it was found that thelinearised deformation equations could not be solved exactlyand hence the use of the spectral method Below we givea systematic procedure for defining the linear operatorsfor various nondifferential equations selected for numericalexperimentation
41 Example 1 Darcy-Brinkman-Forchheimer Equation Weconsider the following Darcy-Brinkman-Forchheimer equa-tion that models the steady state pressure driven fully devel-oped parallel flow through a horizontal channel that is filledwith porous media
11991010158401015840(119909) minus 119904
2119910 (119909) minus 119865119904119910(119909)
2+1
119872= 0
119910 (minus1) = 0 119910 (1) = 0
(22)
where 119865 is the dimensionless Forchheimer number and 119904is the porous media shape parameter This problem waspreviously solved using the SHAM with the auxiliary linearoperator selected using themethod of linear partitionmatch-ing L2 in [5] The equivalent problem cast in cylindricalcoordinates was subsequently solved in [10] where the linearoperator was selected as the linear part of the governingequation excluding the linear termswith variable coefficientsFor this reason we remark that the linear operator used in[10] is not part of the class of linear operators considered inthis study Using the systematic choices described above weset
L3 [119910] = 11991010158401015840minus 1199042119910 minus 119865119904119910
L4 [119911] = 11991110158401015840minus 1199042119911 minus 21198651199041199100119911
L5 [119910] = 11991010158401015840minus 1199042119910 minus 1198651199041199100119910
(23)
The linear operator L4 is obtained by using the linearpartition mapping after using the transformation 119910(119909) =119911(119909) + 1199100(119909) The resulting equation is
11991110158401015840minus 1199042119911 minus 21198651199041199100119911 minus 119865119904119911
2= minus(119910
10158401015840
0minus 11990421199100 minus 119865119904119910
2
0+1
119872)
(24)
The function 1199100(119909) is chosen as a function that satisfies theboundary conditions and the linear part of the (22) That is1199100(119909)must be a solution of
11991010158401015840
0minus 11990421199100 = minus
1
119872 (25)
Thus the initial approximation is chosen to be
1199100 (119909) =1
1199042119872(1 minus
cosh (119904119909)cosh (119904)
) (26)
The initial approximation used in the SHAM algorithms thatuse the linear operators L1 L2 and L3 is obtained bysolving the linear equations
Thus the initial approximations corresponding to theSHAM withL1L2 andL3 are given respectively by
1199100 (119909) =1
2119872(1 minus 119909
2) 1199100 (119909) =
1
1199042119872(1 minus
cosh (119904119909)cosh (119904)
)
1199100 (119909) =1
11990421119872(1 minus
cosh (1199041119909)cosh (1199041)
)
(28)
where 1199041 = radic1199042 + 119865119904The initial approximation 1199110(119909) to use in conjunction
withL4 is obtained as a solution of the differential equationformed from the linear part of the (24) That is we solve
11991110158401015840
0minus 11990421199110 minus 211986511990411991001199110 = minus(119910
10158401015840
0minus 11990421199100 minus 119865119904119910
2
0+1
119872)
1199110 (minus1) = 1199110 (1) = 0
(29)
This equation is a linear equation with variable coefficientsThus it is solved using the spectral method as described inthe earlier section above Similarly the initial approximationto use with L5 is obtained as a solution of the differentialequation
11991010158401015840
0minus 11990421199100 minus 11986511990411991001199100 = minus
1
119872 1199100 (minus1) = 1199100 (1) = 0 (30)
with 1199100 given by (26)With the nonlinear operator 119873119894 defined as the homoge-
neous part of the governing nonlinear equation in each casewe obtain the following higher order deformation equations
L119894 [119910119898 (119909) minus 120594119898119910119898minus1 (119909)]
1198774119898minus1 =L4 (119911119898minus1) minus (1 minus 120594119898) 1206011 (119909) minus 119865119904119898minus1
sum
119899=0
119911119899119911119898minus1minus119899
(34)
and 1206011(119909) is the right hand side of (29)
42 Example 2 Jeffery-Hamel Equation Here we considerthe Jeffery-Hamel equation that models the steady two-dimensional flow of an incompressible conducting viscousfluid between two rigid plane walls that meet at an angle 2120572The rigid walls are considered to be divergent if 120572 gt 0 andconvergent if 120572 lt 0 The governing equation is defined as
where Re is the Reynolds number This problem was inves-tigated using the standard HAM with the highest orderdifferential matching linear operator L1(119910) in [11 12] Arelatedmethod called optimal homotopy asymptoticmethod(OHAM) was used to solve the same problem in [13] againwith L1(119910) In Motsa et al [6] the SHAM was used withthe auxiliary linear operator defined using the transformedlinear partition mapping methodL4(119911) In this example weexplore the auxiliary linear operators defined as
The function 1199100(119909) is chosen as a function that satisfiesthe boundary conditions and the linear part of (35) Thus1199100(119909) is
1199100 (119909) =cos (2120572) minus cos (2120572119909)
cos (2120572) minus 1 (38)
The linear operator L4 is obtained by using the linearpartition mapping after using the transformation 119910(119909) =119911(119909) + 1199100(119909) The resulting equation is
Thus the initial approximations corresponding to the SHAMwithL1L2 andL3 are given respectively by
1199100 (119909) = 1 minus 1199092 1199100 (119909) =
cos (2120572) minus cos (2120572119909)cos (2120572) minus 1
1199100 (119909) =cos (21205721) minus cos (21205721119909)
cos (21205721) minus 1
(43)
where 1199041 = radic41205722 + 2120572Re The initial approximation 1199110(119909)to use with L4 is obtained as a solution of the differentialequation formed from the linear part of (39)That is we solve
119911101584010158401015840
0+ 412057221199111015840
0+ 2120572Re1199100119911
1015840
0+ 2120572Re1199101015840
01199110 = 1206012 (119909) (44)
Since the above equation has variable coefficients it is solvedusing the spectral method Similarly the initial approxima-tion to use withL5 is obtained as a solution of the differentialequation
119910101584010158401015840
0+ 412057221199101015840
0+ 2120572Re1199100119910
1015840
0= 0
1199100 (0) = 1 1199101015840
0(0) = 0 1199100 (1) = 0
(45)
with 1199100 given by (38)With the nonlinear operator 119873119894 defined as the homoge-
neous part of the governing nonlinear equation in each casewe obtain the following higher order deformation equations
L119894 [119910119898 (119909) minus 120594119898119910119898minus1 (119909)]
1198774119898minus1 =L4 (119911119898minus1) minus (1 minus 120594119898) 1206012 (119909) + 2120572Re119898minus1
sum
119899=0
1199111198991199111015840
119898minus1minus119899
(49)
Mathematical Problems in Engineering 7
43 Example 3 Laminar Viscous Flow in a Semiporous Chan-nel Subject to a Transverse Magnetic Field In this section weconsider the problem of laminar viscous flow in a semiporouschannel subject to a transverse magnetic field given by
1199101015840101584010158401015840minusHa211991010158401015840 + Re (119910119910101584010158401015840 minus 119910101584011991010158401015840) = 0 (50)
subject to the boundary conditions
119910 = 0 1199101015840= 0 119909 = 0
119910 = 1 1199101015840= 0 119909 = 1
(51)
This example was previously investigated using the stan-dardHAMwith the highest order differential matching linearoperatorL1(119910) in [14 15] In [16] a blend between the SHAMand a spectral method based quasilinearisation method wasused to solve the same problem In this investigation weconsider the following auxiliary linear operators
Thus the initial approximations corresponding to the SHAMwithL1 andL2 are given by
1199100 (119909) = 31199092minus 21199092
1199100 (119909) =Ha119909 minus 119890Ha119909 + 119890HaminusHa119909 + 119890Ha (Ha119909 minus 1) + 1
119890Ha (Ha minus 2) +Ha + 2
(54)
We note that in this example 1199100 is the second solution in(54)The initial approximations corresponding toL3 andL5are obtained by solving (53) numerically using the spectralmethod Similarly 1199110(119909) is obtained by numerically solving
1199111015840101584010158401015840
0minusHa211991110158401015840
0
+ Re (1199100119911101584010158401015840
0+ 119910101584010158401015840
01199110 minus 119910
1015840
011991110158401015840
0minus 11991010158401015840
01199111015840
0) = 1206013 (119909)
(55)
subject to the boundary conditions
1199110 (0) = 0 1199111015840
0(0) = 0 1199110 (1) = 0 119911
1015840
0(1) = 0
(56)
where
1206013 (119909) = minus (1199101015840101584010158401015840
0minusHa211991010158401015840
0+ Re (1199100119910
101584010158401015840
0minus 1199101015840
011991010158401015840
0)) (57)
The higher order deformation equations are defined as
L119894 [119910119898 (119909) minus 120594119898119910119898minus1 (119909)]
1198774119898minus1 = L4 (119911119898minus1) minus (1 minus 120594119898) 1206013 (119909)
+ Re119898minus1
sum
119899=0
(119911119899119911101584010158401015840
119898minus1minus119899minus 1199111015840
11989911991110158401015840
119898minus1minus119899)
(61)
44 Example 4 Two-Dimensional Viscous Flow in a Rectan-gular Domain Bounded by Two Moving Porous Walls In thissection we consider the problem of two-dimensional viscousflow in a rectangular domain bounded by twomoving porouswalls The governing equations are given in [17 23] as
101584010158401015840+ 311991010158401015840) + Re (119910119910101584010158401015840 minus 119910101584011991010158401015840) = 0 (62)
subject to the boundary conditions
119910 = 0 11991010158401015840= 0 at 119909 = 0
119910 = 1 1199101015840= 0 at 119909 = 1
(63)
where 120572 is the nondimensional wall dilation rate defined tobe positive for expansion and negative for contraction andRe is the permeation Reynolds number defined positive forinjection and negative for suction through the walls Thenonlinear equation (62) was investigated using the standardHAM with the highest order differential matching linearoperatorL1(119910) in [17 18] Rashidi et al [19] used the optimalhomotopy asymptotic method (OHAM) to solve the sameproblem again using L1(119910) as linear operator The SHAMwith transformed linear partition methodL4(119911) was used in[20]
8 Mathematical Problems in Engineering
In this investigation we consider the following auxiliarylinear operators
1198774119898minus1 = L4 (119911119898minus1) minus (1 minus 120594119898) 1206014 (119909)
+ Re119898minus1
sum
119899=0
(119911119899119911101584010158401015840
119898minus1minus119899minus 1199111015840
11989911991110158401015840
119898minus1minus119899)
(72)
5 Results and Discussion
The SHAM algorithm using the proposed auxiliary linearoperators was applied to the several test problems presentedin the last section In this section we present the numeri-cal results that highlight the difference in the accuracy ofthe approximate solutions and convergence of the differentSHAM algorithms All the simulations were conducted using119872 = 20 collocation points The accuracy and convergenceof the SHAM approximate series solution depend on carefulselection of the auxiliary parameter ℏ The residual errorwas used to determine the optimal values of the auxiliaryparameter Using finite terms of the SHAM series we define119870th order approximation at the collocation points 119909119895 for 119895 =0 1 2 119872 as
Y =119870
sum
119896=0
119910119896 (119909119895) =
119870
sum
119896=0
119884119896 (73)
where 119884119896 = [119910119896(1199090)119910119896(1199091) sdot sdot sdot 119910119896(119909119872)]119905 Given that Y is the
SHAM approximate solution at the collocation points theresidual error at the collocation points is defined as
Re 119904 (119884) =N [ (Y ℏ)] (74)
where N is the nonlinear operator defining the governingnonlinear equation (see (4)) We define the infinity norm ofthe residual error as
119864119903 (ℏ) = N[ (Y ℏ)]infin (75)
The optimal ℏ that gives the best accuracy was identified tobe the value of ℏ located at the minimum of the graph of thevariation of the infinity norm against ℏ
Figure 1 gives the typical residual error curves that areobtained using the 20th order SHAM approximate solutionof the Darcy-Brinkman-Forchheimer equation (Example 1)The minimum of the curve gives the best possible residualerror for each auxiliary linear operator used in the SHAMalgorithm A comparison of the minima of the residual errorcurves gives an indication of which auxiliary linear operatoris likely to give better accuracy when used with the optimalauxiliary linear operator in the SHAM algorithm It can beseen from Figure 1 that the best accuracy at the 20th orderSHAM approximation is obtained when using the linearpartition mapping after transformation (L4) The highestorder differential mapping (L1) gives the least accuracyFigure 2 shows the variation of the norm of the residual errorat the optimal value of ℏ as a function of the SHAMorderThisgraph gives an indication of how the SHAM series convergeswith an increase in the number of iterations (terms used inthe series) Rapid convergence is determined by attaining
Maximum Residual error 1198641199031 1599119890 minus 14 4108119890 minus 15 5329119890 minus 15 4441119890 minus 16 8993119890 minus 15
5 3375119890 minus 14 6439119890 minus 15 3442119890 minus 15 1443119890 minus 15 4663119890 minus 15
10 3952119890 minus 14 9770119890 minus 15 3442119890 minus 15 1776119890 minus 15 5218119890 minus 15
15 5529119890 minus 14 3397119890 minus 14 2887119890 minus 15 3553119890 minus 15 3331119890 minus 15
20 1612119890 minus 13 2620119890 minus 14 4885119890 minus 15 3664119890 minus 15 3331119890 minus 15
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 1 Residual ℏ-curve for Example 1 using the 20th orderSHAM with119872 = 1 119904 = 1 and 119865 = 2
the lowest possible residual error after few iterations In thisexample it can be seen from Figure 2 that the highest orderdifferential mapping linear operator L1 is the slowest interms of convergence followed by the complete differentialmappingL3The operator that yields the fastest convergencein this example isL4 which gives full convergence after only10 iterations It can also be seen from Figure 2 that L2L3andL5 all converge to the same level which is slightly lowerthan that of L1 but slightly higher than L4 This resultsuggests that the SHAM algorithm that applies L4 gives thebest accuracy and when usingL1 the accuracy is the lowestin comparison with the other 4 choices of linear operators inthis example
Table 1 gives the minimum number of terms of theSHAM series that are required to attain full convergenceof the SHAM algorithm using the optimal values of theconvergence controlling parameter ℏ The correspondingmaximum residual errors are also tabulated It can be seen
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 2 Variation of residual error with SHAM order for Example1 when119872 = 1 119904 = 1 and 119865 = 2
from Table 1 that the least error is obtained when using L1and the other linear operators give more or less comparableresidual error values The SHAM with L4 requires thefewest number terms of the SHAM series to attain maximumaccuracy The SHAM with L5 comes second followed byL2 L3 and L1 in terms of the minimum SHAM orderrequired to give best accuracy This observation is in accordwith the trends displayed in Figure 2 The table also indicatesthat the number of terms of the SHAM series required for theSHAMapproximation to attainmaximum accuracy increaseswith an increase in the value of the Forchheimer numberparameter 119865 It is worth noting from the governing equation(22) that the parameter119865multiplies the nonlinear componentof the equation Table 2 shows the effect of increasing theporousmedia shape parameter 119904 on the SHAMorder requiredto attain the best accuracy It can be seen from the table thatthe effect of 119904 on convergence and accuracy is similar to that of119865 In particular it can be noted that as 119904 increasesmore SHAM
Maximum residual error 1198641199031 1599119890 minus 14 4108119890 minus 15 5329119890 minus 15 4441119890 minus 16 8993119890 minus 15
5 3375119890 minus 14 6439119890 minus 15 3442119890 minus 15 1443119890 minus 15 4663119890 minus 15
10 3952119890 minus 14 9770119890 minus 15 3442119890 minus 15 1776119890 minus 15 5218119890 minus 15
15 5529119890 minus 14 3397119890 minus 14 2887119890 minus 15 3553119890 minus 15 3331119890 minus 15
20 1612119890 minus 13 2620119890 minus 14 4885119890 minus 15 3664119890 minus 15 3331119890 minus 15
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 3 Residual ℏ-curve for Example 2 using the 20th orderSHAM with Re = 100 and 120572 = minus5∘
series terms are required to attain the maximum accuracy inall linear operators
Figure 3 shows the ℏ-curves for the Jeffery-Hamel prob-lem (Example 2) corresponding to the different SHAMschemes generated using all the linear operators using 20terms of the SHAM approximation series It can be seenfrom Figure 3 that after 20 iterationsL4 would give the bestaccuracy and L1 gives the least accuracy This is in accordwith the observation made earlier in Example 1 It can alsobe observed that the behaviour of the SHAM with L1 isapproximately the same as that of the SHAM with L2 Thiscan be noted from the observation that the ℏ-curve forL1 lieson top of that forL2 In Figure 4 the variation of the residualerror is shown as a function of the SHAM order It can beobserved that in all test linear operators the residual errordecreases with the number of terms used in the SHAM seriesThis demonstrates that the SHAM approximation converges
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 4 Variation of residual error with SHAM order for Example2 when Re = 100 and 120572 = minus5∘
in all cases considered in this example The convergence isfastest for L4 followed by L5 and is slowest when L1 isused in the SHAM algorithm It can also be observed that theresults for L1 L2 L3 and L5 converge to approximatelythe same level while L4 converges to a level lower than thatof the other linear operators Thus in addition to convergingmuch faster L4 gives results that are slightly more accuratethan those obtained using the other linear operators
Table 3 presents the minimum SHAM order required togive maximum accuracy at the the optimal values of theconvergence controlling parameter ℏ and the correspondingresidual errorsThe results are generated for increasing valuesof the Reynolds number (Re) It can be seen from Table 3that the minimum order of the SHAM required graduallyincreases with an increase in Re for both L1 and L2 Thereis no significant increase in the minimum SHAM orderrequired in the case of L3 L4 and L5 The results from
Maximum residual error 1198641199031 8451119890 minus 12 8987119890 minus 12 1074119890 minus 11 4886119890 minus 14 1256119890 minus 11
50 8636119890 minus 12 8887119890 minus 12 1148119890 minus 11 1357119890 minus 12 7468119890 minus 12
100 7930119890 minus 12 8759119890 minus 12 1303119890 minus 11 1886119890 minus 12 1633119890 minus 11
150 7815119890 minus 12 8092119890 minus 12 2546119890 minus 11 2973119890 minus 12 1808119890 minus 11
200 9128119890 minus 12 8421119890 minus 12 3209119890 minus 11 4435119890 minus 12 1176119890 minus 11
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 5 Residual ℏ-curve for Example 3 using the 20th orderSHAM with Re = 10 and Ha = 1
Table 3 also reveal that the required minimum SHAM orderof L1 and L2 is comparable particularly for larger valuesof Re This confirms the observation made in Figures 3 and4 Thus it can be inferred that the behaviour of the SHAMwith higher order differential mapping (L1) is comparablewith that of SHAMwith linear partitionmapping (L2) in thesolution of the Jeffery-Hamel equation
Figure 5 shows the ℏ-curves corresponding to the exam-ple of laminar viscous flow in a semiporous channel subjectto a transverse magnetic field (Example 3) Again it can beseen from Figure 5 that the residual error is lowest whenthe SHAM is used with L4 and largest when used with L1which as was observed in the case of Jeffery-Hamel flowis comparable with L2 The same observation can be madein Figure 6 where the residual error is plotted against theSHAM order From Figure 6 it can also be seen that L4gives the best convergence followed by L5 and the slowest
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 6 Variation of residual error with SHAM order for Example3 when Re = 10 and Ha = 1
convergence is obtained when usingL1 which is comparabletoL2
Table 4 shows effect of increasing the Reynolds number(Re) on the minimum SHAM orders and the correspondingresidual errors It can be observed from Table 4 that theminimumSHAMorders increasewith an increase inRewhenthe SHAM is used with L1 L2 and L3 It can also beseen that there is no significant increase in the minimumSHAM orders when using L4 and L5 Furthermore theresidual error obtained using L4 is lower than that of theother linear operatorsThis is in linewith similar observationsmade earlier
Table 5 shows the effect of increasing the Hartmannnumber (Ha) on the minimum SHAM orders and thecorresponding residual errors in Example 3 In this case itcan be observed that the minimum SHAM orders increasewith an increase in Ha only in the case of L1 In the case of
Maximum Residual error 1198641199032 1484119890 minus 08 1323119890 minus 08 9815119890 minus 09 3391119890 minus 12 2954119890 minus 09
4 8618119890 minus 09 3334119890 minus 09 3888119890 minus 09 5698119890 minus 12 2855119890 minus 09
6 2258119890 minus 09 6625119890 minus 09 1281119890 minus 08 1461119890 minus 11 6068119890 minus 09
8 1172119890 minus 09 2022119890 minus 09 1012119890 minus 08 1690119890 minus 11 7852119890 minus 09
10 8887119890 minus 09 5535119890 minus 09 1261119890 minus 08 5424119890 minus 11 2253119890 minus 09
15 3325119890 minus 09 1870119890 minus 09 5521119890 minus 09 2964119890 minus 10 5871119890 minus 09
the other linear operators theminimum SHAMorders do notincrease with an increase in HaThis means that convergenceof the SHAM series is independent of the value of Ha whenthe SHAM is used with L2 L3 L4 and L5 It can also benoted from Table 5 that the residual error is smallest whenL4 is used If can be inferred that the SHAM is most accuratewhen used withL4
Figure 7 shows the ℏ-curves corresponding to the exam-ple of two-dimensional viscous flow in a rectangular domainbounded by two moving porous walls (Example 4) It canbe noted from Figure 7 that the residual error is the lowestwhen the SHAM is used with L4 and the largest whenused with L1 after 20 iterations Figure 8 illustrates thevariation of the residual error against the SHAM order It canbe seen from Figure 8 that L4 gives the best convergencefollowed by L5 and the slowest convergence is obtainedwhen using L1 The linear operators L1 L2 L3 andL5 converge to the same level of residual error whichis higher than that of L4 Again it can be inferred that
L4 gives superior accuracy compared to the other linearoperators
Table 6 shows the effect of increasing the Reynolds num-ber (Re) on the minimum SHAM orders and correspondingresidual errors in Example 4 It can be observed that theminimumSHAMorders increasewith an increase inRewhenthe SHAM is used with L1 L2 and L3 In the case of L4andL5 it seems possible that an increase in Re has no stronginfluence in the convergence speed of the SHAM It can alsobe noted from Table 6 that the residual error is the smallestwhen L4 is used This is in accord with the trends observedin all the other examples considered in this work Table 7shows the effect of the nondimensional wall dilation rate 120572on the minimum SHAM order and corresponding residualerrors It can be observed from Table 7 that an increasein 120572 results in a corresponding significant increase in theminimum SHAMorder forL1Theminimum SHAMordersare not significantly influenced by 120572 when the SHAM is usedwithL2L3L4 andL5 Furthermore the best accuracy isobtained when usingL4
Maximum residual error 1198641199031 2103119890 minus 09 4065119890 minus 09 1354119890 minus 08 4946119890 minus 13 6441119890 minus 09
2 1198119890 minus 09 1109119890 minus 08 7087119890 minus 10 7009119890 minus 12 9531119890 minus 09
4 1954119890 minus 09 1017119890 minus 08 1604119890 minus 08 3667119890 minus 11 3063119890 minus 09
6 4419119890 minus 09 5935119890 minus 09 1772119890 minus 08 7262119890 minus 11 1388119890 minus 08
8 1930119890 minus 09 2864119890 minus 09 1449119890 minus 08 9716119890 minus 11 1221119890 minus 09
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 7 Residual ℏ-curve for Example 4 using the 20th orderSHAM with Re = 10 and 120572 = 1
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 8 Variation of residual error with SHAM order for Example4 when Re = 10 and 120572 = 1
14 Mathematical Problems in Engineering
6 Conclusion
This study gives a systematic way of choosing initial approx-imations and linear operators that can be employed inthe spectral homotopy analysis method (SHAM) algorithmThe main aim of the study was to determine the linearoperator definition that gives approximate solutions withoptimal convergence and accuracy A comparison betweenthe accuracy and convergence of SHAM algorithms usingcommonly used linear operators and newly proposed linearoperators was conducted Simulations were conducted onselected nonlinear boundary value problems defined onbounded domains One of the more significant findings toemerge from this study is that the linear operator definedby the highest order differential matching which is themost commonly used linear operator for solving boundaryvalue problems defined on bounded domains results inthe slowest converging and least accurate SHAM algorithmThe SHAM algorithm that uses a linear operator definedby the method of linear partition matching was found toperform slightly better than the SHAMwith the highest orderdifferential matching in some cases However in other casesthe methods were found to be comparable It was also shownthat the SHAMwith linear operator defined by themethod ofcomplete differential matching performed much better thanthe SHAMwith linear partitionmapping or the highest orderdifferential matching The study proposed a modification ofthe complete differential method approach which unlike theoriginal approach does not ignore the nonlinear factors of thederivative terms in the definition of the linear operator Thisnew definition of linear operator was found to perform betterthan the original complete differential method approachThe most optimal linear operator was found to be one thatuses the linear partition mapping after a transformation thatexpresses the unknown function as a sum of a new unknownplus some initial guess that satisfies the underlying boundaryconditions
The current findings add substantially to our under-standing of how homotopy analysis based methods canbe modified to yield approximate solutions with optimumaccuracy and convergence Further work needs to be done toinvestigate the application of the linear operators proposed inthis study in solving nonlinear partial differential equationsIt would also be very interesting to extend the current studyto systems of nonlinear differential equations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is based on the research supported in part by theNational Research Foundation of South Africa (Grant no85596)
References
[1] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[2] S J Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer Berlin Germany 2012
[3] S J Liao Advances in Homotopy Analysis Method WorldScientific Singapore 2014
[5] S S Motsa P Sibanda and S Shateyi ldquoA new spectral-homotopy analysismethod for solving a nonlinear second orderBVPrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 9 pp 2293ndash2302 2010
[6] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers and Fluids vol 39 no 7 pp 1219ndash1225 2010
[7] R A Van Gorder and K Vajravelu ldquoOn the selection of auxil-iary functions operators and convergence control parametersin the application of the homotopy analysismethod to nonlineardifferential equations a general approachrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 12 pp4078ndash4089 2009
[8] Y Shan and T Chaolu ldquoA method to select the initial guesssolution auxiliary linear operator and set of basic functions ofhomotopy analysis methodrdquo in Proceedings of the IEEE Inter-national Conference on Intelligent Computing and IntegratedSystems (ICISS rsquo10) pp 410ndash412 Guilin China October 2010
[9] R A van Gorder ldquoStability of auxiliary linear operatorand convergence-control parameter in the homotopy analysismethodrdquo in Advances in Homotopy Analysis Method S J LiaoEd pp 123ndash180 World Scientific Singapore 2014
[10] S M Rassoulinejad-Mousavi and S Abbasbandy ldquoAnalysisof forced convection in a circular tube filled with a darcy-brinkman-forchheimer porous medium using spectral homo-topy analysis methodrdquo Journal of Fluids Engineering Transac-tions of the ASME vol 133 no 10 Article ID 101207 2011
[11] A A Joneidi G Domairry andM Babaelahi ldquoThree analyticalmethods applied to Jeffery-Hamel flowrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 11 pp3423ndash3434 2010
[12] S M Moghimi G Domairry S Soleimani E Ghasemi and HBararnia ldquoApplication of homotopy analysis method to solveMHD Jeffery-Hamel flows in non-parallel wallsrdquo Advances inEngineering Software vol 42 no 3 pp 108ndash113 2011
[13] M Esmaeilpour and D D Ganji ldquoSolution of the Jeffery-Hamel flowproblemby optimal homotopy asymptoticmethodrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3405ndash3411 2010
[14] A Basiri Parsa M M Rashidi O Anwar Beg and S M SadrildquoSemi-computational simulation of magneto-hemodynamicflow in a semi-porous channel using optimal homotopy anddifferential transform methodsrdquo Computers in Biology andMedicine vol 43 no 9 pp 1142ndash1153 2013
[15] Z Ziabakhsh andGDomairry ldquoSolution of the laminar viscousflow in a semi-porous channel in the presence of a uniformmagnetic field by using the homotopy analysis methodrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 4 pp 1284ndash1294 2009
Mathematical Problems in Engineering 15
[16] S SMotsa S Shateyi GMarewo andP Sibanda ldquoAn improvedspectral homotopy analysis method for MHD flow in a semi-porous channelrdquo Numerical Algorithms vol 60 no 3 pp 463ndash481 2012
[17] S Dinarvand and M M Rashidi ldquoA reliable treatment of ahomotopy analysis method for two-dimensional viscous flowin a rectangular domain bounded by twomoving porous wallsrdquoNonlinear Analysis Real World Applications vol 11 no 3 pp1502ndash1512 2010
[18] H Xu Z Lin S Liao JWu and JMajdalani ldquoHomotopy basedsolutions of the Navier-Stokes equations for a porous channelwith orthogonally moving wallsrdquo Physics of Fluids vol 22 no5 Article ID 053601 pp 1ndash18 2010
[19] M M Rashidi E Erfani and B Rostami ldquoOptimal HomotopyAsymptotic Method for solving viscous ow through expandingor contracting gaps with permeable wallsrdquo Transaction on IoTand Cloud Computing vol 2 no 1 pp 85ndash111 2014
[20] Z GMakukula P Sibanda and S S Motsa ldquoA novel numericaltechnique for two-dimensional laminar flow between twomoving porous wallsrdquo Mathematical Problems in Engineeringvol 2010 Article ID 528956 15 pages 2010
[21] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Series in Com-putational Physics Springer New York NY USA 1988
[22] L N Trefethen Spectral Methods in MATLAB SIAM 2000[23] J Majdalani C Zhou and C A Dawson ldquoTwo-dimensional
viscous flow between slowly expanding or contracting wallswith weak permeabilityrdquo Journal of Biomechanics vol 35 no10 pp 1399ndash1403 2002
the accuracy and convergence of the approximate solutionsobtained using their proposed linear operators was not con-sidered Shan and Chaolu [8] proposed a linear operator con-structed by forming a differential equation using the guessedinitial solution that satisfies the boundary conditions Themethod proposed in [8] was demonstrated using theThomasFermi equation and a nonlinear heat transfer equation VanGorder [9] discussed methods for selecting auxiliary linearoperators in solving the Painleve equations Lane-Emdenequation and a third order equation that models a nonlinearstretching sheet in a fluid flow problem It was noted thatfor some auxiliary linear operators the rate of convergencemay be slow and for others the rate of convergence wasrapid All the studies reviewed so far however suffer fromthe fact that the linear operators have to be carefully selectedin such a way that the solution method gives analyticalsolutions made up of simple functions such as polynomialsdecaying exponential sines cosines rational functions andother elementary functions or products of such functionswhich are fairly easy to integrate A particular method ofselecting a linear operator such as the method of completedifferential matching used in [7 9] can only be used in veryfew nonlinear equations if the output solution is expectedto be a simple series function Despite being suggested in[7] as a choice of linear operator that potentially can givebetter convergence and accuracy the method of completedifferential matching has not been widely used in the HAMliterature because it often leads to higher order decomposedequations that cannot be integrated analytically
In seeking to identify auxiliary parameters and operatorsthat give the best accuracy and convergence of the HAMalgorithm a more suitable study would seek to identify theoptimal parameters from the infinitely many possibilities andnot to search from the small set of options that guaranteeexact solutions of the decomposed equations The aim of thispaper is to determine the auxiliary linear operator and initialapproximation that give the optimal accuracy and conver-gence in the HAM solution of highly nonlinear boundaryvalue problems defined in bounded domainsThe study givesa systematic way of selecting initial approximations auxiliarylinear operators and convergence controlling parameterNew methods of defining linear operators in the contextof the discrete spectral method based version of the HAM(SHAM) are proposed The accuracy and convergence of theSHAM algorithms derived using the proposed new linearoperators are validated on the well-known nonlinear differ-ential equations that have been previously solved using theHAMSHAM algorithms with the standard linear operatorsIn particular we consider the Darcy-Brinkman-Forchheimermodel [5 10] Jeffery-Hamel equation [6 11ndash13] the laminarviscous flow in a semiporous channel subject to a transversemagnetic field [14ndash16] and the model of two-dimensionalviscous flow in a rectangular domain bounded by twomovingporous walls [17ndash20]
The remainder of the paper is organized as followsIn Section 2 we give a brief description of the SHAM forgeneral nonlinear ordinary differential equations Section 3introduces the auxiliary linear operators that are used inthe SHAM algorithm Section 4 describes the application
of the SHAM with the proposed linear operators in theproblems that are selected for numerical experimentationThe numerical simulations and results are presented inSection 5 Finally we conclude and describe future work inSection 6
2 Description of the Spectral HomotopyAnalysis Method in a General Setting
In this section we give the basic description of the homotopyanalysis method for the solution of a general nonlineardifferential equation of order 119901 (where 119901 is a positive integer)given by
where 120572119895(119909) and Φ(119909) are known functions of the indepen-dent variable 119909 and 119910(119909) is an unknown functionThe primesin (1) denote differentiation with respect to 119909 and 119910(0) equiv 119910(119909)The function 119873 represents the nonlinear component of thegoverning equation For illustrative purposes we assume that(1) is to be solved in the domain 119909 isin [119886 119887] subject to theseparated boundary conditions
where 119861119886 and 119861119887 are linear operatorsIn the framework of the homotopy analysis method
(HAM) [1ndash4] we define the following zeroth order deforma-tion equations
(1 minus 119902)L [119884 (119909 119902) minus 1199100 (119909)] = 119902ℏ N [119884 (119909 119902)] minus Φ (119909)
(3)
where 119902 isin [0 1] denotes an embedding parameter 119884(119909 119902) isa kind of continuous mapping function of 119910(119909) and ℏ is theconvergence-controlling parameter The nonlinear operatorN is defined from the governing equation (1) as
By differentiating the zeroth order equations (3) 119898 timeswith respect to 119902 setting 119902 = 0 and finally dividing theresulting equations by119898 we obtain the following119898th orderdeformation equations
L [119910119898 (119909) minus (120594119898 + ℏ) 119910119898minus1 (119909)] = ℏ119877119898minus1 [1199100 1199101 119910119898minus1]
From the solutions of (6) the approximate solution for119910(119909) is determined as the series solution
119910 (119909) =
+infin
sum
119896=0
119910119896 (119909) (8)
AHAMsolution is said to be of order119870 if the above seriesis truncated at 119896 = 119870 that is if
119910 (119909) =
119870
sum
119898=0
119910119898 (119909) (9)
For nontrivial linear operators the higher order equa-tions (6) cannot be integrated using analytical means Con-sequently numerical approaches are employed When theChebyshev spectral collocation method is used to solve (6)the method is called spectral homotopy analysis method[3 5 6]
In using the SHAM the initial guess is obtained simplyas a solution of the linear part of the governing equation (1)subject to the underlying boundary conditions (2)That is wesolve
In essence collocation approximates the solution 119910(119909)using an interpolating polynomial of degree119872which satisfiesthe boundary conditions and the differential equations atall points called the collocation points 119909119895 where 119895 =
0 1 119872 In the Chebyshev spectral collocation methodthe collocation points are chosen to be the extrema ofChebyshev polynomials of degree 119879119872 on the interval minus1 le120578 le 1 defined as
120578119895 = cos(120587119895
119872) 119895 = 0 1 119872 (11)
We use the transformation 119909 = (119887 minus 119886)(120578 + 1)2 to map theinterval [119886 119887] to [minus1 1] The so-called differentiation matrix119863 is used to approximate the derivatives of the unknownvariables 119910(119909) at the collocation points as the matrix vectorproduct
Thematrix119863 is of size (119872+1)times(119872+1) and its entries aredefined in [21 22] In the SHAM algorithm the continuousderivatives of the higher order deformation equations arereplaced by the discrete Chebyshev differentiation matricesAs a result the higher order deformation equations reduceto matrix equations and are solved using standard techniquesfor solving linear systems of equations
3 Definition and Selection of LinearAuxiliary Operators
In this section we describe the different approaches used todefine the auxiliary linear operators in the SHAM algorithmIn order to highlight the subtle differences between thevariety of linear operators it is convenient to express thenonlinear operator of the governing equation (1) using a sumformula of the derivative products as follows
where 120573119903119895(119909) (119895 = 0 1 119901) are coefficients of thenonlinear terms containing 119910119903 as the highest derivative Inthis way (1) becomes
The selection of auxiliary linear operators in the appli-cation of the homotopy analysis method was described byVan-Gorder and Vajravelu [7] in a fairly general setting Inparticular three methods namely the method of highest
4 Mathematical Problems in Engineering
order differential matching linear partition matching andcomplete differential matching were defined in [7] Below wegive a definition of the linear operator selection of [7] andpresent other options that can be implemented under theSHAM algorithm for any nonlinear differential equation thatcan be represented by (16)
(i) Method of the Highest Order Differential Matching Thisapproach defines the auxiliary linear operator using only thehighest order derivative This approach is the most widelyused approach in the HAM solution of nonlinear differentialequations defined in finite domains With reference to (16)the linear operator is selected as
L1 [119910] =119889119901119910
119889119909119901 (17)
(ii) Method of Linear Partition Matching This approachdefines the auxiliary linear operator to be the collection ofall linear terms in the governing equation With reference to(16) we set
L2 [119910] =
119901
sum
119895=0
120572119895 (119909)119889119895119910
119889119909119895 (18)
(iii) Method of Complete Differential Matching In thisapproach the collection of all linear operators and somenonlinear factors of the governing nonlinear equations areused to define the linear operator The aim is to ensurethat all the terms of the governing nonlinear differentialequation contribute to the auxiliary linear operator selectedThe following rules were defined in [7]
(a) in the case where we have a termwhich is the productof derivatives we take the higher order derivative inthe term
(b) if the term has a product of derivatives with functionsof the unknown function we again take the highestorder derivative in the term
(c) in the case where we have a nonlinear expression injust the unknown function we take the function itself
Thus applying the above rules to (16) we set
L3 [119910] =
119901
sum
119895=0
120572119895 (119909)119889119895119910
119889119909119895+
119901
sum
119903=0
119889119903119910
119889119909119903
119903
sum
119895=0
120573119903119895 (19)
(iv) Linear Partition Mapping after Transformation Thisapproach uses linear partition mapping after the transforma-tion 119910(119909) = 119911(119909)+1199100(119909) where the function 1199100(119909) is carefullychosen to satisfy the underlying boundary conditions Inearlier versions of the SHAM[5 6] it was suggested that1199100(119909)be defined as a solution of (10) which is solved subject to
the given boundary conditions Substituting in the governingequation (1) to express the equation in terms of 119911(119909) gives
119901
sum
119895=0
120572119895 (119909) 119911(119895)(119909) +
120597119873
120597119911(1199100 119910
1015840
0 119910
(119901)
0) 119911
+120597119873
1205971199111015840(1199100 119910
1015840
0 119910
(119901)
0) 1199111015840
+ sdot sdot sdot +120597119873
120597119911(119901)(1199100 119910
1015840
0 119910
(119901)
0) 119911(119901)+Nonlinear Terms
= Φ (119909) minus
119901
sum
119895=0
120572119895 (119909) 119910(119895)
0(119909) minus 119873 (1199100 119910
1015840
0 119910
(119901)
0)
(20)
Thus the linear operator is set to be
L4 [119911] =
119901
sum
119895=0
120572119895 (119909) 119911(119895)(119909) +
120597119873
120597119911(1199100 119910
1015840
0 119910
(119901)
0) 119911
+ sdot sdot sdot +120597119873
120597119911(119901)(1199100 119910
1015840
0 119910
(119901)
0) 119911(119901)
(21)
(v) Modified Complete Differential Matching Method Thisis a new approach that is proposed as a modification ofthe method of complete differential matching of [7] In theoriginal approach [7] from the group of nonlinear termsof the governing equation only the highest derivative isselected In the proposed approach we want to ensure thatall the terms of the nonlinear groups that make up the termsof the differential equation contribute to the linear operatorWe define the following rules which are a modification of therules set in [7]
(a) In the case where we have a termwhich is the productof derivatives we take the higher order derivativein the term and approximate each function in theremaining derivative product by 1199100(119909) For exampleif the nonlinear product is 1199101015840(119909)11991010158401015840(119909)119910101584010158401015840(119909) we set1199101015840
0(119909)11991010158401015840
0(119909)119910101584010158401015840(119909) where 1199100(119909) is the solution of (10)
(b) If the term has a product of derivatives with func-tions of the unknown function we again take thehighest order derivative in the term and approximateeach function in the remaining derivative productby 1199100(119909) For example if the nonlinear product is1199102(119909)119910101584010158401015840(119909) we set 1199102
0(119909)119910101584010158401015840(119909) where 1199100(119909) is the
solution of (10)(c) In the case where we have a nonlinear expression in
just the unknown function we take the function itselfand approximate the rest of the functions by 1199100(119909)For example if the nonlinear product is 1199102(119909) we set1199100(119909)119910(119909) where 1199100(119909) is the solution of (10)
4 Numerical Experiments
In this section we illustrate how using a different linearoperator can significantly improve convergence and accuracy
Mathematical Problems in Engineering 5
of the SHAM For illustration purposes we consider examplesof nonlinear differential equations that have previously beensolved in the published literature using theHAMSHAMwithstandard linear operators It is worth mentioning that in thepreviousHAM-based investigations the linear operators werechosen in such a way that
(i) the obtained approximate solution is a continuousanalytical expression
(ii) the HAM series solution conforms to a predefinedrule of solution expression
The spectral method based approach of [5 6] was aimedat removing the above restrictions by considering linearoperators which are defined from part of the governingdifferential equation In this case it was found that thelinearised deformation equations could not be solved exactlyand hence the use of the spectral method Below we givea systematic procedure for defining the linear operatorsfor various nondifferential equations selected for numericalexperimentation
41 Example 1 Darcy-Brinkman-Forchheimer Equation Weconsider the following Darcy-Brinkman-Forchheimer equa-tion that models the steady state pressure driven fully devel-oped parallel flow through a horizontal channel that is filledwith porous media
11991010158401015840(119909) minus 119904
2119910 (119909) minus 119865119904119910(119909)
2+1
119872= 0
119910 (minus1) = 0 119910 (1) = 0
(22)
where 119865 is the dimensionless Forchheimer number and 119904is the porous media shape parameter This problem waspreviously solved using the SHAM with the auxiliary linearoperator selected using themethod of linear partitionmatch-ing L2 in [5] The equivalent problem cast in cylindricalcoordinates was subsequently solved in [10] where the linearoperator was selected as the linear part of the governingequation excluding the linear termswith variable coefficientsFor this reason we remark that the linear operator used in[10] is not part of the class of linear operators considered inthis study Using the systematic choices described above weset
L3 [119910] = 11991010158401015840minus 1199042119910 minus 119865119904119910
L4 [119911] = 11991110158401015840minus 1199042119911 minus 21198651199041199100119911
L5 [119910] = 11991010158401015840minus 1199042119910 minus 1198651199041199100119910
(23)
The linear operator L4 is obtained by using the linearpartition mapping after using the transformation 119910(119909) =119911(119909) + 1199100(119909) The resulting equation is
11991110158401015840minus 1199042119911 minus 21198651199041199100119911 minus 119865119904119911
2= minus(119910
10158401015840
0minus 11990421199100 minus 119865119904119910
2
0+1
119872)
(24)
The function 1199100(119909) is chosen as a function that satisfies theboundary conditions and the linear part of the (22) That is1199100(119909)must be a solution of
11991010158401015840
0minus 11990421199100 = minus
1
119872 (25)
Thus the initial approximation is chosen to be
1199100 (119909) =1
1199042119872(1 minus
cosh (119904119909)cosh (119904)
) (26)
The initial approximation used in the SHAM algorithms thatuse the linear operators L1 L2 and L3 is obtained bysolving the linear equations
Thus the initial approximations corresponding to theSHAM withL1L2 andL3 are given respectively by
1199100 (119909) =1
2119872(1 minus 119909
2) 1199100 (119909) =
1
1199042119872(1 minus
cosh (119904119909)cosh (119904)
)
1199100 (119909) =1
11990421119872(1 minus
cosh (1199041119909)cosh (1199041)
)
(28)
where 1199041 = radic1199042 + 119865119904The initial approximation 1199110(119909) to use in conjunction
withL4 is obtained as a solution of the differential equationformed from the linear part of the (24) That is we solve
11991110158401015840
0minus 11990421199110 minus 211986511990411991001199110 = minus(119910
10158401015840
0minus 11990421199100 minus 119865119904119910
2
0+1
119872)
1199110 (minus1) = 1199110 (1) = 0
(29)
This equation is a linear equation with variable coefficientsThus it is solved using the spectral method as described inthe earlier section above Similarly the initial approximationto use with L5 is obtained as a solution of the differentialequation
11991010158401015840
0minus 11990421199100 minus 11986511990411991001199100 = minus
1
119872 1199100 (minus1) = 1199100 (1) = 0 (30)
with 1199100 given by (26)With the nonlinear operator 119873119894 defined as the homoge-
neous part of the governing nonlinear equation in each casewe obtain the following higher order deformation equations
L119894 [119910119898 (119909) minus 120594119898119910119898minus1 (119909)]
1198774119898minus1 =L4 (119911119898minus1) minus (1 minus 120594119898) 1206011 (119909) minus 119865119904119898minus1
sum
119899=0
119911119899119911119898minus1minus119899
(34)
and 1206011(119909) is the right hand side of (29)
42 Example 2 Jeffery-Hamel Equation Here we considerthe Jeffery-Hamel equation that models the steady two-dimensional flow of an incompressible conducting viscousfluid between two rigid plane walls that meet at an angle 2120572The rigid walls are considered to be divergent if 120572 gt 0 andconvergent if 120572 lt 0 The governing equation is defined as
where Re is the Reynolds number This problem was inves-tigated using the standard HAM with the highest orderdifferential matching linear operator L1(119910) in [11 12] Arelatedmethod called optimal homotopy asymptoticmethod(OHAM) was used to solve the same problem in [13] againwith L1(119910) In Motsa et al [6] the SHAM was used withthe auxiliary linear operator defined using the transformedlinear partition mapping methodL4(119911) In this example weexplore the auxiliary linear operators defined as
The function 1199100(119909) is chosen as a function that satisfiesthe boundary conditions and the linear part of (35) Thus1199100(119909) is
1199100 (119909) =cos (2120572) minus cos (2120572119909)
cos (2120572) minus 1 (38)
The linear operator L4 is obtained by using the linearpartition mapping after using the transformation 119910(119909) =119911(119909) + 1199100(119909) The resulting equation is
Thus the initial approximations corresponding to the SHAMwithL1L2 andL3 are given respectively by
1199100 (119909) = 1 minus 1199092 1199100 (119909) =
cos (2120572) minus cos (2120572119909)cos (2120572) minus 1
1199100 (119909) =cos (21205721) minus cos (21205721119909)
cos (21205721) minus 1
(43)
where 1199041 = radic41205722 + 2120572Re The initial approximation 1199110(119909)to use with L4 is obtained as a solution of the differentialequation formed from the linear part of (39)That is we solve
119911101584010158401015840
0+ 412057221199111015840
0+ 2120572Re1199100119911
1015840
0+ 2120572Re1199101015840
01199110 = 1206012 (119909) (44)
Since the above equation has variable coefficients it is solvedusing the spectral method Similarly the initial approxima-tion to use withL5 is obtained as a solution of the differentialequation
119910101584010158401015840
0+ 412057221199101015840
0+ 2120572Re1199100119910
1015840
0= 0
1199100 (0) = 1 1199101015840
0(0) = 0 1199100 (1) = 0
(45)
with 1199100 given by (38)With the nonlinear operator 119873119894 defined as the homoge-
neous part of the governing nonlinear equation in each casewe obtain the following higher order deformation equations
L119894 [119910119898 (119909) minus 120594119898119910119898minus1 (119909)]
1198774119898minus1 =L4 (119911119898minus1) minus (1 minus 120594119898) 1206012 (119909) + 2120572Re119898minus1
sum
119899=0
1199111198991199111015840
119898minus1minus119899
(49)
Mathematical Problems in Engineering 7
43 Example 3 Laminar Viscous Flow in a Semiporous Chan-nel Subject to a Transverse Magnetic Field In this section weconsider the problem of laminar viscous flow in a semiporouschannel subject to a transverse magnetic field given by
1199101015840101584010158401015840minusHa211991010158401015840 + Re (119910119910101584010158401015840 minus 119910101584011991010158401015840) = 0 (50)
subject to the boundary conditions
119910 = 0 1199101015840= 0 119909 = 0
119910 = 1 1199101015840= 0 119909 = 1
(51)
This example was previously investigated using the stan-dardHAMwith the highest order differential matching linearoperatorL1(119910) in [14 15] In [16] a blend between the SHAMand a spectral method based quasilinearisation method wasused to solve the same problem In this investigation weconsider the following auxiliary linear operators
Thus the initial approximations corresponding to the SHAMwithL1 andL2 are given by
1199100 (119909) = 31199092minus 21199092
1199100 (119909) =Ha119909 minus 119890Ha119909 + 119890HaminusHa119909 + 119890Ha (Ha119909 minus 1) + 1
119890Ha (Ha minus 2) +Ha + 2
(54)
We note that in this example 1199100 is the second solution in(54)The initial approximations corresponding toL3 andL5are obtained by solving (53) numerically using the spectralmethod Similarly 1199110(119909) is obtained by numerically solving
1199111015840101584010158401015840
0minusHa211991110158401015840
0
+ Re (1199100119911101584010158401015840
0+ 119910101584010158401015840
01199110 minus 119910
1015840
011991110158401015840
0minus 11991010158401015840
01199111015840
0) = 1206013 (119909)
(55)
subject to the boundary conditions
1199110 (0) = 0 1199111015840
0(0) = 0 1199110 (1) = 0 119911
1015840
0(1) = 0
(56)
where
1206013 (119909) = minus (1199101015840101584010158401015840
0minusHa211991010158401015840
0+ Re (1199100119910
101584010158401015840
0minus 1199101015840
011991010158401015840
0)) (57)
The higher order deformation equations are defined as
L119894 [119910119898 (119909) minus 120594119898119910119898minus1 (119909)]
1198774119898minus1 = L4 (119911119898minus1) minus (1 minus 120594119898) 1206013 (119909)
+ Re119898minus1
sum
119899=0
(119911119899119911101584010158401015840
119898minus1minus119899minus 1199111015840
11989911991110158401015840
119898minus1minus119899)
(61)
44 Example 4 Two-Dimensional Viscous Flow in a Rectan-gular Domain Bounded by Two Moving Porous Walls In thissection we consider the problem of two-dimensional viscousflow in a rectangular domain bounded by twomoving porouswalls The governing equations are given in [17 23] as
101584010158401015840+ 311991010158401015840) + Re (119910119910101584010158401015840 minus 119910101584011991010158401015840) = 0 (62)
subject to the boundary conditions
119910 = 0 11991010158401015840= 0 at 119909 = 0
119910 = 1 1199101015840= 0 at 119909 = 1
(63)
where 120572 is the nondimensional wall dilation rate defined tobe positive for expansion and negative for contraction andRe is the permeation Reynolds number defined positive forinjection and negative for suction through the walls Thenonlinear equation (62) was investigated using the standardHAM with the highest order differential matching linearoperatorL1(119910) in [17 18] Rashidi et al [19] used the optimalhomotopy asymptotic method (OHAM) to solve the sameproblem again using L1(119910) as linear operator The SHAMwith transformed linear partition methodL4(119911) was used in[20]
8 Mathematical Problems in Engineering
In this investigation we consider the following auxiliarylinear operators
1198774119898minus1 = L4 (119911119898minus1) minus (1 minus 120594119898) 1206014 (119909)
+ Re119898minus1
sum
119899=0
(119911119899119911101584010158401015840
119898minus1minus119899minus 1199111015840
11989911991110158401015840
119898minus1minus119899)
(72)
5 Results and Discussion
The SHAM algorithm using the proposed auxiliary linearoperators was applied to the several test problems presentedin the last section In this section we present the numeri-cal results that highlight the difference in the accuracy ofthe approximate solutions and convergence of the differentSHAM algorithms All the simulations were conducted using119872 = 20 collocation points The accuracy and convergenceof the SHAM approximate series solution depend on carefulselection of the auxiliary parameter ℏ The residual errorwas used to determine the optimal values of the auxiliaryparameter Using finite terms of the SHAM series we define119870th order approximation at the collocation points 119909119895 for 119895 =0 1 2 119872 as
Y =119870
sum
119896=0
119910119896 (119909119895) =
119870
sum
119896=0
119884119896 (73)
where 119884119896 = [119910119896(1199090)119910119896(1199091) sdot sdot sdot 119910119896(119909119872)]119905 Given that Y is the
SHAM approximate solution at the collocation points theresidual error at the collocation points is defined as
Re 119904 (119884) =N [ (Y ℏ)] (74)
where N is the nonlinear operator defining the governingnonlinear equation (see (4)) We define the infinity norm ofthe residual error as
119864119903 (ℏ) = N[ (Y ℏ)]infin (75)
The optimal ℏ that gives the best accuracy was identified tobe the value of ℏ located at the minimum of the graph of thevariation of the infinity norm against ℏ
Figure 1 gives the typical residual error curves that areobtained using the 20th order SHAM approximate solutionof the Darcy-Brinkman-Forchheimer equation (Example 1)The minimum of the curve gives the best possible residualerror for each auxiliary linear operator used in the SHAMalgorithm A comparison of the minima of the residual errorcurves gives an indication of which auxiliary linear operatoris likely to give better accuracy when used with the optimalauxiliary linear operator in the SHAM algorithm It can beseen from Figure 1 that the best accuracy at the 20th orderSHAM approximation is obtained when using the linearpartition mapping after transformation (L4) The highestorder differential mapping (L1) gives the least accuracyFigure 2 shows the variation of the norm of the residual errorat the optimal value of ℏ as a function of the SHAMorderThisgraph gives an indication of how the SHAM series convergeswith an increase in the number of iterations (terms used inthe series) Rapid convergence is determined by attaining
Maximum Residual error 1198641199031 1599119890 minus 14 4108119890 minus 15 5329119890 minus 15 4441119890 minus 16 8993119890 minus 15
5 3375119890 minus 14 6439119890 minus 15 3442119890 minus 15 1443119890 minus 15 4663119890 minus 15
10 3952119890 minus 14 9770119890 minus 15 3442119890 minus 15 1776119890 minus 15 5218119890 minus 15
15 5529119890 minus 14 3397119890 minus 14 2887119890 minus 15 3553119890 minus 15 3331119890 minus 15
20 1612119890 minus 13 2620119890 minus 14 4885119890 minus 15 3664119890 minus 15 3331119890 minus 15
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 1 Residual ℏ-curve for Example 1 using the 20th orderSHAM with119872 = 1 119904 = 1 and 119865 = 2
the lowest possible residual error after few iterations In thisexample it can be seen from Figure 2 that the highest orderdifferential mapping linear operator L1 is the slowest interms of convergence followed by the complete differentialmappingL3The operator that yields the fastest convergencein this example isL4 which gives full convergence after only10 iterations It can also be seen from Figure 2 that L2L3andL5 all converge to the same level which is slightly lowerthan that of L1 but slightly higher than L4 This resultsuggests that the SHAM algorithm that applies L4 gives thebest accuracy and when usingL1 the accuracy is the lowestin comparison with the other 4 choices of linear operators inthis example
Table 1 gives the minimum number of terms of theSHAM series that are required to attain full convergenceof the SHAM algorithm using the optimal values of theconvergence controlling parameter ℏ The correspondingmaximum residual errors are also tabulated It can be seen
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 2 Variation of residual error with SHAM order for Example1 when119872 = 1 119904 = 1 and 119865 = 2
from Table 1 that the least error is obtained when using L1and the other linear operators give more or less comparableresidual error values The SHAM with L4 requires thefewest number terms of the SHAM series to attain maximumaccuracy The SHAM with L5 comes second followed byL2 L3 and L1 in terms of the minimum SHAM orderrequired to give best accuracy This observation is in accordwith the trends displayed in Figure 2 The table also indicatesthat the number of terms of the SHAM series required for theSHAMapproximation to attainmaximum accuracy increaseswith an increase in the value of the Forchheimer numberparameter 119865 It is worth noting from the governing equation(22) that the parameter119865multiplies the nonlinear componentof the equation Table 2 shows the effect of increasing theporousmedia shape parameter 119904 on the SHAMorder requiredto attain the best accuracy It can be seen from the table thatthe effect of 119904 on convergence and accuracy is similar to that of119865 In particular it can be noted that as 119904 increasesmore SHAM
Maximum residual error 1198641199031 1599119890 minus 14 4108119890 minus 15 5329119890 minus 15 4441119890 minus 16 8993119890 minus 15
5 3375119890 minus 14 6439119890 minus 15 3442119890 minus 15 1443119890 minus 15 4663119890 minus 15
10 3952119890 minus 14 9770119890 minus 15 3442119890 minus 15 1776119890 minus 15 5218119890 minus 15
15 5529119890 minus 14 3397119890 minus 14 2887119890 minus 15 3553119890 minus 15 3331119890 minus 15
20 1612119890 minus 13 2620119890 minus 14 4885119890 minus 15 3664119890 minus 15 3331119890 minus 15
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 3 Residual ℏ-curve for Example 2 using the 20th orderSHAM with Re = 100 and 120572 = minus5∘
series terms are required to attain the maximum accuracy inall linear operators
Figure 3 shows the ℏ-curves for the Jeffery-Hamel prob-lem (Example 2) corresponding to the different SHAMschemes generated using all the linear operators using 20terms of the SHAM approximation series It can be seenfrom Figure 3 that after 20 iterationsL4 would give the bestaccuracy and L1 gives the least accuracy This is in accordwith the observation made earlier in Example 1 It can alsobe observed that the behaviour of the SHAM with L1 isapproximately the same as that of the SHAM with L2 Thiscan be noted from the observation that the ℏ-curve forL1 lieson top of that forL2 In Figure 4 the variation of the residualerror is shown as a function of the SHAM order It can beobserved that in all test linear operators the residual errordecreases with the number of terms used in the SHAM seriesThis demonstrates that the SHAM approximation converges
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 4 Variation of residual error with SHAM order for Example2 when Re = 100 and 120572 = minus5∘
in all cases considered in this example The convergence isfastest for L4 followed by L5 and is slowest when L1 isused in the SHAM algorithm It can also be observed that theresults for L1 L2 L3 and L5 converge to approximatelythe same level while L4 converges to a level lower than thatof the other linear operators Thus in addition to convergingmuch faster L4 gives results that are slightly more accuratethan those obtained using the other linear operators
Table 3 presents the minimum SHAM order required togive maximum accuracy at the the optimal values of theconvergence controlling parameter ℏ and the correspondingresidual errorsThe results are generated for increasing valuesof the Reynolds number (Re) It can be seen from Table 3that the minimum order of the SHAM required graduallyincreases with an increase in Re for both L1 and L2 Thereis no significant increase in the minimum SHAM orderrequired in the case of L3 L4 and L5 The results from
Maximum residual error 1198641199031 8451119890 minus 12 8987119890 minus 12 1074119890 minus 11 4886119890 minus 14 1256119890 minus 11
50 8636119890 minus 12 8887119890 minus 12 1148119890 minus 11 1357119890 minus 12 7468119890 minus 12
100 7930119890 minus 12 8759119890 minus 12 1303119890 minus 11 1886119890 minus 12 1633119890 minus 11
150 7815119890 minus 12 8092119890 minus 12 2546119890 minus 11 2973119890 minus 12 1808119890 minus 11
200 9128119890 minus 12 8421119890 minus 12 3209119890 minus 11 4435119890 minus 12 1176119890 minus 11
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 5 Residual ℏ-curve for Example 3 using the 20th orderSHAM with Re = 10 and Ha = 1
Table 3 also reveal that the required minimum SHAM orderof L1 and L2 is comparable particularly for larger valuesof Re This confirms the observation made in Figures 3 and4 Thus it can be inferred that the behaviour of the SHAMwith higher order differential mapping (L1) is comparablewith that of SHAMwith linear partitionmapping (L2) in thesolution of the Jeffery-Hamel equation
Figure 5 shows the ℏ-curves corresponding to the exam-ple of laminar viscous flow in a semiporous channel subjectto a transverse magnetic field (Example 3) Again it can beseen from Figure 5 that the residual error is lowest whenthe SHAM is used with L4 and largest when used with L1which as was observed in the case of Jeffery-Hamel flowis comparable with L2 The same observation can be madein Figure 6 where the residual error is plotted against theSHAM order From Figure 6 it can also be seen that L4gives the best convergence followed by L5 and the slowest
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 6 Variation of residual error with SHAM order for Example3 when Re = 10 and Ha = 1
convergence is obtained when usingL1 which is comparabletoL2
Table 4 shows effect of increasing the Reynolds number(Re) on the minimum SHAM orders and the correspondingresidual errors It can be observed from Table 4 that theminimumSHAMorders increasewith an increase inRewhenthe SHAM is used with L1 L2 and L3 It can also beseen that there is no significant increase in the minimumSHAM orders when using L4 and L5 Furthermore theresidual error obtained using L4 is lower than that of theother linear operatorsThis is in linewith similar observationsmade earlier
Table 5 shows the effect of increasing the Hartmannnumber (Ha) on the minimum SHAM orders and thecorresponding residual errors in Example 3 In this case itcan be observed that the minimum SHAM orders increasewith an increase in Ha only in the case of L1 In the case of
Maximum Residual error 1198641199032 1484119890 minus 08 1323119890 minus 08 9815119890 minus 09 3391119890 minus 12 2954119890 minus 09
4 8618119890 minus 09 3334119890 minus 09 3888119890 minus 09 5698119890 minus 12 2855119890 minus 09
6 2258119890 minus 09 6625119890 minus 09 1281119890 minus 08 1461119890 minus 11 6068119890 minus 09
8 1172119890 minus 09 2022119890 minus 09 1012119890 minus 08 1690119890 minus 11 7852119890 minus 09
10 8887119890 minus 09 5535119890 minus 09 1261119890 minus 08 5424119890 minus 11 2253119890 minus 09
15 3325119890 minus 09 1870119890 minus 09 5521119890 minus 09 2964119890 minus 10 5871119890 minus 09
the other linear operators theminimum SHAMorders do notincrease with an increase in HaThis means that convergenceof the SHAM series is independent of the value of Ha whenthe SHAM is used with L2 L3 L4 and L5 It can also benoted from Table 5 that the residual error is smallest whenL4 is used If can be inferred that the SHAM is most accuratewhen used withL4
Figure 7 shows the ℏ-curves corresponding to the exam-ple of two-dimensional viscous flow in a rectangular domainbounded by two moving porous walls (Example 4) It canbe noted from Figure 7 that the residual error is the lowestwhen the SHAM is used with L4 and the largest whenused with L1 after 20 iterations Figure 8 illustrates thevariation of the residual error against the SHAM order It canbe seen from Figure 8 that L4 gives the best convergencefollowed by L5 and the slowest convergence is obtainedwhen using L1 The linear operators L1 L2 L3 andL5 converge to the same level of residual error whichis higher than that of L4 Again it can be inferred that
L4 gives superior accuracy compared to the other linearoperators
Table 6 shows the effect of increasing the Reynolds num-ber (Re) on the minimum SHAM orders and correspondingresidual errors in Example 4 It can be observed that theminimumSHAMorders increasewith an increase inRewhenthe SHAM is used with L1 L2 and L3 In the case of L4andL5 it seems possible that an increase in Re has no stronginfluence in the convergence speed of the SHAM It can alsobe noted from Table 6 that the residual error is the smallestwhen L4 is used This is in accord with the trends observedin all the other examples considered in this work Table 7shows the effect of the nondimensional wall dilation rate 120572on the minimum SHAM order and corresponding residualerrors It can be observed from Table 7 that an increasein 120572 results in a corresponding significant increase in theminimum SHAMorder forL1Theminimum SHAMordersare not significantly influenced by 120572 when the SHAM is usedwithL2L3L4 andL5 Furthermore the best accuracy isobtained when usingL4
Maximum residual error 1198641199031 2103119890 minus 09 4065119890 minus 09 1354119890 minus 08 4946119890 minus 13 6441119890 minus 09
2 1198119890 minus 09 1109119890 minus 08 7087119890 minus 10 7009119890 minus 12 9531119890 minus 09
4 1954119890 minus 09 1017119890 minus 08 1604119890 minus 08 3667119890 minus 11 3063119890 minus 09
6 4419119890 minus 09 5935119890 minus 09 1772119890 minus 08 7262119890 minus 11 1388119890 minus 08
8 1930119890 minus 09 2864119890 minus 09 1449119890 minus 08 9716119890 minus 11 1221119890 minus 09
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 7 Residual ℏ-curve for Example 4 using the 20th orderSHAM with Re = 10 and 120572 = 1
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 8 Variation of residual error with SHAM order for Example4 when Re = 10 and 120572 = 1
14 Mathematical Problems in Engineering
6 Conclusion
This study gives a systematic way of choosing initial approx-imations and linear operators that can be employed inthe spectral homotopy analysis method (SHAM) algorithmThe main aim of the study was to determine the linearoperator definition that gives approximate solutions withoptimal convergence and accuracy A comparison betweenthe accuracy and convergence of SHAM algorithms usingcommonly used linear operators and newly proposed linearoperators was conducted Simulations were conducted onselected nonlinear boundary value problems defined onbounded domains One of the more significant findings toemerge from this study is that the linear operator definedby the highest order differential matching which is themost commonly used linear operator for solving boundaryvalue problems defined on bounded domains results inthe slowest converging and least accurate SHAM algorithmThe SHAM algorithm that uses a linear operator definedby the method of linear partition matching was found toperform slightly better than the SHAMwith the highest orderdifferential matching in some cases However in other casesthe methods were found to be comparable It was also shownthat the SHAMwith linear operator defined by themethod ofcomplete differential matching performed much better thanthe SHAMwith linear partitionmapping or the highest orderdifferential matching The study proposed a modification ofthe complete differential method approach which unlike theoriginal approach does not ignore the nonlinear factors of thederivative terms in the definition of the linear operator Thisnew definition of linear operator was found to perform betterthan the original complete differential method approachThe most optimal linear operator was found to be one thatuses the linear partition mapping after a transformation thatexpresses the unknown function as a sum of a new unknownplus some initial guess that satisfies the underlying boundaryconditions
The current findings add substantially to our under-standing of how homotopy analysis based methods canbe modified to yield approximate solutions with optimumaccuracy and convergence Further work needs to be done toinvestigate the application of the linear operators proposed inthis study in solving nonlinear partial differential equationsIt would also be very interesting to extend the current studyto systems of nonlinear differential equations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is based on the research supported in part by theNational Research Foundation of South Africa (Grant no85596)
References
[1] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[2] S J Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer Berlin Germany 2012
[3] S J Liao Advances in Homotopy Analysis Method WorldScientific Singapore 2014
[5] S S Motsa P Sibanda and S Shateyi ldquoA new spectral-homotopy analysismethod for solving a nonlinear second orderBVPrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 9 pp 2293ndash2302 2010
[6] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers and Fluids vol 39 no 7 pp 1219ndash1225 2010
[7] R A Van Gorder and K Vajravelu ldquoOn the selection of auxil-iary functions operators and convergence control parametersin the application of the homotopy analysismethod to nonlineardifferential equations a general approachrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 12 pp4078ndash4089 2009
[8] Y Shan and T Chaolu ldquoA method to select the initial guesssolution auxiliary linear operator and set of basic functions ofhomotopy analysis methodrdquo in Proceedings of the IEEE Inter-national Conference on Intelligent Computing and IntegratedSystems (ICISS rsquo10) pp 410ndash412 Guilin China October 2010
[9] R A van Gorder ldquoStability of auxiliary linear operatorand convergence-control parameter in the homotopy analysismethodrdquo in Advances in Homotopy Analysis Method S J LiaoEd pp 123ndash180 World Scientific Singapore 2014
[10] S M Rassoulinejad-Mousavi and S Abbasbandy ldquoAnalysisof forced convection in a circular tube filled with a darcy-brinkman-forchheimer porous medium using spectral homo-topy analysis methodrdquo Journal of Fluids Engineering Transac-tions of the ASME vol 133 no 10 Article ID 101207 2011
[11] A A Joneidi G Domairry andM Babaelahi ldquoThree analyticalmethods applied to Jeffery-Hamel flowrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 11 pp3423ndash3434 2010
[12] S M Moghimi G Domairry S Soleimani E Ghasemi and HBararnia ldquoApplication of homotopy analysis method to solveMHD Jeffery-Hamel flows in non-parallel wallsrdquo Advances inEngineering Software vol 42 no 3 pp 108ndash113 2011
[13] M Esmaeilpour and D D Ganji ldquoSolution of the Jeffery-Hamel flowproblemby optimal homotopy asymptoticmethodrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3405ndash3411 2010
[14] A Basiri Parsa M M Rashidi O Anwar Beg and S M SadrildquoSemi-computational simulation of magneto-hemodynamicflow in a semi-porous channel using optimal homotopy anddifferential transform methodsrdquo Computers in Biology andMedicine vol 43 no 9 pp 1142ndash1153 2013
[15] Z Ziabakhsh andGDomairry ldquoSolution of the laminar viscousflow in a semi-porous channel in the presence of a uniformmagnetic field by using the homotopy analysis methodrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 4 pp 1284ndash1294 2009
Mathematical Problems in Engineering 15
[16] S SMotsa S Shateyi GMarewo andP Sibanda ldquoAn improvedspectral homotopy analysis method for MHD flow in a semi-porous channelrdquo Numerical Algorithms vol 60 no 3 pp 463ndash481 2012
[17] S Dinarvand and M M Rashidi ldquoA reliable treatment of ahomotopy analysis method for two-dimensional viscous flowin a rectangular domain bounded by twomoving porous wallsrdquoNonlinear Analysis Real World Applications vol 11 no 3 pp1502ndash1512 2010
[18] H Xu Z Lin S Liao JWu and JMajdalani ldquoHomotopy basedsolutions of the Navier-Stokes equations for a porous channelwith orthogonally moving wallsrdquo Physics of Fluids vol 22 no5 Article ID 053601 pp 1ndash18 2010
[19] M M Rashidi E Erfani and B Rostami ldquoOptimal HomotopyAsymptotic Method for solving viscous ow through expandingor contracting gaps with permeable wallsrdquo Transaction on IoTand Cloud Computing vol 2 no 1 pp 85ndash111 2014
[20] Z GMakukula P Sibanda and S S Motsa ldquoA novel numericaltechnique for two-dimensional laminar flow between twomoving porous wallsrdquo Mathematical Problems in Engineeringvol 2010 Article ID 528956 15 pages 2010
[21] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Series in Com-putational Physics Springer New York NY USA 1988
[22] L N Trefethen Spectral Methods in MATLAB SIAM 2000[23] J Majdalani C Zhou and C A Dawson ldquoTwo-dimensional
viscous flow between slowly expanding or contracting wallswith weak permeabilityrdquo Journal of Biomechanics vol 35 no10 pp 1399ndash1403 2002
From the solutions of (6) the approximate solution for119910(119909) is determined as the series solution
119910 (119909) =
+infin
sum
119896=0
119910119896 (119909) (8)
AHAMsolution is said to be of order119870 if the above seriesis truncated at 119896 = 119870 that is if
119910 (119909) =
119870
sum
119898=0
119910119898 (119909) (9)
For nontrivial linear operators the higher order equa-tions (6) cannot be integrated using analytical means Con-sequently numerical approaches are employed When theChebyshev spectral collocation method is used to solve (6)the method is called spectral homotopy analysis method[3 5 6]
In using the SHAM the initial guess is obtained simplyas a solution of the linear part of the governing equation (1)subject to the underlying boundary conditions (2)That is wesolve
In essence collocation approximates the solution 119910(119909)using an interpolating polynomial of degree119872which satisfiesthe boundary conditions and the differential equations atall points called the collocation points 119909119895 where 119895 =
0 1 119872 In the Chebyshev spectral collocation methodthe collocation points are chosen to be the extrema ofChebyshev polynomials of degree 119879119872 on the interval minus1 le120578 le 1 defined as
120578119895 = cos(120587119895
119872) 119895 = 0 1 119872 (11)
We use the transformation 119909 = (119887 minus 119886)(120578 + 1)2 to map theinterval [119886 119887] to [minus1 1] The so-called differentiation matrix119863 is used to approximate the derivatives of the unknownvariables 119910(119909) at the collocation points as the matrix vectorproduct
Thematrix119863 is of size (119872+1)times(119872+1) and its entries aredefined in [21 22] In the SHAM algorithm the continuousderivatives of the higher order deformation equations arereplaced by the discrete Chebyshev differentiation matricesAs a result the higher order deformation equations reduceto matrix equations and are solved using standard techniquesfor solving linear systems of equations
3 Definition and Selection of LinearAuxiliary Operators
In this section we describe the different approaches used todefine the auxiliary linear operators in the SHAM algorithmIn order to highlight the subtle differences between thevariety of linear operators it is convenient to express thenonlinear operator of the governing equation (1) using a sumformula of the derivative products as follows
where 120573119903119895(119909) (119895 = 0 1 119901) are coefficients of thenonlinear terms containing 119910119903 as the highest derivative Inthis way (1) becomes
The selection of auxiliary linear operators in the appli-cation of the homotopy analysis method was described byVan-Gorder and Vajravelu [7] in a fairly general setting Inparticular three methods namely the method of highest
4 Mathematical Problems in Engineering
order differential matching linear partition matching andcomplete differential matching were defined in [7] Below wegive a definition of the linear operator selection of [7] andpresent other options that can be implemented under theSHAM algorithm for any nonlinear differential equation thatcan be represented by (16)
(i) Method of the Highest Order Differential Matching Thisapproach defines the auxiliary linear operator using only thehighest order derivative This approach is the most widelyused approach in the HAM solution of nonlinear differentialequations defined in finite domains With reference to (16)the linear operator is selected as
L1 [119910] =119889119901119910
119889119909119901 (17)
(ii) Method of Linear Partition Matching This approachdefines the auxiliary linear operator to be the collection ofall linear terms in the governing equation With reference to(16) we set
L2 [119910] =
119901
sum
119895=0
120572119895 (119909)119889119895119910
119889119909119895 (18)
(iii) Method of Complete Differential Matching In thisapproach the collection of all linear operators and somenonlinear factors of the governing nonlinear equations areused to define the linear operator The aim is to ensurethat all the terms of the governing nonlinear differentialequation contribute to the auxiliary linear operator selectedThe following rules were defined in [7]
(a) in the case where we have a termwhich is the productof derivatives we take the higher order derivative inthe term
(b) if the term has a product of derivatives with functionsof the unknown function we again take the highestorder derivative in the term
(c) in the case where we have a nonlinear expression injust the unknown function we take the function itself
Thus applying the above rules to (16) we set
L3 [119910] =
119901
sum
119895=0
120572119895 (119909)119889119895119910
119889119909119895+
119901
sum
119903=0
119889119903119910
119889119909119903
119903
sum
119895=0
120573119903119895 (19)
(iv) Linear Partition Mapping after Transformation Thisapproach uses linear partition mapping after the transforma-tion 119910(119909) = 119911(119909)+1199100(119909) where the function 1199100(119909) is carefullychosen to satisfy the underlying boundary conditions Inearlier versions of the SHAM[5 6] it was suggested that1199100(119909)be defined as a solution of (10) which is solved subject to
the given boundary conditions Substituting in the governingequation (1) to express the equation in terms of 119911(119909) gives
119901
sum
119895=0
120572119895 (119909) 119911(119895)(119909) +
120597119873
120597119911(1199100 119910
1015840
0 119910
(119901)
0) 119911
+120597119873
1205971199111015840(1199100 119910
1015840
0 119910
(119901)
0) 1199111015840
+ sdot sdot sdot +120597119873
120597119911(119901)(1199100 119910
1015840
0 119910
(119901)
0) 119911(119901)+Nonlinear Terms
= Φ (119909) minus
119901
sum
119895=0
120572119895 (119909) 119910(119895)
0(119909) minus 119873 (1199100 119910
1015840
0 119910
(119901)
0)
(20)
Thus the linear operator is set to be
L4 [119911] =
119901
sum
119895=0
120572119895 (119909) 119911(119895)(119909) +
120597119873
120597119911(1199100 119910
1015840
0 119910
(119901)
0) 119911
+ sdot sdot sdot +120597119873
120597119911(119901)(1199100 119910
1015840
0 119910
(119901)
0) 119911(119901)
(21)
(v) Modified Complete Differential Matching Method Thisis a new approach that is proposed as a modification ofthe method of complete differential matching of [7] In theoriginal approach [7] from the group of nonlinear termsof the governing equation only the highest derivative isselected In the proposed approach we want to ensure thatall the terms of the nonlinear groups that make up the termsof the differential equation contribute to the linear operatorWe define the following rules which are a modification of therules set in [7]
(a) In the case where we have a termwhich is the productof derivatives we take the higher order derivativein the term and approximate each function in theremaining derivative product by 1199100(119909) For exampleif the nonlinear product is 1199101015840(119909)11991010158401015840(119909)119910101584010158401015840(119909) we set1199101015840
0(119909)11991010158401015840
0(119909)119910101584010158401015840(119909) where 1199100(119909) is the solution of (10)
(b) If the term has a product of derivatives with func-tions of the unknown function we again take thehighest order derivative in the term and approximateeach function in the remaining derivative productby 1199100(119909) For example if the nonlinear product is1199102(119909)119910101584010158401015840(119909) we set 1199102
0(119909)119910101584010158401015840(119909) where 1199100(119909) is the
solution of (10)(c) In the case where we have a nonlinear expression in
just the unknown function we take the function itselfand approximate the rest of the functions by 1199100(119909)For example if the nonlinear product is 1199102(119909) we set1199100(119909)119910(119909) where 1199100(119909) is the solution of (10)
4 Numerical Experiments
In this section we illustrate how using a different linearoperator can significantly improve convergence and accuracy
Mathematical Problems in Engineering 5
of the SHAM For illustration purposes we consider examplesof nonlinear differential equations that have previously beensolved in the published literature using theHAMSHAMwithstandard linear operators It is worth mentioning that in thepreviousHAM-based investigations the linear operators werechosen in such a way that
(i) the obtained approximate solution is a continuousanalytical expression
(ii) the HAM series solution conforms to a predefinedrule of solution expression
The spectral method based approach of [5 6] was aimedat removing the above restrictions by considering linearoperators which are defined from part of the governingdifferential equation In this case it was found that thelinearised deformation equations could not be solved exactlyand hence the use of the spectral method Below we givea systematic procedure for defining the linear operatorsfor various nondifferential equations selected for numericalexperimentation
41 Example 1 Darcy-Brinkman-Forchheimer Equation Weconsider the following Darcy-Brinkman-Forchheimer equa-tion that models the steady state pressure driven fully devel-oped parallel flow through a horizontal channel that is filledwith porous media
11991010158401015840(119909) minus 119904
2119910 (119909) minus 119865119904119910(119909)
2+1
119872= 0
119910 (minus1) = 0 119910 (1) = 0
(22)
where 119865 is the dimensionless Forchheimer number and 119904is the porous media shape parameter This problem waspreviously solved using the SHAM with the auxiliary linearoperator selected using themethod of linear partitionmatch-ing L2 in [5] The equivalent problem cast in cylindricalcoordinates was subsequently solved in [10] where the linearoperator was selected as the linear part of the governingequation excluding the linear termswith variable coefficientsFor this reason we remark that the linear operator used in[10] is not part of the class of linear operators considered inthis study Using the systematic choices described above weset
L3 [119910] = 11991010158401015840minus 1199042119910 minus 119865119904119910
L4 [119911] = 11991110158401015840minus 1199042119911 minus 21198651199041199100119911
L5 [119910] = 11991010158401015840minus 1199042119910 minus 1198651199041199100119910
(23)
The linear operator L4 is obtained by using the linearpartition mapping after using the transformation 119910(119909) =119911(119909) + 1199100(119909) The resulting equation is
11991110158401015840minus 1199042119911 minus 21198651199041199100119911 minus 119865119904119911
2= minus(119910
10158401015840
0minus 11990421199100 minus 119865119904119910
2
0+1
119872)
(24)
The function 1199100(119909) is chosen as a function that satisfies theboundary conditions and the linear part of the (22) That is1199100(119909)must be a solution of
11991010158401015840
0minus 11990421199100 = minus
1
119872 (25)
Thus the initial approximation is chosen to be
1199100 (119909) =1
1199042119872(1 minus
cosh (119904119909)cosh (119904)
) (26)
The initial approximation used in the SHAM algorithms thatuse the linear operators L1 L2 and L3 is obtained bysolving the linear equations
Thus the initial approximations corresponding to theSHAM withL1L2 andL3 are given respectively by
1199100 (119909) =1
2119872(1 minus 119909
2) 1199100 (119909) =
1
1199042119872(1 minus
cosh (119904119909)cosh (119904)
)
1199100 (119909) =1
11990421119872(1 minus
cosh (1199041119909)cosh (1199041)
)
(28)
where 1199041 = radic1199042 + 119865119904The initial approximation 1199110(119909) to use in conjunction
withL4 is obtained as a solution of the differential equationformed from the linear part of the (24) That is we solve
11991110158401015840
0minus 11990421199110 minus 211986511990411991001199110 = minus(119910
10158401015840
0minus 11990421199100 minus 119865119904119910
2
0+1
119872)
1199110 (minus1) = 1199110 (1) = 0
(29)
This equation is a linear equation with variable coefficientsThus it is solved using the spectral method as described inthe earlier section above Similarly the initial approximationto use with L5 is obtained as a solution of the differentialequation
11991010158401015840
0minus 11990421199100 minus 11986511990411991001199100 = minus
1
119872 1199100 (minus1) = 1199100 (1) = 0 (30)
with 1199100 given by (26)With the nonlinear operator 119873119894 defined as the homoge-
neous part of the governing nonlinear equation in each casewe obtain the following higher order deformation equations
L119894 [119910119898 (119909) minus 120594119898119910119898minus1 (119909)]
1198774119898minus1 =L4 (119911119898minus1) minus (1 minus 120594119898) 1206011 (119909) minus 119865119904119898minus1
sum
119899=0
119911119899119911119898minus1minus119899
(34)
and 1206011(119909) is the right hand side of (29)
42 Example 2 Jeffery-Hamel Equation Here we considerthe Jeffery-Hamel equation that models the steady two-dimensional flow of an incompressible conducting viscousfluid between two rigid plane walls that meet at an angle 2120572The rigid walls are considered to be divergent if 120572 gt 0 andconvergent if 120572 lt 0 The governing equation is defined as
where Re is the Reynolds number This problem was inves-tigated using the standard HAM with the highest orderdifferential matching linear operator L1(119910) in [11 12] Arelatedmethod called optimal homotopy asymptoticmethod(OHAM) was used to solve the same problem in [13] againwith L1(119910) In Motsa et al [6] the SHAM was used withthe auxiliary linear operator defined using the transformedlinear partition mapping methodL4(119911) In this example weexplore the auxiliary linear operators defined as
The function 1199100(119909) is chosen as a function that satisfiesthe boundary conditions and the linear part of (35) Thus1199100(119909) is
1199100 (119909) =cos (2120572) minus cos (2120572119909)
cos (2120572) minus 1 (38)
The linear operator L4 is obtained by using the linearpartition mapping after using the transformation 119910(119909) =119911(119909) + 1199100(119909) The resulting equation is
Thus the initial approximations corresponding to the SHAMwithL1L2 andL3 are given respectively by
1199100 (119909) = 1 minus 1199092 1199100 (119909) =
cos (2120572) minus cos (2120572119909)cos (2120572) minus 1
1199100 (119909) =cos (21205721) minus cos (21205721119909)
cos (21205721) minus 1
(43)
where 1199041 = radic41205722 + 2120572Re The initial approximation 1199110(119909)to use with L4 is obtained as a solution of the differentialequation formed from the linear part of (39)That is we solve
119911101584010158401015840
0+ 412057221199111015840
0+ 2120572Re1199100119911
1015840
0+ 2120572Re1199101015840
01199110 = 1206012 (119909) (44)
Since the above equation has variable coefficients it is solvedusing the spectral method Similarly the initial approxima-tion to use withL5 is obtained as a solution of the differentialequation
119910101584010158401015840
0+ 412057221199101015840
0+ 2120572Re1199100119910
1015840
0= 0
1199100 (0) = 1 1199101015840
0(0) = 0 1199100 (1) = 0
(45)
with 1199100 given by (38)With the nonlinear operator 119873119894 defined as the homoge-
neous part of the governing nonlinear equation in each casewe obtain the following higher order deformation equations
L119894 [119910119898 (119909) minus 120594119898119910119898minus1 (119909)]
1198774119898minus1 =L4 (119911119898minus1) minus (1 minus 120594119898) 1206012 (119909) + 2120572Re119898minus1
sum
119899=0
1199111198991199111015840
119898minus1minus119899
(49)
Mathematical Problems in Engineering 7
43 Example 3 Laminar Viscous Flow in a Semiporous Chan-nel Subject to a Transverse Magnetic Field In this section weconsider the problem of laminar viscous flow in a semiporouschannel subject to a transverse magnetic field given by
1199101015840101584010158401015840minusHa211991010158401015840 + Re (119910119910101584010158401015840 minus 119910101584011991010158401015840) = 0 (50)
subject to the boundary conditions
119910 = 0 1199101015840= 0 119909 = 0
119910 = 1 1199101015840= 0 119909 = 1
(51)
This example was previously investigated using the stan-dardHAMwith the highest order differential matching linearoperatorL1(119910) in [14 15] In [16] a blend between the SHAMand a spectral method based quasilinearisation method wasused to solve the same problem In this investigation weconsider the following auxiliary linear operators
Thus the initial approximations corresponding to the SHAMwithL1 andL2 are given by
1199100 (119909) = 31199092minus 21199092
1199100 (119909) =Ha119909 minus 119890Ha119909 + 119890HaminusHa119909 + 119890Ha (Ha119909 minus 1) + 1
119890Ha (Ha minus 2) +Ha + 2
(54)
We note that in this example 1199100 is the second solution in(54)The initial approximations corresponding toL3 andL5are obtained by solving (53) numerically using the spectralmethod Similarly 1199110(119909) is obtained by numerically solving
1199111015840101584010158401015840
0minusHa211991110158401015840
0
+ Re (1199100119911101584010158401015840
0+ 119910101584010158401015840
01199110 minus 119910
1015840
011991110158401015840
0minus 11991010158401015840
01199111015840
0) = 1206013 (119909)
(55)
subject to the boundary conditions
1199110 (0) = 0 1199111015840
0(0) = 0 1199110 (1) = 0 119911
1015840
0(1) = 0
(56)
where
1206013 (119909) = minus (1199101015840101584010158401015840
0minusHa211991010158401015840
0+ Re (1199100119910
101584010158401015840
0minus 1199101015840
011991010158401015840
0)) (57)
The higher order deformation equations are defined as
L119894 [119910119898 (119909) minus 120594119898119910119898minus1 (119909)]
1198774119898minus1 = L4 (119911119898minus1) minus (1 minus 120594119898) 1206013 (119909)
+ Re119898minus1
sum
119899=0
(119911119899119911101584010158401015840
119898minus1minus119899minus 1199111015840
11989911991110158401015840
119898minus1minus119899)
(61)
44 Example 4 Two-Dimensional Viscous Flow in a Rectan-gular Domain Bounded by Two Moving Porous Walls In thissection we consider the problem of two-dimensional viscousflow in a rectangular domain bounded by twomoving porouswalls The governing equations are given in [17 23] as
101584010158401015840+ 311991010158401015840) + Re (119910119910101584010158401015840 minus 119910101584011991010158401015840) = 0 (62)
subject to the boundary conditions
119910 = 0 11991010158401015840= 0 at 119909 = 0
119910 = 1 1199101015840= 0 at 119909 = 1
(63)
where 120572 is the nondimensional wall dilation rate defined tobe positive for expansion and negative for contraction andRe is the permeation Reynolds number defined positive forinjection and negative for suction through the walls Thenonlinear equation (62) was investigated using the standardHAM with the highest order differential matching linearoperatorL1(119910) in [17 18] Rashidi et al [19] used the optimalhomotopy asymptotic method (OHAM) to solve the sameproblem again using L1(119910) as linear operator The SHAMwith transformed linear partition methodL4(119911) was used in[20]
8 Mathematical Problems in Engineering
In this investigation we consider the following auxiliarylinear operators
1198774119898minus1 = L4 (119911119898minus1) minus (1 minus 120594119898) 1206014 (119909)
+ Re119898minus1
sum
119899=0
(119911119899119911101584010158401015840
119898minus1minus119899minus 1199111015840
11989911991110158401015840
119898minus1minus119899)
(72)
5 Results and Discussion
The SHAM algorithm using the proposed auxiliary linearoperators was applied to the several test problems presentedin the last section In this section we present the numeri-cal results that highlight the difference in the accuracy ofthe approximate solutions and convergence of the differentSHAM algorithms All the simulations were conducted using119872 = 20 collocation points The accuracy and convergenceof the SHAM approximate series solution depend on carefulselection of the auxiliary parameter ℏ The residual errorwas used to determine the optimal values of the auxiliaryparameter Using finite terms of the SHAM series we define119870th order approximation at the collocation points 119909119895 for 119895 =0 1 2 119872 as
Y =119870
sum
119896=0
119910119896 (119909119895) =
119870
sum
119896=0
119884119896 (73)
where 119884119896 = [119910119896(1199090)119910119896(1199091) sdot sdot sdot 119910119896(119909119872)]119905 Given that Y is the
SHAM approximate solution at the collocation points theresidual error at the collocation points is defined as
Re 119904 (119884) =N [ (Y ℏ)] (74)
where N is the nonlinear operator defining the governingnonlinear equation (see (4)) We define the infinity norm ofthe residual error as
119864119903 (ℏ) = N[ (Y ℏ)]infin (75)
The optimal ℏ that gives the best accuracy was identified tobe the value of ℏ located at the minimum of the graph of thevariation of the infinity norm against ℏ
Figure 1 gives the typical residual error curves that areobtained using the 20th order SHAM approximate solutionof the Darcy-Brinkman-Forchheimer equation (Example 1)The minimum of the curve gives the best possible residualerror for each auxiliary linear operator used in the SHAMalgorithm A comparison of the minima of the residual errorcurves gives an indication of which auxiliary linear operatoris likely to give better accuracy when used with the optimalauxiliary linear operator in the SHAM algorithm It can beseen from Figure 1 that the best accuracy at the 20th orderSHAM approximation is obtained when using the linearpartition mapping after transformation (L4) The highestorder differential mapping (L1) gives the least accuracyFigure 2 shows the variation of the norm of the residual errorat the optimal value of ℏ as a function of the SHAMorderThisgraph gives an indication of how the SHAM series convergeswith an increase in the number of iterations (terms used inthe series) Rapid convergence is determined by attaining
Maximum Residual error 1198641199031 1599119890 minus 14 4108119890 minus 15 5329119890 minus 15 4441119890 minus 16 8993119890 minus 15
5 3375119890 minus 14 6439119890 minus 15 3442119890 minus 15 1443119890 minus 15 4663119890 minus 15
10 3952119890 minus 14 9770119890 minus 15 3442119890 minus 15 1776119890 minus 15 5218119890 minus 15
15 5529119890 minus 14 3397119890 minus 14 2887119890 minus 15 3553119890 minus 15 3331119890 minus 15
20 1612119890 minus 13 2620119890 minus 14 4885119890 minus 15 3664119890 minus 15 3331119890 minus 15
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 1 Residual ℏ-curve for Example 1 using the 20th orderSHAM with119872 = 1 119904 = 1 and 119865 = 2
the lowest possible residual error after few iterations In thisexample it can be seen from Figure 2 that the highest orderdifferential mapping linear operator L1 is the slowest interms of convergence followed by the complete differentialmappingL3The operator that yields the fastest convergencein this example isL4 which gives full convergence after only10 iterations It can also be seen from Figure 2 that L2L3andL5 all converge to the same level which is slightly lowerthan that of L1 but slightly higher than L4 This resultsuggests that the SHAM algorithm that applies L4 gives thebest accuracy and when usingL1 the accuracy is the lowestin comparison with the other 4 choices of linear operators inthis example
Table 1 gives the minimum number of terms of theSHAM series that are required to attain full convergenceof the SHAM algorithm using the optimal values of theconvergence controlling parameter ℏ The correspondingmaximum residual errors are also tabulated It can be seen
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 2 Variation of residual error with SHAM order for Example1 when119872 = 1 119904 = 1 and 119865 = 2
from Table 1 that the least error is obtained when using L1and the other linear operators give more or less comparableresidual error values The SHAM with L4 requires thefewest number terms of the SHAM series to attain maximumaccuracy The SHAM with L5 comes second followed byL2 L3 and L1 in terms of the minimum SHAM orderrequired to give best accuracy This observation is in accordwith the trends displayed in Figure 2 The table also indicatesthat the number of terms of the SHAM series required for theSHAMapproximation to attainmaximum accuracy increaseswith an increase in the value of the Forchheimer numberparameter 119865 It is worth noting from the governing equation(22) that the parameter119865multiplies the nonlinear componentof the equation Table 2 shows the effect of increasing theporousmedia shape parameter 119904 on the SHAMorder requiredto attain the best accuracy It can be seen from the table thatthe effect of 119904 on convergence and accuracy is similar to that of119865 In particular it can be noted that as 119904 increasesmore SHAM
Maximum residual error 1198641199031 1599119890 minus 14 4108119890 minus 15 5329119890 minus 15 4441119890 minus 16 8993119890 minus 15
5 3375119890 minus 14 6439119890 minus 15 3442119890 minus 15 1443119890 minus 15 4663119890 minus 15
10 3952119890 minus 14 9770119890 minus 15 3442119890 minus 15 1776119890 minus 15 5218119890 minus 15
15 5529119890 minus 14 3397119890 minus 14 2887119890 minus 15 3553119890 minus 15 3331119890 minus 15
20 1612119890 minus 13 2620119890 minus 14 4885119890 minus 15 3664119890 minus 15 3331119890 minus 15
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 3 Residual ℏ-curve for Example 2 using the 20th orderSHAM with Re = 100 and 120572 = minus5∘
series terms are required to attain the maximum accuracy inall linear operators
Figure 3 shows the ℏ-curves for the Jeffery-Hamel prob-lem (Example 2) corresponding to the different SHAMschemes generated using all the linear operators using 20terms of the SHAM approximation series It can be seenfrom Figure 3 that after 20 iterationsL4 would give the bestaccuracy and L1 gives the least accuracy This is in accordwith the observation made earlier in Example 1 It can alsobe observed that the behaviour of the SHAM with L1 isapproximately the same as that of the SHAM with L2 Thiscan be noted from the observation that the ℏ-curve forL1 lieson top of that forL2 In Figure 4 the variation of the residualerror is shown as a function of the SHAM order It can beobserved that in all test linear operators the residual errordecreases with the number of terms used in the SHAM seriesThis demonstrates that the SHAM approximation converges
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 4 Variation of residual error with SHAM order for Example2 when Re = 100 and 120572 = minus5∘
in all cases considered in this example The convergence isfastest for L4 followed by L5 and is slowest when L1 isused in the SHAM algorithm It can also be observed that theresults for L1 L2 L3 and L5 converge to approximatelythe same level while L4 converges to a level lower than thatof the other linear operators Thus in addition to convergingmuch faster L4 gives results that are slightly more accuratethan those obtained using the other linear operators
Table 3 presents the minimum SHAM order required togive maximum accuracy at the the optimal values of theconvergence controlling parameter ℏ and the correspondingresidual errorsThe results are generated for increasing valuesof the Reynolds number (Re) It can be seen from Table 3that the minimum order of the SHAM required graduallyincreases with an increase in Re for both L1 and L2 Thereis no significant increase in the minimum SHAM orderrequired in the case of L3 L4 and L5 The results from
Maximum residual error 1198641199031 8451119890 minus 12 8987119890 minus 12 1074119890 minus 11 4886119890 minus 14 1256119890 minus 11
50 8636119890 minus 12 8887119890 minus 12 1148119890 minus 11 1357119890 minus 12 7468119890 minus 12
100 7930119890 minus 12 8759119890 minus 12 1303119890 minus 11 1886119890 minus 12 1633119890 minus 11
150 7815119890 minus 12 8092119890 minus 12 2546119890 minus 11 2973119890 minus 12 1808119890 minus 11
200 9128119890 minus 12 8421119890 minus 12 3209119890 minus 11 4435119890 minus 12 1176119890 minus 11
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 5 Residual ℏ-curve for Example 3 using the 20th orderSHAM with Re = 10 and Ha = 1
Table 3 also reveal that the required minimum SHAM orderof L1 and L2 is comparable particularly for larger valuesof Re This confirms the observation made in Figures 3 and4 Thus it can be inferred that the behaviour of the SHAMwith higher order differential mapping (L1) is comparablewith that of SHAMwith linear partitionmapping (L2) in thesolution of the Jeffery-Hamel equation
Figure 5 shows the ℏ-curves corresponding to the exam-ple of laminar viscous flow in a semiporous channel subjectto a transverse magnetic field (Example 3) Again it can beseen from Figure 5 that the residual error is lowest whenthe SHAM is used with L4 and largest when used with L1which as was observed in the case of Jeffery-Hamel flowis comparable with L2 The same observation can be madein Figure 6 where the residual error is plotted against theSHAM order From Figure 6 it can also be seen that L4gives the best convergence followed by L5 and the slowest
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 6 Variation of residual error with SHAM order for Example3 when Re = 10 and Ha = 1
convergence is obtained when usingL1 which is comparabletoL2
Table 4 shows effect of increasing the Reynolds number(Re) on the minimum SHAM orders and the correspondingresidual errors It can be observed from Table 4 that theminimumSHAMorders increasewith an increase inRewhenthe SHAM is used with L1 L2 and L3 It can also beseen that there is no significant increase in the minimumSHAM orders when using L4 and L5 Furthermore theresidual error obtained using L4 is lower than that of theother linear operatorsThis is in linewith similar observationsmade earlier
Table 5 shows the effect of increasing the Hartmannnumber (Ha) on the minimum SHAM orders and thecorresponding residual errors in Example 3 In this case itcan be observed that the minimum SHAM orders increasewith an increase in Ha only in the case of L1 In the case of
Maximum Residual error 1198641199032 1484119890 minus 08 1323119890 minus 08 9815119890 minus 09 3391119890 minus 12 2954119890 minus 09
4 8618119890 minus 09 3334119890 minus 09 3888119890 minus 09 5698119890 minus 12 2855119890 minus 09
6 2258119890 minus 09 6625119890 minus 09 1281119890 minus 08 1461119890 minus 11 6068119890 minus 09
8 1172119890 minus 09 2022119890 minus 09 1012119890 minus 08 1690119890 minus 11 7852119890 minus 09
10 8887119890 minus 09 5535119890 minus 09 1261119890 minus 08 5424119890 minus 11 2253119890 minus 09
15 3325119890 minus 09 1870119890 minus 09 5521119890 minus 09 2964119890 minus 10 5871119890 minus 09
the other linear operators theminimum SHAMorders do notincrease with an increase in HaThis means that convergenceof the SHAM series is independent of the value of Ha whenthe SHAM is used with L2 L3 L4 and L5 It can also benoted from Table 5 that the residual error is smallest whenL4 is used If can be inferred that the SHAM is most accuratewhen used withL4
Figure 7 shows the ℏ-curves corresponding to the exam-ple of two-dimensional viscous flow in a rectangular domainbounded by two moving porous walls (Example 4) It canbe noted from Figure 7 that the residual error is the lowestwhen the SHAM is used with L4 and the largest whenused with L1 after 20 iterations Figure 8 illustrates thevariation of the residual error against the SHAM order It canbe seen from Figure 8 that L4 gives the best convergencefollowed by L5 and the slowest convergence is obtainedwhen using L1 The linear operators L1 L2 L3 andL5 converge to the same level of residual error whichis higher than that of L4 Again it can be inferred that
L4 gives superior accuracy compared to the other linearoperators
Table 6 shows the effect of increasing the Reynolds num-ber (Re) on the minimum SHAM orders and correspondingresidual errors in Example 4 It can be observed that theminimumSHAMorders increasewith an increase inRewhenthe SHAM is used with L1 L2 and L3 In the case of L4andL5 it seems possible that an increase in Re has no stronginfluence in the convergence speed of the SHAM It can alsobe noted from Table 6 that the residual error is the smallestwhen L4 is used This is in accord with the trends observedin all the other examples considered in this work Table 7shows the effect of the nondimensional wall dilation rate 120572on the minimum SHAM order and corresponding residualerrors It can be observed from Table 7 that an increasein 120572 results in a corresponding significant increase in theminimum SHAMorder forL1Theminimum SHAMordersare not significantly influenced by 120572 when the SHAM is usedwithL2L3L4 andL5 Furthermore the best accuracy isobtained when usingL4
Maximum residual error 1198641199031 2103119890 minus 09 4065119890 minus 09 1354119890 minus 08 4946119890 minus 13 6441119890 minus 09
2 1198119890 minus 09 1109119890 minus 08 7087119890 minus 10 7009119890 minus 12 9531119890 minus 09
4 1954119890 minus 09 1017119890 minus 08 1604119890 minus 08 3667119890 minus 11 3063119890 minus 09
6 4419119890 minus 09 5935119890 minus 09 1772119890 minus 08 7262119890 minus 11 1388119890 minus 08
8 1930119890 minus 09 2864119890 minus 09 1449119890 minus 08 9716119890 minus 11 1221119890 minus 09
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 7 Residual ℏ-curve for Example 4 using the 20th orderSHAM with Re = 10 and 120572 = 1
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 8 Variation of residual error with SHAM order for Example4 when Re = 10 and 120572 = 1
14 Mathematical Problems in Engineering
6 Conclusion
This study gives a systematic way of choosing initial approx-imations and linear operators that can be employed inthe spectral homotopy analysis method (SHAM) algorithmThe main aim of the study was to determine the linearoperator definition that gives approximate solutions withoptimal convergence and accuracy A comparison betweenthe accuracy and convergence of SHAM algorithms usingcommonly used linear operators and newly proposed linearoperators was conducted Simulations were conducted onselected nonlinear boundary value problems defined onbounded domains One of the more significant findings toemerge from this study is that the linear operator definedby the highest order differential matching which is themost commonly used linear operator for solving boundaryvalue problems defined on bounded domains results inthe slowest converging and least accurate SHAM algorithmThe SHAM algorithm that uses a linear operator definedby the method of linear partition matching was found toperform slightly better than the SHAMwith the highest orderdifferential matching in some cases However in other casesthe methods were found to be comparable It was also shownthat the SHAMwith linear operator defined by themethod ofcomplete differential matching performed much better thanthe SHAMwith linear partitionmapping or the highest orderdifferential matching The study proposed a modification ofthe complete differential method approach which unlike theoriginal approach does not ignore the nonlinear factors of thederivative terms in the definition of the linear operator Thisnew definition of linear operator was found to perform betterthan the original complete differential method approachThe most optimal linear operator was found to be one thatuses the linear partition mapping after a transformation thatexpresses the unknown function as a sum of a new unknownplus some initial guess that satisfies the underlying boundaryconditions
The current findings add substantially to our under-standing of how homotopy analysis based methods canbe modified to yield approximate solutions with optimumaccuracy and convergence Further work needs to be done toinvestigate the application of the linear operators proposed inthis study in solving nonlinear partial differential equationsIt would also be very interesting to extend the current studyto systems of nonlinear differential equations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is based on the research supported in part by theNational Research Foundation of South Africa (Grant no85596)
References
[1] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[2] S J Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer Berlin Germany 2012
[3] S J Liao Advances in Homotopy Analysis Method WorldScientific Singapore 2014
[5] S S Motsa P Sibanda and S Shateyi ldquoA new spectral-homotopy analysismethod for solving a nonlinear second orderBVPrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 9 pp 2293ndash2302 2010
[6] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers and Fluids vol 39 no 7 pp 1219ndash1225 2010
[7] R A Van Gorder and K Vajravelu ldquoOn the selection of auxil-iary functions operators and convergence control parametersin the application of the homotopy analysismethod to nonlineardifferential equations a general approachrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 12 pp4078ndash4089 2009
[8] Y Shan and T Chaolu ldquoA method to select the initial guesssolution auxiliary linear operator and set of basic functions ofhomotopy analysis methodrdquo in Proceedings of the IEEE Inter-national Conference on Intelligent Computing and IntegratedSystems (ICISS rsquo10) pp 410ndash412 Guilin China October 2010
[9] R A van Gorder ldquoStability of auxiliary linear operatorand convergence-control parameter in the homotopy analysismethodrdquo in Advances in Homotopy Analysis Method S J LiaoEd pp 123ndash180 World Scientific Singapore 2014
[10] S M Rassoulinejad-Mousavi and S Abbasbandy ldquoAnalysisof forced convection in a circular tube filled with a darcy-brinkman-forchheimer porous medium using spectral homo-topy analysis methodrdquo Journal of Fluids Engineering Transac-tions of the ASME vol 133 no 10 Article ID 101207 2011
[11] A A Joneidi G Domairry andM Babaelahi ldquoThree analyticalmethods applied to Jeffery-Hamel flowrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 11 pp3423ndash3434 2010
[12] S M Moghimi G Domairry S Soleimani E Ghasemi and HBararnia ldquoApplication of homotopy analysis method to solveMHD Jeffery-Hamel flows in non-parallel wallsrdquo Advances inEngineering Software vol 42 no 3 pp 108ndash113 2011
[13] M Esmaeilpour and D D Ganji ldquoSolution of the Jeffery-Hamel flowproblemby optimal homotopy asymptoticmethodrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3405ndash3411 2010
[14] A Basiri Parsa M M Rashidi O Anwar Beg and S M SadrildquoSemi-computational simulation of magneto-hemodynamicflow in a semi-porous channel using optimal homotopy anddifferential transform methodsrdquo Computers in Biology andMedicine vol 43 no 9 pp 1142ndash1153 2013
[15] Z Ziabakhsh andGDomairry ldquoSolution of the laminar viscousflow in a semi-porous channel in the presence of a uniformmagnetic field by using the homotopy analysis methodrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 4 pp 1284ndash1294 2009
Mathematical Problems in Engineering 15
[16] S SMotsa S Shateyi GMarewo andP Sibanda ldquoAn improvedspectral homotopy analysis method for MHD flow in a semi-porous channelrdquo Numerical Algorithms vol 60 no 3 pp 463ndash481 2012
[17] S Dinarvand and M M Rashidi ldquoA reliable treatment of ahomotopy analysis method for two-dimensional viscous flowin a rectangular domain bounded by twomoving porous wallsrdquoNonlinear Analysis Real World Applications vol 11 no 3 pp1502ndash1512 2010
[18] H Xu Z Lin S Liao JWu and JMajdalani ldquoHomotopy basedsolutions of the Navier-Stokes equations for a porous channelwith orthogonally moving wallsrdquo Physics of Fluids vol 22 no5 Article ID 053601 pp 1ndash18 2010
[19] M M Rashidi E Erfani and B Rostami ldquoOptimal HomotopyAsymptotic Method for solving viscous ow through expandingor contracting gaps with permeable wallsrdquo Transaction on IoTand Cloud Computing vol 2 no 1 pp 85ndash111 2014
[20] Z GMakukula P Sibanda and S S Motsa ldquoA novel numericaltechnique for two-dimensional laminar flow between twomoving porous wallsrdquo Mathematical Problems in Engineeringvol 2010 Article ID 528956 15 pages 2010
[21] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Series in Com-putational Physics Springer New York NY USA 1988
[22] L N Trefethen Spectral Methods in MATLAB SIAM 2000[23] J Majdalani C Zhou and C A Dawson ldquoTwo-dimensional
viscous flow between slowly expanding or contracting wallswith weak permeabilityrdquo Journal of Biomechanics vol 35 no10 pp 1399ndash1403 2002
order differential matching linear partition matching andcomplete differential matching were defined in [7] Below wegive a definition of the linear operator selection of [7] andpresent other options that can be implemented under theSHAM algorithm for any nonlinear differential equation thatcan be represented by (16)
(i) Method of the Highest Order Differential Matching Thisapproach defines the auxiliary linear operator using only thehighest order derivative This approach is the most widelyused approach in the HAM solution of nonlinear differentialequations defined in finite domains With reference to (16)the linear operator is selected as
L1 [119910] =119889119901119910
119889119909119901 (17)
(ii) Method of Linear Partition Matching This approachdefines the auxiliary linear operator to be the collection ofall linear terms in the governing equation With reference to(16) we set
L2 [119910] =
119901
sum
119895=0
120572119895 (119909)119889119895119910
119889119909119895 (18)
(iii) Method of Complete Differential Matching In thisapproach the collection of all linear operators and somenonlinear factors of the governing nonlinear equations areused to define the linear operator The aim is to ensurethat all the terms of the governing nonlinear differentialequation contribute to the auxiliary linear operator selectedThe following rules were defined in [7]
(a) in the case where we have a termwhich is the productof derivatives we take the higher order derivative inthe term
(b) if the term has a product of derivatives with functionsof the unknown function we again take the highestorder derivative in the term
(c) in the case where we have a nonlinear expression injust the unknown function we take the function itself
Thus applying the above rules to (16) we set
L3 [119910] =
119901
sum
119895=0
120572119895 (119909)119889119895119910
119889119909119895+
119901
sum
119903=0
119889119903119910
119889119909119903
119903
sum
119895=0
120573119903119895 (19)
(iv) Linear Partition Mapping after Transformation Thisapproach uses linear partition mapping after the transforma-tion 119910(119909) = 119911(119909)+1199100(119909) where the function 1199100(119909) is carefullychosen to satisfy the underlying boundary conditions Inearlier versions of the SHAM[5 6] it was suggested that1199100(119909)be defined as a solution of (10) which is solved subject to
the given boundary conditions Substituting in the governingequation (1) to express the equation in terms of 119911(119909) gives
119901
sum
119895=0
120572119895 (119909) 119911(119895)(119909) +
120597119873
120597119911(1199100 119910
1015840
0 119910
(119901)
0) 119911
+120597119873
1205971199111015840(1199100 119910
1015840
0 119910
(119901)
0) 1199111015840
+ sdot sdot sdot +120597119873
120597119911(119901)(1199100 119910
1015840
0 119910
(119901)
0) 119911(119901)+Nonlinear Terms
= Φ (119909) minus
119901
sum
119895=0
120572119895 (119909) 119910(119895)
0(119909) minus 119873 (1199100 119910
1015840
0 119910
(119901)
0)
(20)
Thus the linear operator is set to be
L4 [119911] =
119901
sum
119895=0
120572119895 (119909) 119911(119895)(119909) +
120597119873
120597119911(1199100 119910
1015840
0 119910
(119901)
0) 119911
+ sdot sdot sdot +120597119873
120597119911(119901)(1199100 119910
1015840
0 119910
(119901)
0) 119911(119901)
(21)
(v) Modified Complete Differential Matching Method Thisis a new approach that is proposed as a modification ofthe method of complete differential matching of [7] In theoriginal approach [7] from the group of nonlinear termsof the governing equation only the highest derivative isselected In the proposed approach we want to ensure thatall the terms of the nonlinear groups that make up the termsof the differential equation contribute to the linear operatorWe define the following rules which are a modification of therules set in [7]
(a) In the case where we have a termwhich is the productof derivatives we take the higher order derivativein the term and approximate each function in theremaining derivative product by 1199100(119909) For exampleif the nonlinear product is 1199101015840(119909)11991010158401015840(119909)119910101584010158401015840(119909) we set1199101015840
0(119909)11991010158401015840
0(119909)119910101584010158401015840(119909) where 1199100(119909) is the solution of (10)
(b) If the term has a product of derivatives with func-tions of the unknown function we again take thehighest order derivative in the term and approximateeach function in the remaining derivative productby 1199100(119909) For example if the nonlinear product is1199102(119909)119910101584010158401015840(119909) we set 1199102
0(119909)119910101584010158401015840(119909) where 1199100(119909) is the
solution of (10)(c) In the case where we have a nonlinear expression in
just the unknown function we take the function itselfand approximate the rest of the functions by 1199100(119909)For example if the nonlinear product is 1199102(119909) we set1199100(119909)119910(119909) where 1199100(119909) is the solution of (10)
4 Numerical Experiments
In this section we illustrate how using a different linearoperator can significantly improve convergence and accuracy
Mathematical Problems in Engineering 5
of the SHAM For illustration purposes we consider examplesof nonlinear differential equations that have previously beensolved in the published literature using theHAMSHAMwithstandard linear operators It is worth mentioning that in thepreviousHAM-based investigations the linear operators werechosen in such a way that
(i) the obtained approximate solution is a continuousanalytical expression
(ii) the HAM series solution conforms to a predefinedrule of solution expression
The spectral method based approach of [5 6] was aimedat removing the above restrictions by considering linearoperators which are defined from part of the governingdifferential equation In this case it was found that thelinearised deformation equations could not be solved exactlyand hence the use of the spectral method Below we givea systematic procedure for defining the linear operatorsfor various nondifferential equations selected for numericalexperimentation
41 Example 1 Darcy-Brinkman-Forchheimer Equation Weconsider the following Darcy-Brinkman-Forchheimer equa-tion that models the steady state pressure driven fully devel-oped parallel flow through a horizontal channel that is filledwith porous media
11991010158401015840(119909) minus 119904
2119910 (119909) minus 119865119904119910(119909)
2+1
119872= 0
119910 (minus1) = 0 119910 (1) = 0
(22)
where 119865 is the dimensionless Forchheimer number and 119904is the porous media shape parameter This problem waspreviously solved using the SHAM with the auxiliary linearoperator selected using themethod of linear partitionmatch-ing L2 in [5] The equivalent problem cast in cylindricalcoordinates was subsequently solved in [10] where the linearoperator was selected as the linear part of the governingequation excluding the linear termswith variable coefficientsFor this reason we remark that the linear operator used in[10] is not part of the class of linear operators considered inthis study Using the systematic choices described above weset
L3 [119910] = 11991010158401015840minus 1199042119910 minus 119865119904119910
L4 [119911] = 11991110158401015840minus 1199042119911 minus 21198651199041199100119911
L5 [119910] = 11991010158401015840minus 1199042119910 minus 1198651199041199100119910
(23)
The linear operator L4 is obtained by using the linearpartition mapping after using the transformation 119910(119909) =119911(119909) + 1199100(119909) The resulting equation is
11991110158401015840minus 1199042119911 minus 21198651199041199100119911 minus 119865119904119911
2= minus(119910
10158401015840
0minus 11990421199100 minus 119865119904119910
2
0+1
119872)
(24)
The function 1199100(119909) is chosen as a function that satisfies theboundary conditions and the linear part of the (22) That is1199100(119909)must be a solution of
11991010158401015840
0minus 11990421199100 = minus
1
119872 (25)
Thus the initial approximation is chosen to be
1199100 (119909) =1
1199042119872(1 minus
cosh (119904119909)cosh (119904)
) (26)
The initial approximation used in the SHAM algorithms thatuse the linear operators L1 L2 and L3 is obtained bysolving the linear equations
Thus the initial approximations corresponding to theSHAM withL1L2 andL3 are given respectively by
1199100 (119909) =1
2119872(1 minus 119909
2) 1199100 (119909) =
1
1199042119872(1 minus
cosh (119904119909)cosh (119904)
)
1199100 (119909) =1
11990421119872(1 minus
cosh (1199041119909)cosh (1199041)
)
(28)
where 1199041 = radic1199042 + 119865119904The initial approximation 1199110(119909) to use in conjunction
withL4 is obtained as a solution of the differential equationformed from the linear part of the (24) That is we solve
11991110158401015840
0minus 11990421199110 minus 211986511990411991001199110 = minus(119910
10158401015840
0minus 11990421199100 minus 119865119904119910
2
0+1
119872)
1199110 (minus1) = 1199110 (1) = 0
(29)
This equation is a linear equation with variable coefficientsThus it is solved using the spectral method as described inthe earlier section above Similarly the initial approximationto use with L5 is obtained as a solution of the differentialequation
11991010158401015840
0minus 11990421199100 minus 11986511990411991001199100 = minus
1
119872 1199100 (minus1) = 1199100 (1) = 0 (30)
with 1199100 given by (26)With the nonlinear operator 119873119894 defined as the homoge-
neous part of the governing nonlinear equation in each casewe obtain the following higher order deformation equations
L119894 [119910119898 (119909) minus 120594119898119910119898minus1 (119909)]
1198774119898minus1 =L4 (119911119898minus1) minus (1 minus 120594119898) 1206011 (119909) minus 119865119904119898minus1
sum
119899=0
119911119899119911119898minus1minus119899
(34)
and 1206011(119909) is the right hand side of (29)
42 Example 2 Jeffery-Hamel Equation Here we considerthe Jeffery-Hamel equation that models the steady two-dimensional flow of an incompressible conducting viscousfluid between two rigid plane walls that meet at an angle 2120572The rigid walls are considered to be divergent if 120572 gt 0 andconvergent if 120572 lt 0 The governing equation is defined as
where Re is the Reynolds number This problem was inves-tigated using the standard HAM with the highest orderdifferential matching linear operator L1(119910) in [11 12] Arelatedmethod called optimal homotopy asymptoticmethod(OHAM) was used to solve the same problem in [13] againwith L1(119910) In Motsa et al [6] the SHAM was used withthe auxiliary linear operator defined using the transformedlinear partition mapping methodL4(119911) In this example weexplore the auxiliary linear operators defined as
The function 1199100(119909) is chosen as a function that satisfiesthe boundary conditions and the linear part of (35) Thus1199100(119909) is
1199100 (119909) =cos (2120572) minus cos (2120572119909)
cos (2120572) minus 1 (38)
The linear operator L4 is obtained by using the linearpartition mapping after using the transformation 119910(119909) =119911(119909) + 1199100(119909) The resulting equation is
Thus the initial approximations corresponding to the SHAMwithL1L2 andL3 are given respectively by
1199100 (119909) = 1 minus 1199092 1199100 (119909) =
cos (2120572) minus cos (2120572119909)cos (2120572) minus 1
1199100 (119909) =cos (21205721) minus cos (21205721119909)
cos (21205721) minus 1
(43)
where 1199041 = radic41205722 + 2120572Re The initial approximation 1199110(119909)to use with L4 is obtained as a solution of the differentialequation formed from the linear part of (39)That is we solve
119911101584010158401015840
0+ 412057221199111015840
0+ 2120572Re1199100119911
1015840
0+ 2120572Re1199101015840
01199110 = 1206012 (119909) (44)
Since the above equation has variable coefficients it is solvedusing the spectral method Similarly the initial approxima-tion to use withL5 is obtained as a solution of the differentialequation
119910101584010158401015840
0+ 412057221199101015840
0+ 2120572Re1199100119910
1015840
0= 0
1199100 (0) = 1 1199101015840
0(0) = 0 1199100 (1) = 0
(45)
with 1199100 given by (38)With the nonlinear operator 119873119894 defined as the homoge-
neous part of the governing nonlinear equation in each casewe obtain the following higher order deformation equations
L119894 [119910119898 (119909) minus 120594119898119910119898minus1 (119909)]
1198774119898minus1 =L4 (119911119898minus1) minus (1 minus 120594119898) 1206012 (119909) + 2120572Re119898minus1
sum
119899=0
1199111198991199111015840
119898minus1minus119899
(49)
Mathematical Problems in Engineering 7
43 Example 3 Laminar Viscous Flow in a Semiporous Chan-nel Subject to a Transverse Magnetic Field In this section weconsider the problem of laminar viscous flow in a semiporouschannel subject to a transverse magnetic field given by
1199101015840101584010158401015840minusHa211991010158401015840 + Re (119910119910101584010158401015840 minus 119910101584011991010158401015840) = 0 (50)
subject to the boundary conditions
119910 = 0 1199101015840= 0 119909 = 0
119910 = 1 1199101015840= 0 119909 = 1
(51)
This example was previously investigated using the stan-dardHAMwith the highest order differential matching linearoperatorL1(119910) in [14 15] In [16] a blend between the SHAMand a spectral method based quasilinearisation method wasused to solve the same problem In this investigation weconsider the following auxiliary linear operators
Thus the initial approximations corresponding to the SHAMwithL1 andL2 are given by
1199100 (119909) = 31199092minus 21199092
1199100 (119909) =Ha119909 minus 119890Ha119909 + 119890HaminusHa119909 + 119890Ha (Ha119909 minus 1) + 1
119890Ha (Ha minus 2) +Ha + 2
(54)
We note that in this example 1199100 is the second solution in(54)The initial approximations corresponding toL3 andL5are obtained by solving (53) numerically using the spectralmethod Similarly 1199110(119909) is obtained by numerically solving
1199111015840101584010158401015840
0minusHa211991110158401015840
0
+ Re (1199100119911101584010158401015840
0+ 119910101584010158401015840
01199110 minus 119910
1015840
011991110158401015840
0minus 11991010158401015840
01199111015840
0) = 1206013 (119909)
(55)
subject to the boundary conditions
1199110 (0) = 0 1199111015840
0(0) = 0 1199110 (1) = 0 119911
1015840
0(1) = 0
(56)
where
1206013 (119909) = minus (1199101015840101584010158401015840
0minusHa211991010158401015840
0+ Re (1199100119910
101584010158401015840
0minus 1199101015840
011991010158401015840
0)) (57)
The higher order deformation equations are defined as
L119894 [119910119898 (119909) minus 120594119898119910119898minus1 (119909)]
1198774119898minus1 = L4 (119911119898minus1) minus (1 minus 120594119898) 1206013 (119909)
+ Re119898minus1
sum
119899=0
(119911119899119911101584010158401015840
119898minus1minus119899minus 1199111015840
11989911991110158401015840
119898minus1minus119899)
(61)
44 Example 4 Two-Dimensional Viscous Flow in a Rectan-gular Domain Bounded by Two Moving Porous Walls In thissection we consider the problem of two-dimensional viscousflow in a rectangular domain bounded by twomoving porouswalls The governing equations are given in [17 23] as
101584010158401015840+ 311991010158401015840) + Re (119910119910101584010158401015840 minus 119910101584011991010158401015840) = 0 (62)
subject to the boundary conditions
119910 = 0 11991010158401015840= 0 at 119909 = 0
119910 = 1 1199101015840= 0 at 119909 = 1
(63)
where 120572 is the nondimensional wall dilation rate defined tobe positive for expansion and negative for contraction andRe is the permeation Reynolds number defined positive forinjection and negative for suction through the walls Thenonlinear equation (62) was investigated using the standardHAM with the highest order differential matching linearoperatorL1(119910) in [17 18] Rashidi et al [19] used the optimalhomotopy asymptotic method (OHAM) to solve the sameproblem again using L1(119910) as linear operator The SHAMwith transformed linear partition methodL4(119911) was used in[20]
8 Mathematical Problems in Engineering
In this investigation we consider the following auxiliarylinear operators
1198774119898minus1 = L4 (119911119898minus1) minus (1 minus 120594119898) 1206014 (119909)
+ Re119898minus1
sum
119899=0
(119911119899119911101584010158401015840
119898minus1minus119899minus 1199111015840
11989911991110158401015840
119898minus1minus119899)
(72)
5 Results and Discussion
The SHAM algorithm using the proposed auxiliary linearoperators was applied to the several test problems presentedin the last section In this section we present the numeri-cal results that highlight the difference in the accuracy ofthe approximate solutions and convergence of the differentSHAM algorithms All the simulations were conducted using119872 = 20 collocation points The accuracy and convergenceof the SHAM approximate series solution depend on carefulselection of the auxiliary parameter ℏ The residual errorwas used to determine the optimal values of the auxiliaryparameter Using finite terms of the SHAM series we define119870th order approximation at the collocation points 119909119895 for 119895 =0 1 2 119872 as
Y =119870
sum
119896=0
119910119896 (119909119895) =
119870
sum
119896=0
119884119896 (73)
where 119884119896 = [119910119896(1199090)119910119896(1199091) sdot sdot sdot 119910119896(119909119872)]119905 Given that Y is the
SHAM approximate solution at the collocation points theresidual error at the collocation points is defined as
Re 119904 (119884) =N [ (Y ℏ)] (74)
where N is the nonlinear operator defining the governingnonlinear equation (see (4)) We define the infinity norm ofthe residual error as
119864119903 (ℏ) = N[ (Y ℏ)]infin (75)
The optimal ℏ that gives the best accuracy was identified tobe the value of ℏ located at the minimum of the graph of thevariation of the infinity norm against ℏ
Figure 1 gives the typical residual error curves that areobtained using the 20th order SHAM approximate solutionof the Darcy-Brinkman-Forchheimer equation (Example 1)The minimum of the curve gives the best possible residualerror for each auxiliary linear operator used in the SHAMalgorithm A comparison of the minima of the residual errorcurves gives an indication of which auxiliary linear operatoris likely to give better accuracy when used with the optimalauxiliary linear operator in the SHAM algorithm It can beseen from Figure 1 that the best accuracy at the 20th orderSHAM approximation is obtained when using the linearpartition mapping after transformation (L4) The highestorder differential mapping (L1) gives the least accuracyFigure 2 shows the variation of the norm of the residual errorat the optimal value of ℏ as a function of the SHAMorderThisgraph gives an indication of how the SHAM series convergeswith an increase in the number of iterations (terms used inthe series) Rapid convergence is determined by attaining
Maximum Residual error 1198641199031 1599119890 minus 14 4108119890 minus 15 5329119890 minus 15 4441119890 minus 16 8993119890 minus 15
5 3375119890 minus 14 6439119890 minus 15 3442119890 minus 15 1443119890 minus 15 4663119890 minus 15
10 3952119890 minus 14 9770119890 minus 15 3442119890 minus 15 1776119890 minus 15 5218119890 minus 15
15 5529119890 minus 14 3397119890 minus 14 2887119890 minus 15 3553119890 minus 15 3331119890 minus 15
20 1612119890 minus 13 2620119890 minus 14 4885119890 minus 15 3664119890 minus 15 3331119890 minus 15
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 1 Residual ℏ-curve for Example 1 using the 20th orderSHAM with119872 = 1 119904 = 1 and 119865 = 2
the lowest possible residual error after few iterations In thisexample it can be seen from Figure 2 that the highest orderdifferential mapping linear operator L1 is the slowest interms of convergence followed by the complete differentialmappingL3The operator that yields the fastest convergencein this example isL4 which gives full convergence after only10 iterations It can also be seen from Figure 2 that L2L3andL5 all converge to the same level which is slightly lowerthan that of L1 but slightly higher than L4 This resultsuggests that the SHAM algorithm that applies L4 gives thebest accuracy and when usingL1 the accuracy is the lowestin comparison with the other 4 choices of linear operators inthis example
Table 1 gives the minimum number of terms of theSHAM series that are required to attain full convergenceof the SHAM algorithm using the optimal values of theconvergence controlling parameter ℏ The correspondingmaximum residual errors are also tabulated It can be seen
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 2 Variation of residual error with SHAM order for Example1 when119872 = 1 119904 = 1 and 119865 = 2
from Table 1 that the least error is obtained when using L1and the other linear operators give more or less comparableresidual error values The SHAM with L4 requires thefewest number terms of the SHAM series to attain maximumaccuracy The SHAM with L5 comes second followed byL2 L3 and L1 in terms of the minimum SHAM orderrequired to give best accuracy This observation is in accordwith the trends displayed in Figure 2 The table also indicatesthat the number of terms of the SHAM series required for theSHAMapproximation to attainmaximum accuracy increaseswith an increase in the value of the Forchheimer numberparameter 119865 It is worth noting from the governing equation(22) that the parameter119865multiplies the nonlinear componentof the equation Table 2 shows the effect of increasing theporousmedia shape parameter 119904 on the SHAMorder requiredto attain the best accuracy It can be seen from the table thatthe effect of 119904 on convergence and accuracy is similar to that of119865 In particular it can be noted that as 119904 increasesmore SHAM
Maximum residual error 1198641199031 1599119890 minus 14 4108119890 minus 15 5329119890 minus 15 4441119890 minus 16 8993119890 minus 15
5 3375119890 minus 14 6439119890 minus 15 3442119890 minus 15 1443119890 minus 15 4663119890 minus 15
10 3952119890 minus 14 9770119890 minus 15 3442119890 minus 15 1776119890 minus 15 5218119890 minus 15
15 5529119890 minus 14 3397119890 minus 14 2887119890 minus 15 3553119890 minus 15 3331119890 minus 15
20 1612119890 minus 13 2620119890 minus 14 4885119890 minus 15 3664119890 minus 15 3331119890 minus 15
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 3 Residual ℏ-curve for Example 2 using the 20th orderSHAM with Re = 100 and 120572 = minus5∘
series terms are required to attain the maximum accuracy inall linear operators
Figure 3 shows the ℏ-curves for the Jeffery-Hamel prob-lem (Example 2) corresponding to the different SHAMschemes generated using all the linear operators using 20terms of the SHAM approximation series It can be seenfrom Figure 3 that after 20 iterationsL4 would give the bestaccuracy and L1 gives the least accuracy This is in accordwith the observation made earlier in Example 1 It can alsobe observed that the behaviour of the SHAM with L1 isapproximately the same as that of the SHAM with L2 Thiscan be noted from the observation that the ℏ-curve forL1 lieson top of that forL2 In Figure 4 the variation of the residualerror is shown as a function of the SHAM order It can beobserved that in all test linear operators the residual errordecreases with the number of terms used in the SHAM seriesThis demonstrates that the SHAM approximation converges
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 4 Variation of residual error with SHAM order for Example2 when Re = 100 and 120572 = minus5∘
in all cases considered in this example The convergence isfastest for L4 followed by L5 and is slowest when L1 isused in the SHAM algorithm It can also be observed that theresults for L1 L2 L3 and L5 converge to approximatelythe same level while L4 converges to a level lower than thatof the other linear operators Thus in addition to convergingmuch faster L4 gives results that are slightly more accuratethan those obtained using the other linear operators
Table 3 presents the minimum SHAM order required togive maximum accuracy at the the optimal values of theconvergence controlling parameter ℏ and the correspondingresidual errorsThe results are generated for increasing valuesof the Reynolds number (Re) It can be seen from Table 3that the minimum order of the SHAM required graduallyincreases with an increase in Re for both L1 and L2 Thereis no significant increase in the minimum SHAM orderrequired in the case of L3 L4 and L5 The results from
Maximum residual error 1198641199031 8451119890 minus 12 8987119890 minus 12 1074119890 minus 11 4886119890 minus 14 1256119890 minus 11
50 8636119890 minus 12 8887119890 minus 12 1148119890 minus 11 1357119890 minus 12 7468119890 minus 12
100 7930119890 minus 12 8759119890 minus 12 1303119890 minus 11 1886119890 minus 12 1633119890 minus 11
150 7815119890 minus 12 8092119890 minus 12 2546119890 minus 11 2973119890 minus 12 1808119890 minus 11
200 9128119890 minus 12 8421119890 minus 12 3209119890 minus 11 4435119890 minus 12 1176119890 minus 11
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 5 Residual ℏ-curve for Example 3 using the 20th orderSHAM with Re = 10 and Ha = 1
Table 3 also reveal that the required minimum SHAM orderof L1 and L2 is comparable particularly for larger valuesof Re This confirms the observation made in Figures 3 and4 Thus it can be inferred that the behaviour of the SHAMwith higher order differential mapping (L1) is comparablewith that of SHAMwith linear partitionmapping (L2) in thesolution of the Jeffery-Hamel equation
Figure 5 shows the ℏ-curves corresponding to the exam-ple of laminar viscous flow in a semiporous channel subjectto a transverse magnetic field (Example 3) Again it can beseen from Figure 5 that the residual error is lowest whenthe SHAM is used with L4 and largest when used with L1which as was observed in the case of Jeffery-Hamel flowis comparable with L2 The same observation can be madein Figure 6 where the residual error is plotted against theSHAM order From Figure 6 it can also be seen that L4gives the best convergence followed by L5 and the slowest
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 6 Variation of residual error with SHAM order for Example3 when Re = 10 and Ha = 1
convergence is obtained when usingL1 which is comparabletoL2
Table 4 shows effect of increasing the Reynolds number(Re) on the minimum SHAM orders and the correspondingresidual errors It can be observed from Table 4 that theminimumSHAMorders increasewith an increase inRewhenthe SHAM is used with L1 L2 and L3 It can also beseen that there is no significant increase in the minimumSHAM orders when using L4 and L5 Furthermore theresidual error obtained using L4 is lower than that of theother linear operatorsThis is in linewith similar observationsmade earlier
Table 5 shows the effect of increasing the Hartmannnumber (Ha) on the minimum SHAM orders and thecorresponding residual errors in Example 3 In this case itcan be observed that the minimum SHAM orders increasewith an increase in Ha only in the case of L1 In the case of
Maximum Residual error 1198641199032 1484119890 minus 08 1323119890 minus 08 9815119890 minus 09 3391119890 minus 12 2954119890 minus 09
4 8618119890 minus 09 3334119890 minus 09 3888119890 minus 09 5698119890 minus 12 2855119890 minus 09
6 2258119890 minus 09 6625119890 minus 09 1281119890 minus 08 1461119890 minus 11 6068119890 minus 09
8 1172119890 minus 09 2022119890 minus 09 1012119890 minus 08 1690119890 minus 11 7852119890 minus 09
10 8887119890 minus 09 5535119890 minus 09 1261119890 minus 08 5424119890 minus 11 2253119890 minus 09
15 3325119890 minus 09 1870119890 minus 09 5521119890 minus 09 2964119890 minus 10 5871119890 minus 09
the other linear operators theminimum SHAMorders do notincrease with an increase in HaThis means that convergenceof the SHAM series is independent of the value of Ha whenthe SHAM is used with L2 L3 L4 and L5 It can also benoted from Table 5 that the residual error is smallest whenL4 is used If can be inferred that the SHAM is most accuratewhen used withL4
Figure 7 shows the ℏ-curves corresponding to the exam-ple of two-dimensional viscous flow in a rectangular domainbounded by two moving porous walls (Example 4) It canbe noted from Figure 7 that the residual error is the lowestwhen the SHAM is used with L4 and the largest whenused with L1 after 20 iterations Figure 8 illustrates thevariation of the residual error against the SHAM order It canbe seen from Figure 8 that L4 gives the best convergencefollowed by L5 and the slowest convergence is obtainedwhen using L1 The linear operators L1 L2 L3 andL5 converge to the same level of residual error whichis higher than that of L4 Again it can be inferred that
L4 gives superior accuracy compared to the other linearoperators
Table 6 shows the effect of increasing the Reynolds num-ber (Re) on the minimum SHAM orders and correspondingresidual errors in Example 4 It can be observed that theminimumSHAMorders increasewith an increase inRewhenthe SHAM is used with L1 L2 and L3 In the case of L4andL5 it seems possible that an increase in Re has no stronginfluence in the convergence speed of the SHAM It can alsobe noted from Table 6 that the residual error is the smallestwhen L4 is used This is in accord with the trends observedin all the other examples considered in this work Table 7shows the effect of the nondimensional wall dilation rate 120572on the minimum SHAM order and corresponding residualerrors It can be observed from Table 7 that an increasein 120572 results in a corresponding significant increase in theminimum SHAMorder forL1Theminimum SHAMordersare not significantly influenced by 120572 when the SHAM is usedwithL2L3L4 andL5 Furthermore the best accuracy isobtained when usingL4
Maximum residual error 1198641199031 2103119890 minus 09 4065119890 minus 09 1354119890 minus 08 4946119890 minus 13 6441119890 minus 09
2 1198119890 minus 09 1109119890 minus 08 7087119890 minus 10 7009119890 minus 12 9531119890 minus 09
4 1954119890 minus 09 1017119890 minus 08 1604119890 minus 08 3667119890 minus 11 3063119890 minus 09
6 4419119890 minus 09 5935119890 minus 09 1772119890 minus 08 7262119890 minus 11 1388119890 minus 08
8 1930119890 minus 09 2864119890 minus 09 1449119890 minus 08 9716119890 minus 11 1221119890 minus 09
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 7 Residual ℏ-curve for Example 4 using the 20th orderSHAM with Re = 10 and 120572 = 1
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 8 Variation of residual error with SHAM order for Example4 when Re = 10 and 120572 = 1
14 Mathematical Problems in Engineering
6 Conclusion
This study gives a systematic way of choosing initial approx-imations and linear operators that can be employed inthe spectral homotopy analysis method (SHAM) algorithmThe main aim of the study was to determine the linearoperator definition that gives approximate solutions withoptimal convergence and accuracy A comparison betweenthe accuracy and convergence of SHAM algorithms usingcommonly used linear operators and newly proposed linearoperators was conducted Simulations were conducted onselected nonlinear boundary value problems defined onbounded domains One of the more significant findings toemerge from this study is that the linear operator definedby the highest order differential matching which is themost commonly used linear operator for solving boundaryvalue problems defined on bounded domains results inthe slowest converging and least accurate SHAM algorithmThe SHAM algorithm that uses a linear operator definedby the method of linear partition matching was found toperform slightly better than the SHAMwith the highest orderdifferential matching in some cases However in other casesthe methods were found to be comparable It was also shownthat the SHAMwith linear operator defined by themethod ofcomplete differential matching performed much better thanthe SHAMwith linear partitionmapping or the highest orderdifferential matching The study proposed a modification ofthe complete differential method approach which unlike theoriginal approach does not ignore the nonlinear factors of thederivative terms in the definition of the linear operator Thisnew definition of linear operator was found to perform betterthan the original complete differential method approachThe most optimal linear operator was found to be one thatuses the linear partition mapping after a transformation thatexpresses the unknown function as a sum of a new unknownplus some initial guess that satisfies the underlying boundaryconditions
The current findings add substantially to our under-standing of how homotopy analysis based methods canbe modified to yield approximate solutions with optimumaccuracy and convergence Further work needs to be done toinvestigate the application of the linear operators proposed inthis study in solving nonlinear partial differential equationsIt would also be very interesting to extend the current studyto systems of nonlinear differential equations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is based on the research supported in part by theNational Research Foundation of South Africa (Grant no85596)
References
[1] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[2] S J Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer Berlin Germany 2012
[3] S J Liao Advances in Homotopy Analysis Method WorldScientific Singapore 2014
[5] S S Motsa P Sibanda and S Shateyi ldquoA new spectral-homotopy analysismethod for solving a nonlinear second orderBVPrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 9 pp 2293ndash2302 2010
[6] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers and Fluids vol 39 no 7 pp 1219ndash1225 2010
[7] R A Van Gorder and K Vajravelu ldquoOn the selection of auxil-iary functions operators and convergence control parametersin the application of the homotopy analysismethod to nonlineardifferential equations a general approachrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 12 pp4078ndash4089 2009
[8] Y Shan and T Chaolu ldquoA method to select the initial guesssolution auxiliary linear operator and set of basic functions ofhomotopy analysis methodrdquo in Proceedings of the IEEE Inter-national Conference on Intelligent Computing and IntegratedSystems (ICISS rsquo10) pp 410ndash412 Guilin China October 2010
[9] R A van Gorder ldquoStability of auxiliary linear operatorand convergence-control parameter in the homotopy analysismethodrdquo in Advances in Homotopy Analysis Method S J LiaoEd pp 123ndash180 World Scientific Singapore 2014
[10] S M Rassoulinejad-Mousavi and S Abbasbandy ldquoAnalysisof forced convection in a circular tube filled with a darcy-brinkman-forchheimer porous medium using spectral homo-topy analysis methodrdquo Journal of Fluids Engineering Transac-tions of the ASME vol 133 no 10 Article ID 101207 2011
[11] A A Joneidi G Domairry andM Babaelahi ldquoThree analyticalmethods applied to Jeffery-Hamel flowrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 11 pp3423ndash3434 2010
[12] S M Moghimi G Domairry S Soleimani E Ghasemi and HBararnia ldquoApplication of homotopy analysis method to solveMHD Jeffery-Hamel flows in non-parallel wallsrdquo Advances inEngineering Software vol 42 no 3 pp 108ndash113 2011
[13] M Esmaeilpour and D D Ganji ldquoSolution of the Jeffery-Hamel flowproblemby optimal homotopy asymptoticmethodrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3405ndash3411 2010
[14] A Basiri Parsa M M Rashidi O Anwar Beg and S M SadrildquoSemi-computational simulation of magneto-hemodynamicflow in a semi-porous channel using optimal homotopy anddifferential transform methodsrdquo Computers in Biology andMedicine vol 43 no 9 pp 1142ndash1153 2013
[15] Z Ziabakhsh andGDomairry ldquoSolution of the laminar viscousflow in a semi-porous channel in the presence of a uniformmagnetic field by using the homotopy analysis methodrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 4 pp 1284ndash1294 2009
Mathematical Problems in Engineering 15
[16] S SMotsa S Shateyi GMarewo andP Sibanda ldquoAn improvedspectral homotopy analysis method for MHD flow in a semi-porous channelrdquo Numerical Algorithms vol 60 no 3 pp 463ndash481 2012
[17] S Dinarvand and M M Rashidi ldquoA reliable treatment of ahomotopy analysis method for two-dimensional viscous flowin a rectangular domain bounded by twomoving porous wallsrdquoNonlinear Analysis Real World Applications vol 11 no 3 pp1502ndash1512 2010
[18] H Xu Z Lin S Liao JWu and JMajdalani ldquoHomotopy basedsolutions of the Navier-Stokes equations for a porous channelwith orthogonally moving wallsrdquo Physics of Fluids vol 22 no5 Article ID 053601 pp 1ndash18 2010
[19] M M Rashidi E Erfani and B Rostami ldquoOptimal HomotopyAsymptotic Method for solving viscous ow through expandingor contracting gaps with permeable wallsrdquo Transaction on IoTand Cloud Computing vol 2 no 1 pp 85ndash111 2014
[20] Z GMakukula P Sibanda and S S Motsa ldquoA novel numericaltechnique for two-dimensional laminar flow between twomoving porous wallsrdquo Mathematical Problems in Engineeringvol 2010 Article ID 528956 15 pages 2010
[21] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Series in Com-putational Physics Springer New York NY USA 1988
[22] L N Trefethen Spectral Methods in MATLAB SIAM 2000[23] J Majdalani C Zhou and C A Dawson ldquoTwo-dimensional
viscous flow between slowly expanding or contracting wallswith weak permeabilityrdquo Journal of Biomechanics vol 35 no10 pp 1399ndash1403 2002
of the SHAM For illustration purposes we consider examplesof nonlinear differential equations that have previously beensolved in the published literature using theHAMSHAMwithstandard linear operators It is worth mentioning that in thepreviousHAM-based investigations the linear operators werechosen in such a way that
(i) the obtained approximate solution is a continuousanalytical expression
(ii) the HAM series solution conforms to a predefinedrule of solution expression
The spectral method based approach of [5 6] was aimedat removing the above restrictions by considering linearoperators which are defined from part of the governingdifferential equation In this case it was found that thelinearised deformation equations could not be solved exactlyand hence the use of the spectral method Below we givea systematic procedure for defining the linear operatorsfor various nondifferential equations selected for numericalexperimentation
41 Example 1 Darcy-Brinkman-Forchheimer Equation Weconsider the following Darcy-Brinkman-Forchheimer equa-tion that models the steady state pressure driven fully devel-oped parallel flow through a horizontal channel that is filledwith porous media
11991010158401015840(119909) minus 119904
2119910 (119909) minus 119865119904119910(119909)
2+1
119872= 0
119910 (minus1) = 0 119910 (1) = 0
(22)
where 119865 is the dimensionless Forchheimer number and 119904is the porous media shape parameter This problem waspreviously solved using the SHAM with the auxiliary linearoperator selected using themethod of linear partitionmatch-ing L2 in [5] The equivalent problem cast in cylindricalcoordinates was subsequently solved in [10] where the linearoperator was selected as the linear part of the governingequation excluding the linear termswith variable coefficientsFor this reason we remark that the linear operator used in[10] is not part of the class of linear operators considered inthis study Using the systematic choices described above weset
L3 [119910] = 11991010158401015840minus 1199042119910 minus 119865119904119910
L4 [119911] = 11991110158401015840minus 1199042119911 minus 21198651199041199100119911
L5 [119910] = 11991010158401015840minus 1199042119910 minus 1198651199041199100119910
(23)
The linear operator L4 is obtained by using the linearpartition mapping after using the transformation 119910(119909) =119911(119909) + 1199100(119909) The resulting equation is
11991110158401015840minus 1199042119911 minus 21198651199041199100119911 minus 119865119904119911
2= minus(119910
10158401015840
0minus 11990421199100 minus 119865119904119910
2
0+1
119872)
(24)
The function 1199100(119909) is chosen as a function that satisfies theboundary conditions and the linear part of the (22) That is1199100(119909)must be a solution of
11991010158401015840
0minus 11990421199100 = minus
1
119872 (25)
Thus the initial approximation is chosen to be
1199100 (119909) =1
1199042119872(1 minus
cosh (119904119909)cosh (119904)
) (26)
The initial approximation used in the SHAM algorithms thatuse the linear operators L1 L2 and L3 is obtained bysolving the linear equations
Thus the initial approximations corresponding to theSHAM withL1L2 andL3 are given respectively by
1199100 (119909) =1
2119872(1 minus 119909
2) 1199100 (119909) =
1
1199042119872(1 minus
cosh (119904119909)cosh (119904)
)
1199100 (119909) =1
11990421119872(1 minus
cosh (1199041119909)cosh (1199041)
)
(28)
where 1199041 = radic1199042 + 119865119904The initial approximation 1199110(119909) to use in conjunction
withL4 is obtained as a solution of the differential equationformed from the linear part of the (24) That is we solve
11991110158401015840
0minus 11990421199110 minus 211986511990411991001199110 = minus(119910
10158401015840
0minus 11990421199100 minus 119865119904119910
2
0+1
119872)
1199110 (minus1) = 1199110 (1) = 0
(29)
This equation is a linear equation with variable coefficientsThus it is solved using the spectral method as described inthe earlier section above Similarly the initial approximationto use with L5 is obtained as a solution of the differentialequation
11991010158401015840
0minus 11990421199100 minus 11986511990411991001199100 = minus
1
119872 1199100 (minus1) = 1199100 (1) = 0 (30)
with 1199100 given by (26)With the nonlinear operator 119873119894 defined as the homoge-
neous part of the governing nonlinear equation in each casewe obtain the following higher order deformation equations
L119894 [119910119898 (119909) minus 120594119898119910119898minus1 (119909)]
1198774119898minus1 =L4 (119911119898minus1) minus (1 minus 120594119898) 1206011 (119909) minus 119865119904119898minus1
sum
119899=0
119911119899119911119898minus1minus119899
(34)
and 1206011(119909) is the right hand side of (29)
42 Example 2 Jeffery-Hamel Equation Here we considerthe Jeffery-Hamel equation that models the steady two-dimensional flow of an incompressible conducting viscousfluid between two rigid plane walls that meet at an angle 2120572The rigid walls are considered to be divergent if 120572 gt 0 andconvergent if 120572 lt 0 The governing equation is defined as
where Re is the Reynolds number This problem was inves-tigated using the standard HAM with the highest orderdifferential matching linear operator L1(119910) in [11 12] Arelatedmethod called optimal homotopy asymptoticmethod(OHAM) was used to solve the same problem in [13] againwith L1(119910) In Motsa et al [6] the SHAM was used withthe auxiliary linear operator defined using the transformedlinear partition mapping methodL4(119911) In this example weexplore the auxiliary linear operators defined as
The function 1199100(119909) is chosen as a function that satisfiesthe boundary conditions and the linear part of (35) Thus1199100(119909) is
1199100 (119909) =cos (2120572) minus cos (2120572119909)
cos (2120572) minus 1 (38)
The linear operator L4 is obtained by using the linearpartition mapping after using the transformation 119910(119909) =119911(119909) + 1199100(119909) The resulting equation is
Thus the initial approximations corresponding to the SHAMwithL1L2 andL3 are given respectively by
1199100 (119909) = 1 minus 1199092 1199100 (119909) =
cos (2120572) minus cos (2120572119909)cos (2120572) minus 1
1199100 (119909) =cos (21205721) minus cos (21205721119909)
cos (21205721) minus 1
(43)
where 1199041 = radic41205722 + 2120572Re The initial approximation 1199110(119909)to use with L4 is obtained as a solution of the differentialequation formed from the linear part of (39)That is we solve
119911101584010158401015840
0+ 412057221199111015840
0+ 2120572Re1199100119911
1015840
0+ 2120572Re1199101015840
01199110 = 1206012 (119909) (44)
Since the above equation has variable coefficients it is solvedusing the spectral method Similarly the initial approxima-tion to use withL5 is obtained as a solution of the differentialequation
119910101584010158401015840
0+ 412057221199101015840
0+ 2120572Re1199100119910
1015840
0= 0
1199100 (0) = 1 1199101015840
0(0) = 0 1199100 (1) = 0
(45)
with 1199100 given by (38)With the nonlinear operator 119873119894 defined as the homoge-
neous part of the governing nonlinear equation in each casewe obtain the following higher order deformation equations
L119894 [119910119898 (119909) minus 120594119898119910119898minus1 (119909)]
1198774119898minus1 =L4 (119911119898minus1) minus (1 minus 120594119898) 1206012 (119909) + 2120572Re119898minus1
sum
119899=0
1199111198991199111015840
119898minus1minus119899
(49)
Mathematical Problems in Engineering 7
43 Example 3 Laminar Viscous Flow in a Semiporous Chan-nel Subject to a Transverse Magnetic Field In this section weconsider the problem of laminar viscous flow in a semiporouschannel subject to a transverse magnetic field given by
1199101015840101584010158401015840minusHa211991010158401015840 + Re (119910119910101584010158401015840 minus 119910101584011991010158401015840) = 0 (50)
subject to the boundary conditions
119910 = 0 1199101015840= 0 119909 = 0
119910 = 1 1199101015840= 0 119909 = 1
(51)
This example was previously investigated using the stan-dardHAMwith the highest order differential matching linearoperatorL1(119910) in [14 15] In [16] a blend between the SHAMand a spectral method based quasilinearisation method wasused to solve the same problem In this investigation weconsider the following auxiliary linear operators
Thus the initial approximations corresponding to the SHAMwithL1 andL2 are given by
1199100 (119909) = 31199092minus 21199092
1199100 (119909) =Ha119909 minus 119890Ha119909 + 119890HaminusHa119909 + 119890Ha (Ha119909 minus 1) + 1
119890Ha (Ha minus 2) +Ha + 2
(54)
We note that in this example 1199100 is the second solution in(54)The initial approximations corresponding toL3 andL5are obtained by solving (53) numerically using the spectralmethod Similarly 1199110(119909) is obtained by numerically solving
1199111015840101584010158401015840
0minusHa211991110158401015840
0
+ Re (1199100119911101584010158401015840
0+ 119910101584010158401015840
01199110 minus 119910
1015840
011991110158401015840
0minus 11991010158401015840
01199111015840
0) = 1206013 (119909)
(55)
subject to the boundary conditions
1199110 (0) = 0 1199111015840
0(0) = 0 1199110 (1) = 0 119911
1015840
0(1) = 0
(56)
where
1206013 (119909) = minus (1199101015840101584010158401015840
0minusHa211991010158401015840
0+ Re (1199100119910
101584010158401015840
0minus 1199101015840
011991010158401015840
0)) (57)
The higher order deformation equations are defined as
L119894 [119910119898 (119909) minus 120594119898119910119898minus1 (119909)]
1198774119898minus1 = L4 (119911119898minus1) minus (1 minus 120594119898) 1206013 (119909)
+ Re119898minus1
sum
119899=0
(119911119899119911101584010158401015840
119898minus1minus119899minus 1199111015840
11989911991110158401015840
119898minus1minus119899)
(61)
44 Example 4 Two-Dimensional Viscous Flow in a Rectan-gular Domain Bounded by Two Moving Porous Walls In thissection we consider the problem of two-dimensional viscousflow in a rectangular domain bounded by twomoving porouswalls The governing equations are given in [17 23] as
101584010158401015840+ 311991010158401015840) + Re (119910119910101584010158401015840 minus 119910101584011991010158401015840) = 0 (62)
subject to the boundary conditions
119910 = 0 11991010158401015840= 0 at 119909 = 0
119910 = 1 1199101015840= 0 at 119909 = 1
(63)
where 120572 is the nondimensional wall dilation rate defined tobe positive for expansion and negative for contraction andRe is the permeation Reynolds number defined positive forinjection and negative for suction through the walls Thenonlinear equation (62) was investigated using the standardHAM with the highest order differential matching linearoperatorL1(119910) in [17 18] Rashidi et al [19] used the optimalhomotopy asymptotic method (OHAM) to solve the sameproblem again using L1(119910) as linear operator The SHAMwith transformed linear partition methodL4(119911) was used in[20]
8 Mathematical Problems in Engineering
In this investigation we consider the following auxiliarylinear operators
1198774119898minus1 = L4 (119911119898minus1) minus (1 minus 120594119898) 1206014 (119909)
+ Re119898minus1
sum
119899=0
(119911119899119911101584010158401015840
119898minus1minus119899minus 1199111015840
11989911991110158401015840
119898minus1minus119899)
(72)
5 Results and Discussion
The SHAM algorithm using the proposed auxiliary linearoperators was applied to the several test problems presentedin the last section In this section we present the numeri-cal results that highlight the difference in the accuracy ofthe approximate solutions and convergence of the differentSHAM algorithms All the simulations were conducted using119872 = 20 collocation points The accuracy and convergenceof the SHAM approximate series solution depend on carefulselection of the auxiliary parameter ℏ The residual errorwas used to determine the optimal values of the auxiliaryparameter Using finite terms of the SHAM series we define119870th order approximation at the collocation points 119909119895 for 119895 =0 1 2 119872 as
Y =119870
sum
119896=0
119910119896 (119909119895) =
119870
sum
119896=0
119884119896 (73)
where 119884119896 = [119910119896(1199090)119910119896(1199091) sdot sdot sdot 119910119896(119909119872)]119905 Given that Y is the
SHAM approximate solution at the collocation points theresidual error at the collocation points is defined as
Re 119904 (119884) =N [ (Y ℏ)] (74)
where N is the nonlinear operator defining the governingnonlinear equation (see (4)) We define the infinity norm ofthe residual error as
119864119903 (ℏ) = N[ (Y ℏ)]infin (75)
The optimal ℏ that gives the best accuracy was identified tobe the value of ℏ located at the minimum of the graph of thevariation of the infinity norm against ℏ
Figure 1 gives the typical residual error curves that areobtained using the 20th order SHAM approximate solutionof the Darcy-Brinkman-Forchheimer equation (Example 1)The minimum of the curve gives the best possible residualerror for each auxiliary linear operator used in the SHAMalgorithm A comparison of the minima of the residual errorcurves gives an indication of which auxiliary linear operatoris likely to give better accuracy when used with the optimalauxiliary linear operator in the SHAM algorithm It can beseen from Figure 1 that the best accuracy at the 20th orderSHAM approximation is obtained when using the linearpartition mapping after transformation (L4) The highestorder differential mapping (L1) gives the least accuracyFigure 2 shows the variation of the norm of the residual errorat the optimal value of ℏ as a function of the SHAMorderThisgraph gives an indication of how the SHAM series convergeswith an increase in the number of iterations (terms used inthe series) Rapid convergence is determined by attaining
Maximum Residual error 1198641199031 1599119890 minus 14 4108119890 minus 15 5329119890 minus 15 4441119890 minus 16 8993119890 minus 15
5 3375119890 minus 14 6439119890 minus 15 3442119890 minus 15 1443119890 minus 15 4663119890 minus 15
10 3952119890 minus 14 9770119890 minus 15 3442119890 minus 15 1776119890 minus 15 5218119890 minus 15
15 5529119890 minus 14 3397119890 minus 14 2887119890 minus 15 3553119890 minus 15 3331119890 minus 15
20 1612119890 minus 13 2620119890 minus 14 4885119890 minus 15 3664119890 minus 15 3331119890 minus 15
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 1 Residual ℏ-curve for Example 1 using the 20th orderSHAM with119872 = 1 119904 = 1 and 119865 = 2
the lowest possible residual error after few iterations In thisexample it can be seen from Figure 2 that the highest orderdifferential mapping linear operator L1 is the slowest interms of convergence followed by the complete differentialmappingL3The operator that yields the fastest convergencein this example isL4 which gives full convergence after only10 iterations It can also be seen from Figure 2 that L2L3andL5 all converge to the same level which is slightly lowerthan that of L1 but slightly higher than L4 This resultsuggests that the SHAM algorithm that applies L4 gives thebest accuracy and when usingL1 the accuracy is the lowestin comparison with the other 4 choices of linear operators inthis example
Table 1 gives the minimum number of terms of theSHAM series that are required to attain full convergenceof the SHAM algorithm using the optimal values of theconvergence controlling parameter ℏ The correspondingmaximum residual errors are also tabulated It can be seen
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 2 Variation of residual error with SHAM order for Example1 when119872 = 1 119904 = 1 and 119865 = 2
from Table 1 that the least error is obtained when using L1and the other linear operators give more or less comparableresidual error values The SHAM with L4 requires thefewest number terms of the SHAM series to attain maximumaccuracy The SHAM with L5 comes second followed byL2 L3 and L1 in terms of the minimum SHAM orderrequired to give best accuracy This observation is in accordwith the trends displayed in Figure 2 The table also indicatesthat the number of terms of the SHAM series required for theSHAMapproximation to attainmaximum accuracy increaseswith an increase in the value of the Forchheimer numberparameter 119865 It is worth noting from the governing equation(22) that the parameter119865multiplies the nonlinear componentof the equation Table 2 shows the effect of increasing theporousmedia shape parameter 119904 on the SHAMorder requiredto attain the best accuracy It can be seen from the table thatthe effect of 119904 on convergence and accuracy is similar to that of119865 In particular it can be noted that as 119904 increasesmore SHAM
Maximum residual error 1198641199031 1599119890 minus 14 4108119890 minus 15 5329119890 minus 15 4441119890 minus 16 8993119890 minus 15
5 3375119890 minus 14 6439119890 minus 15 3442119890 minus 15 1443119890 minus 15 4663119890 minus 15
10 3952119890 minus 14 9770119890 minus 15 3442119890 minus 15 1776119890 minus 15 5218119890 minus 15
15 5529119890 minus 14 3397119890 minus 14 2887119890 minus 15 3553119890 minus 15 3331119890 minus 15
20 1612119890 minus 13 2620119890 minus 14 4885119890 minus 15 3664119890 minus 15 3331119890 minus 15
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 3 Residual ℏ-curve for Example 2 using the 20th orderSHAM with Re = 100 and 120572 = minus5∘
series terms are required to attain the maximum accuracy inall linear operators
Figure 3 shows the ℏ-curves for the Jeffery-Hamel prob-lem (Example 2) corresponding to the different SHAMschemes generated using all the linear operators using 20terms of the SHAM approximation series It can be seenfrom Figure 3 that after 20 iterationsL4 would give the bestaccuracy and L1 gives the least accuracy This is in accordwith the observation made earlier in Example 1 It can alsobe observed that the behaviour of the SHAM with L1 isapproximately the same as that of the SHAM with L2 Thiscan be noted from the observation that the ℏ-curve forL1 lieson top of that forL2 In Figure 4 the variation of the residualerror is shown as a function of the SHAM order It can beobserved that in all test linear operators the residual errordecreases with the number of terms used in the SHAM seriesThis demonstrates that the SHAM approximation converges
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 4 Variation of residual error with SHAM order for Example2 when Re = 100 and 120572 = minus5∘
in all cases considered in this example The convergence isfastest for L4 followed by L5 and is slowest when L1 isused in the SHAM algorithm It can also be observed that theresults for L1 L2 L3 and L5 converge to approximatelythe same level while L4 converges to a level lower than thatof the other linear operators Thus in addition to convergingmuch faster L4 gives results that are slightly more accuratethan those obtained using the other linear operators
Table 3 presents the minimum SHAM order required togive maximum accuracy at the the optimal values of theconvergence controlling parameter ℏ and the correspondingresidual errorsThe results are generated for increasing valuesof the Reynolds number (Re) It can be seen from Table 3that the minimum order of the SHAM required graduallyincreases with an increase in Re for both L1 and L2 Thereis no significant increase in the minimum SHAM orderrequired in the case of L3 L4 and L5 The results from
Maximum residual error 1198641199031 8451119890 minus 12 8987119890 minus 12 1074119890 minus 11 4886119890 minus 14 1256119890 minus 11
50 8636119890 minus 12 8887119890 minus 12 1148119890 minus 11 1357119890 minus 12 7468119890 minus 12
100 7930119890 minus 12 8759119890 minus 12 1303119890 minus 11 1886119890 minus 12 1633119890 minus 11
150 7815119890 minus 12 8092119890 minus 12 2546119890 minus 11 2973119890 minus 12 1808119890 minus 11
200 9128119890 minus 12 8421119890 minus 12 3209119890 minus 11 4435119890 minus 12 1176119890 minus 11
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 5 Residual ℏ-curve for Example 3 using the 20th orderSHAM with Re = 10 and Ha = 1
Table 3 also reveal that the required minimum SHAM orderof L1 and L2 is comparable particularly for larger valuesof Re This confirms the observation made in Figures 3 and4 Thus it can be inferred that the behaviour of the SHAMwith higher order differential mapping (L1) is comparablewith that of SHAMwith linear partitionmapping (L2) in thesolution of the Jeffery-Hamel equation
Figure 5 shows the ℏ-curves corresponding to the exam-ple of laminar viscous flow in a semiporous channel subjectto a transverse magnetic field (Example 3) Again it can beseen from Figure 5 that the residual error is lowest whenthe SHAM is used with L4 and largest when used with L1which as was observed in the case of Jeffery-Hamel flowis comparable with L2 The same observation can be madein Figure 6 where the residual error is plotted against theSHAM order From Figure 6 it can also be seen that L4gives the best convergence followed by L5 and the slowest
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 6 Variation of residual error with SHAM order for Example3 when Re = 10 and Ha = 1
convergence is obtained when usingL1 which is comparabletoL2
Table 4 shows effect of increasing the Reynolds number(Re) on the minimum SHAM orders and the correspondingresidual errors It can be observed from Table 4 that theminimumSHAMorders increasewith an increase inRewhenthe SHAM is used with L1 L2 and L3 It can also beseen that there is no significant increase in the minimumSHAM orders when using L4 and L5 Furthermore theresidual error obtained using L4 is lower than that of theother linear operatorsThis is in linewith similar observationsmade earlier
Table 5 shows the effect of increasing the Hartmannnumber (Ha) on the minimum SHAM orders and thecorresponding residual errors in Example 3 In this case itcan be observed that the minimum SHAM orders increasewith an increase in Ha only in the case of L1 In the case of
Maximum Residual error 1198641199032 1484119890 minus 08 1323119890 minus 08 9815119890 minus 09 3391119890 minus 12 2954119890 minus 09
4 8618119890 minus 09 3334119890 minus 09 3888119890 minus 09 5698119890 minus 12 2855119890 minus 09
6 2258119890 minus 09 6625119890 minus 09 1281119890 minus 08 1461119890 minus 11 6068119890 minus 09
8 1172119890 minus 09 2022119890 minus 09 1012119890 minus 08 1690119890 minus 11 7852119890 minus 09
10 8887119890 minus 09 5535119890 minus 09 1261119890 minus 08 5424119890 minus 11 2253119890 minus 09
15 3325119890 minus 09 1870119890 minus 09 5521119890 minus 09 2964119890 minus 10 5871119890 minus 09
the other linear operators theminimum SHAMorders do notincrease with an increase in HaThis means that convergenceof the SHAM series is independent of the value of Ha whenthe SHAM is used with L2 L3 L4 and L5 It can also benoted from Table 5 that the residual error is smallest whenL4 is used If can be inferred that the SHAM is most accuratewhen used withL4
Figure 7 shows the ℏ-curves corresponding to the exam-ple of two-dimensional viscous flow in a rectangular domainbounded by two moving porous walls (Example 4) It canbe noted from Figure 7 that the residual error is the lowestwhen the SHAM is used with L4 and the largest whenused with L1 after 20 iterations Figure 8 illustrates thevariation of the residual error against the SHAM order It canbe seen from Figure 8 that L4 gives the best convergencefollowed by L5 and the slowest convergence is obtainedwhen using L1 The linear operators L1 L2 L3 andL5 converge to the same level of residual error whichis higher than that of L4 Again it can be inferred that
L4 gives superior accuracy compared to the other linearoperators
Table 6 shows the effect of increasing the Reynolds num-ber (Re) on the minimum SHAM orders and correspondingresidual errors in Example 4 It can be observed that theminimumSHAMorders increasewith an increase inRewhenthe SHAM is used with L1 L2 and L3 In the case of L4andL5 it seems possible that an increase in Re has no stronginfluence in the convergence speed of the SHAM It can alsobe noted from Table 6 that the residual error is the smallestwhen L4 is used This is in accord with the trends observedin all the other examples considered in this work Table 7shows the effect of the nondimensional wall dilation rate 120572on the minimum SHAM order and corresponding residualerrors It can be observed from Table 7 that an increasein 120572 results in a corresponding significant increase in theminimum SHAMorder forL1Theminimum SHAMordersare not significantly influenced by 120572 when the SHAM is usedwithL2L3L4 andL5 Furthermore the best accuracy isobtained when usingL4
Maximum residual error 1198641199031 2103119890 minus 09 4065119890 minus 09 1354119890 minus 08 4946119890 minus 13 6441119890 minus 09
2 1198119890 minus 09 1109119890 minus 08 7087119890 minus 10 7009119890 minus 12 9531119890 minus 09
4 1954119890 minus 09 1017119890 minus 08 1604119890 minus 08 3667119890 minus 11 3063119890 minus 09
6 4419119890 minus 09 5935119890 minus 09 1772119890 minus 08 7262119890 minus 11 1388119890 minus 08
8 1930119890 minus 09 2864119890 minus 09 1449119890 minus 08 9716119890 minus 11 1221119890 minus 09
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 7 Residual ℏ-curve for Example 4 using the 20th orderSHAM with Re = 10 and 120572 = 1
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 8 Variation of residual error with SHAM order for Example4 when Re = 10 and 120572 = 1
14 Mathematical Problems in Engineering
6 Conclusion
This study gives a systematic way of choosing initial approx-imations and linear operators that can be employed inthe spectral homotopy analysis method (SHAM) algorithmThe main aim of the study was to determine the linearoperator definition that gives approximate solutions withoptimal convergence and accuracy A comparison betweenthe accuracy and convergence of SHAM algorithms usingcommonly used linear operators and newly proposed linearoperators was conducted Simulations were conducted onselected nonlinear boundary value problems defined onbounded domains One of the more significant findings toemerge from this study is that the linear operator definedby the highest order differential matching which is themost commonly used linear operator for solving boundaryvalue problems defined on bounded domains results inthe slowest converging and least accurate SHAM algorithmThe SHAM algorithm that uses a linear operator definedby the method of linear partition matching was found toperform slightly better than the SHAMwith the highest orderdifferential matching in some cases However in other casesthe methods were found to be comparable It was also shownthat the SHAMwith linear operator defined by themethod ofcomplete differential matching performed much better thanthe SHAMwith linear partitionmapping or the highest orderdifferential matching The study proposed a modification ofthe complete differential method approach which unlike theoriginal approach does not ignore the nonlinear factors of thederivative terms in the definition of the linear operator Thisnew definition of linear operator was found to perform betterthan the original complete differential method approachThe most optimal linear operator was found to be one thatuses the linear partition mapping after a transformation thatexpresses the unknown function as a sum of a new unknownplus some initial guess that satisfies the underlying boundaryconditions
The current findings add substantially to our under-standing of how homotopy analysis based methods canbe modified to yield approximate solutions with optimumaccuracy and convergence Further work needs to be done toinvestigate the application of the linear operators proposed inthis study in solving nonlinear partial differential equationsIt would also be very interesting to extend the current studyto systems of nonlinear differential equations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is based on the research supported in part by theNational Research Foundation of South Africa (Grant no85596)
References
[1] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[2] S J Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer Berlin Germany 2012
[3] S J Liao Advances in Homotopy Analysis Method WorldScientific Singapore 2014
[5] S S Motsa P Sibanda and S Shateyi ldquoA new spectral-homotopy analysismethod for solving a nonlinear second orderBVPrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 9 pp 2293ndash2302 2010
[6] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers and Fluids vol 39 no 7 pp 1219ndash1225 2010
[7] R A Van Gorder and K Vajravelu ldquoOn the selection of auxil-iary functions operators and convergence control parametersin the application of the homotopy analysismethod to nonlineardifferential equations a general approachrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 12 pp4078ndash4089 2009
[8] Y Shan and T Chaolu ldquoA method to select the initial guesssolution auxiliary linear operator and set of basic functions ofhomotopy analysis methodrdquo in Proceedings of the IEEE Inter-national Conference on Intelligent Computing and IntegratedSystems (ICISS rsquo10) pp 410ndash412 Guilin China October 2010
[9] R A van Gorder ldquoStability of auxiliary linear operatorand convergence-control parameter in the homotopy analysismethodrdquo in Advances in Homotopy Analysis Method S J LiaoEd pp 123ndash180 World Scientific Singapore 2014
[10] S M Rassoulinejad-Mousavi and S Abbasbandy ldquoAnalysisof forced convection in a circular tube filled with a darcy-brinkman-forchheimer porous medium using spectral homo-topy analysis methodrdquo Journal of Fluids Engineering Transac-tions of the ASME vol 133 no 10 Article ID 101207 2011
[11] A A Joneidi G Domairry andM Babaelahi ldquoThree analyticalmethods applied to Jeffery-Hamel flowrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 11 pp3423ndash3434 2010
[12] S M Moghimi G Domairry S Soleimani E Ghasemi and HBararnia ldquoApplication of homotopy analysis method to solveMHD Jeffery-Hamel flows in non-parallel wallsrdquo Advances inEngineering Software vol 42 no 3 pp 108ndash113 2011
[13] M Esmaeilpour and D D Ganji ldquoSolution of the Jeffery-Hamel flowproblemby optimal homotopy asymptoticmethodrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3405ndash3411 2010
[14] A Basiri Parsa M M Rashidi O Anwar Beg and S M SadrildquoSemi-computational simulation of magneto-hemodynamicflow in a semi-porous channel using optimal homotopy anddifferential transform methodsrdquo Computers in Biology andMedicine vol 43 no 9 pp 1142ndash1153 2013
[15] Z Ziabakhsh andGDomairry ldquoSolution of the laminar viscousflow in a semi-porous channel in the presence of a uniformmagnetic field by using the homotopy analysis methodrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 4 pp 1284ndash1294 2009
Mathematical Problems in Engineering 15
[16] S SMotsa S Shateyi GMarewo andP Sibanda ldquoAn improvedspectral homotopy analysis method for MHD flow in a semi-porous channelrdquo Numerical Algorithms vol 60 no 3 pp 463ndash481 2012
[17] S Dinarvand and M M Rashidi ldquoA reliable treatment of ahomotopy analysis method for two-dimensional viscous flowin a rectangular domain bounded by twomoving porous wallsrdquoNonlinear Analysis Real World Applications vol 11 no 3 pp1502ndash1512 2010
[18] H Xu Z Lin S Liao JWu and JMajdalani ldquoHomotopy basedsolutions of the Navier-Stokes equations for a porous channelwith orthogonally moving wallsrdquo Physics of Fluids vol 22 no5 Article ID 053601 pp 1ndash18 2010
[19] M M Rashidi E Erfani and B Rostami ldquoOptimal HomotopyAsymptotic Method for solving viscous ow through expandingor contracting gaps with permeable wallsrdquo Transaction on IoTand Cloud Computing vol 2 no 1 pp 85ndash111 2014
[20] Z GMakukula P Sibanda and S S Motsa ldquoA novel numericaltechnique for two-dimensional laminar flow between twomoving porous wallsrdquo Mathematical Problems in Engineeringvol 2010 Article ID 528956 15 pages 2010
[21] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Series in Com-putational Physics Springer New York NY USA 1988
[22] L N Trefethen Spectral Methods in MATLAB SIAM 2000[23] J Majdalani C Zhou and C A Dawson ldquoTwo-dimensional
viscous flow between slowly expanding or contracting wallswith weak permeabilityrdquo Journal of Biomechanics vol 35 no10 pp 1399ndash1403 2002
1198774119898minus1 =L4 (119911119898minus1) minus (1 minus 120594119898) 1206011 (119909) minus 119865119904119898minus1
sum
119899=0
119911119899119911119898minus1minus119899
(34)
and 1206011(119909) is the right hand side of (29)
42 Example 2 Jeffery-Hamel Equation Here we considerthe Jeffery-Hamel equation that models the steady two-dimensional flow of an incompressible conducting viscousfluid between two rigid plane walls that meet at an angle 2120572The rigid walls are considered to be divergent if 120572 gt 0 andconvergent if 120572 lt 0 The governing equation is defined as
where Re is the Reynolds number This problem was inves-tigated using the standard HAM with the highest orderdifferential matching linear operator L1(119910) in [11 12] Arelatedmethod called optimal homotopy asymptoticmethod(OHAM) was used to solve the same problem in [13] againwith L1(119910) In Motsa et al [6] the SHAM was used withthe auxiliary linear operator defined using the transformedlinear partition mapping methodL4(119911) In this example weexplore the auxiliary linear operators defined as
The function 1199100(119909) is chosen as a function that satisfiesthe boundary conditions and the linear part of (35) Thus1199100(119909) is
1199100 (119909) =cos (2120572) minus cos (2120572119909)
cos (2120572) minus 1 (38)
The linear operator L4 is obtained by using the linearpartition mapping after using the transformation 119910(119909) =119911(119909) + 1199100(119909) The resulting equation is
Thus the initial approximations corresponding to the SHAMwithL1L2 andL3 are given respectively by
1199100 (119909) = 1 minus 1199092 1199100 (119909) =
cos (2120572) minus cos (2120572119909)cos (2120572) minus 1
1199100 (119909) =cos (21205721) minus cos (21205721119909)
cos (21205721) minus 1
(43)
where 1199041 = radic41205722 + 2120572Re The initial approximation 1199110(119909)to use with L4 is obtained as a solution of the differentialequation formed from the linear part of (39)That is we solve
119911101584010158401015840
0+ 412057221199111015840
0+ 2120572Re1199100119911
1015840
0+ 2120572Re1199101015840
01199110 = 1206012 (119909) (44)
Since the above equation has variable coefficients it is solvedusing the spectral method Similarly the initial approxima-tion to use withL5 is obtained as a solution of the differentialequation
119910101584010158401015840
0+ 412057221199101015840
0+ 2120572Re1199100119910
1015840
0= 0
1199100 (0) = 1 1199101015840
0(0) = 0 1199100 (1) = 0
(45)
with 1199100 given by (38)With the nonlinear operator 119873119894 defined as the homoge-
neous part of the governing nonlinear equation in each casewe obtain the following higher order deformation equations
L119894 [119910119898 (119909) minus 120594119898119910119898minus1 (119909)]
1198774119898minus1 =L4 (119911119898minus1) minus (1 minus 120594119898) 1206012 (119909) + 2120572Re119898minus1
sum
119899=0
1199111198991199111015840
119898minus1minus119899
(49)
Mathematical Problems in Engineering 7
43 Example 3 Laminar Viscous Flow in a Semiporous Chan-nel Subject to a Transverse Magnetic Field In this section weconsider the problem of laminar viscous flow in a semiporouschannel subject to a transverse magnetic field given by
1199101015840101584010158401015840minusHa211991010158401015840 + Re (119910119910101584010158401015840 minus 119910101584011991010158401015840) = 0 (50)
subject to the boundary conditions
119910 = 0 1199101015840= 0 119909 = 0
119910 = 1 1199101015840= 0 119909 = 1
(51)
This example was previously investigated using the stan-dardHAMwith the highest order differential matching linearoperatorL1(119910) in [14 15] In [16] a blend between the SHAMand a spectral method based quasilinearisation method wasused to solve the same problem In this investigation weconsider the following auxiliary linear operators
Thus the initial approximations corresponding to the SHAMwithL1 andL2 are given by
1199100 (119909) = 31199092minus 21199092
1199100 (119909) =Ha119909 minus 119890Ha119909 + 119890HaminusHa119909 + 119890Ha (Ha119909 minus 1) + 1
119890Ha (Ha minus 2) +Ha + 2
(54)
We note that in this example 1199100 is the second solution in(54)The initial approximations corresponding toL3 andL5are obtained by solving (53) numerically using the spectralmethod Similarly 1199110(119909) is obtained by numerically solving
1199111015840101584010158401015840
0minusHa211991110158401015840
0
+ Re (1199100119911101584010158401015840
0+ 119910101584010158401015840
01199110 minus 119910
1015840
011991110158401015840
0minus 11991010158401015840
01199111015840
0) = 1206013 (119909)
(55)
subject to the boundary conditions
1199110 (0) = 0 1199111015840
0(0) = 0 1199110 (1) = 0 119911
1015840
0(1) = 0
(56)
where
1206013 (119909) = minus (1199101015840101584010158401015840
0minusHa211991010158401015840
0+ Re (1199100119910
101584010158401015840
0minus 1199101015840
011991010158401015840
0)) (57)
The higher order deformation equations are defined as
L119894 [119910119898 (119909) minus 120594119898119910119898minus1 (119909)]
1198774119898minus1 = L4 (119911119898minus1) minus (1 minus 120594119898) 1206013 (119909)
+ Re119898minus1
sum
119899=0
(119911119899119911101584010158401015840
119898minus1minus119899minus 1199111015840
11989911991110158401015840
119898minus1minus119899)
(61)
44 Example 4 Two-Dimensional Viscous Flow in a Rectan-gular Domain Bounded by Two Moving Porous Walls In thissection we consider the problem of two-dimensional viscousflow in a rectangular domain bounded by twomoving porouswalls The governing equations are given in [17 23] as
101584010158401015840+ 311991010158401015840) + Re (119910119910101584010158401015840 minus 119910101584011991010158401015840) = 0 (62)
subject to the boundary conditions
119910 = 0 11991010158401015840= 0 at 119909 = 0
119910 = 1 1199101015840= 0 at 119909 = 1
(63)
where 120572 is the nondimensional wall dilation rate defined tobe positive for expansion and negative for contraction andRe is the permeation Reynolds number defined positive forinjection and negative for suction through the walls Thenonlinear equation (62) was investigated using the standardHAM with the highest order differential matching linearoperatorL1(119910) in [17 18] Rashidi et al [19] used the optimalhomotopy asymptotic method (OHAM) to solve the sameproblem again using L1(119910) as linear operator The SHAMwith transformed linear partition methodL4(119911) was used in[20]
8 Mathematical Problems in Engineering
In this investigation we consider the following auxiliarylinear operators
1198774119898minus1 = L4 (119911119898minus1) minus (1 minus 120594119898) 1206014 (119909)
+ Re119898minus1
sum
119899=0
(119911119899119911101584010158401015840
119898minus1minus119899minus 1199111015840
11989911991110158401015840
119898minus1minus119899)
(72)
5 Results and Discussion
The SHAM algorithm using the proposed auxiliary linearoperators was applied to the several test problems presentedin the last section In this section we present the numeri-cal results that highlight the difference in the accuracy ofthe approximate solutions and convergence of the differentSHAM algorithms All the simulations were conducted using119872 = 20 collocation points The accuracy and convergenceof the SHAM approximate series solution depend on carefulselection of the auxiliary parameter ℏ The residual errorwas used to determine the optimal values of the auxiliaryparameter Using finite terms of the SHAM series we define119870th order approximation at the collocation points 119909119895 for 119895 =0 1 2 119872 as
Y =119870
sum
119896=0
119910119896 (119909119895) =
119870
sum
119896=0
119884119896 (73)
where 119884119896 = [119910119896(1199090)119910119896(1199091) sdot sdot sdot 119910119896(119909119872)]119905 Given that Y is the
SHAM approximate solution at the collocation points theresidual error at the collocation points is defined as
Re 119904 (119884) =N [ (Y ℏ)] (74)
where N is the nonlinear operator defining the governingnonlinear equation (see (4)) We define the infinity norm ofthe residual error as
119864119903 (ℏ) = N[ (Y ℏ)]infin (75)
The optimal ℏ that gives the best accuracy was identified tobe the value of ℏ located at the minimum of the graph of thevariation of the infinity norm against ℏ
Figure 1 gives the typical residual error curves that areobtained using the 20th order SHAM approximate solutionof the Darcy-Brinkman-Forchheimer equation (Example 1)The minimum of the curve gives the best possible residualerror for each auxiliary linear operator used in the SHAMalgorithm A comparison of the minima of the residual errorcurves gives an indication of which auxiliary linear operatoris likely to give better accuracy when used with the optimalauxiliary linear operator in the SHAM algorithm It can beseen from Figure 1 that the best accuracy at the 20th orderSHAM approximation is obtained when using the linearpartition mapping after transformation (L4) The highestorder differential mapping (L1) gives the least accuracyFigure 2 shows the variation of the norm of the residual errorat the optimal value of ℏ as a function of the SHAMorderThisgraph gives an indication of how the SHAM series convergeswith an increase in the number of iterations (terms used inthe series) Rapid convergence is determined by attaining
Maximum Residual error 1198641199031 1599119890 minus 14 4108119890 minus 15 5329119890 minus 15 4441119890 minus 16 8993119890 minus 15
5 3375119890 minus 14 6439119890 minus 15 3442119890 minus 15 1443119890 minus 15 4663119890 minus 15
10 3952119890 minus 14 9770119890 minus 15 3442119890 minus 15 1776119890 minus 15 5218119890 minus 15
15 5529119890 minus 14 3397119890 minus 14 2887119890 minus 15 3553119890 minus 15 3331119890 minus 15
20 1612119890 minus 13 2620119890 minus 14 4885119890 minus 15 3664119890 minus 15 3331119890 minus 15
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 1 Residual ℏ-curve for Example 1 using the 20th orderSHAM with119872 = 1 119904 = 1 and 119865 = 2
the lowest possible residual error after few iterations In thisexample it can be seen from Figure 2 that the highest orderdifferential mapping linear operator L1 is the slowest interms of convergence followed by the complete differentialmappingL3The operator that yields the fastest convergencein this example isL4 which gives full convergence after only10 iterations It can also be seen from Figure 2 that L2L3andL5 all converge to the same level which is slightly lowerthan that of L1 but slightly higher than L4 This resultsuggests that the SHAM algorithm that applies L4 gives thebest accuracy and when usingL1 the accuracy is the lowestin comparison with the other 4 choices of linear operators inthis example
Table 1 gives the minimum number of terms of theSHAM series that are required to attain full convergenceof the SHAM algorithm using the optimal values of theconvergence controlling parameter ℏ The correspondingmaximum residual errors are also tabulated It can be seen
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 2 Variation of residual error with SHAM order for Example1 when119872 = 1 119904 = 1 and 119865 = 2
from Table 1 that the least error is obtained when using L1and the other linear operators give more or less comparableresidual error values The SHAM with L4 requires thefewest number terms of the SHAM series to attain maximumaccuracy The SHAM with L5 comes second followed byL2 L3 and L1 in terms of the minimum SHAM orderrequired to give best accuracy This observation is in accordwith the trends displayed in Figure 2 The table also indicatesthat the number of terms of the SHAM series required for theSHAMapproximation to attainmaximum accuracy increaseswith an increase in the value of the Forchheimer numberparameter 119865 It is worth noting from the governing equation(22) that the parameter119865multiplies the nonlinear componentof the equation Table 2 shows the effect of increasing theporousmedia shape parameter 119904 on the SHAMorder requiredto attain the best accuracy It can be seen from the table thatthe effect of 119904 on convergence and accuracy is similar to that of119865 In particular it can be noted that as 119904 increasesmore SHAM
Maximum residual error 1198641199031 1599119890 minus 14 4108119890 minus 15 5329119890 minus 15 4441119890 minus 16 8993119890 minus 15
5 3375119890 minus 14 6439119890 minus 15 3442119890 minus 15 1443119890 minus 15 4663119890 minus 15
10 3952119890 minus 14 9770119890 minus 15 3442119890 minus 15 1776119890 minus 15 5218119890 minus 15
15 5529119890 minus 14 3397119890 minus 14 2887119890 minus 15 3553119890 minus 15 3331119890 minus 15
20 1612119890 minus 13 2620119890 minus 14 4885119890 minus 15 3664119890 minus 15 3331119890 minus 15
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 3 Residual ℏ-curve for Example 2 using the 20th orderSHAM with Re = 100 and 120572 = minus5∘
series terms are required to attain the maximum accuracy inall linear operators
Figure 3 shows the ℏ-curves for the Jeffery-Hamel prob-lem (Example 2) corresponding to the different SHAMschemes generated using all the linear operators using 20terms of the SHAM approximation series It can be seenfrom Figure 3 that after 20 iterationsL4 would give the bestaccuracy and L1 gives the least accuracy This is in accordwith the observation made earlier in Example 1 It can alsobe observed that the behaviour of the SHAM with L1 isapproximately the same as that of the SHAM with L2 Thiscan be noted from the observation that the ℏ-curve forL1 lieson top of that forL2 In Figure 4 the variation of the residualerror is shown as a function of the SHAM order It can beobserved that in all test linear operators the residual errordecreases with the number of terms used in the SHAM seriesThis demonstrates that the SHAM approximation converges
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 4 Variation of residual error with SHAM order for Example2 when Re = 100 and 120572 = minus5∘
in all cases considered in this example The convergence isfastest for L4 followed by L5 and is slowest when L1 isused in the SHAM algorithm It can also be observed that theresults for L1 L2 L3 and L5 converge to approximatelythe same level while L4 converges to a level lower than thatof the other linear operators Thus in addition to convergingmuch faster L4 gives results that are slightly more accuratethan those obtained using the other linear operators
Table 3 presents the minimum SHAM order required togive maximum accuracy at the the optimal values of theconvergence controlling parameter ℏ and the correspondingresidual errorsThe results are generated for increasing valuesof the Reynolds number (Re) It can be seen from Table 3that the minimum order of the SHAM required graduallyincreases with an increase in Re for both L1 and L2 Thereis no significant increase in the minimum SHAM orderrequired in the case of L3 L4 and L5 The results from
Maximum residual error 1198641199031 8451119890 minus 12 8987119890 minus 12 1074119890 minus 11 4886119890 minus 14 1256119890 minus 11
50 8636119890 minus 12 8887119890 minus 12 1148119890 minus 11 1357119890 minus 12 7468119890 minus 12
100 7930119890 minus 12 8759119890 minus 12 1303119890 minus 11 1886119890 minus 12 1633119890 minus 11
150 7815119890 minus 12 8092119890 minus 12 2546119890 minus 11 2973119890 minus 12 1808119890 minus 11
200 9128119890 minus 12 8421119890 minus 12 3209119890 minus 11 4435119890 minus 12 1176119890 minus 11
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 5 Residual ℏ-curve for Example 3 using the 20th orderSHAM with Re = 10 and Ha = 1
Table 3 also reveal that the required minimum SHAM orderof L1 and L2 is comparable particularly for larger valuesof Re This confirms the observation made in Figures 3 and4 Thus it can be inferred that the behaviour of the SHAMwith higher order differential mapping (L1) is comparablewith that of SHAMwith linear partitionmapping (L2) in thesolution of the Jeffery-Hamel equation
Figure 5 shows the ℏ-curves corresponding to the exam-ple of laminar viscous flow in a semiporous channel subjectto a transverse magnetic field (Example 3) Again it can beseen from Figure 5 that the residual error is lowest whenthe SHAM is used with L4 and largest when used with L1which as was observed in the case of Jeffery-Hamel flowis comparable with L2 The same observation can be madein Figure 6 where the residual error is plotted against theSHAM order From Figure 6 it can also be seen that L4gives the best convergence followed by L5 and the slowest
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 6 Variation of residual error with SHAM order for Example3 when Re = 10 and Ha = 1
convergence is obtained when usingL1 which is comparabletoL2
Table 4 shows effect of increasing the Reynolds number(Re) on the minimum SHAM orders and the correspondingresidual errors It can be observed from Table 4 that theminimumSHAMorders increasewith an increase inRewhenthe SHAM is used with L1 L2 and L3 It can also beseen that there is no significant increase in the minimumSHAM orders when using L4 and L5 Furthermore theresidual error obtained using L4 is lower than that of theother linear operatorsThis is in linewith similar observationsmade earlier
Table 5 shows the effect of increasing the Hartmannnumber (Ha) on the minimum SHAM orders and thecorresponding residual errors in Example 3 In this case itcan be observed that the minimum SHAM orders increasewith an increase in Ha only in the case of L1 In the case of
Maximum Residual error 1198641199032 1484119890 minus 08 1323119890 minus 08 9815119890 minus 09 3391119890 minus 12 2954119890 minus 09
4 8618119890 minus 09 3334119890 minus 09 3888119890 minus 09 5698119890 minus 12 2855119890 minus 09
6 2258119890 minus 09 6625119890 minus 09 1281119890 minus 08 1461119890 minus 11 6068119890 minus 09
8 1172119890 minus 09 2022119890 minus 09 1012119890 minus 08 1690119890 minus 11 7852119890 minus 09
10 8887119890 minus 09 5535119890 minus 09 1261119890 minus 08 5424119890 minus 11 2253119890 minus 09
15 3325119890 minus 09 1870119890 minus 09 5521119890 minus 09 2964119890 minus 10 5871119890 minus 09
the other linear operators theminimum SHAMorders do notincrease with an increase in HaThis means that convergenceof the SHAM series is independent of the value of Ha whenthe SHAM is used with L2 L3 L4 and L5 It can also benoted from Table 5 that the residual error is smallest whenL4 is used If can be inferred that the SHAM is most accuratewhen used withL4
Figure 7 shows the ℏ-curves corresponding to the exam-ple of two-dimensional viscous flow in a rectangular domainbounded by two moving porous walls (Example 4) It canbe noted from Figure 7 that the residual error is the lowestwhen the SHAM is used with L4 and the largest whenused with L1 after 20 iterations Figure 8 illustrates thevariation of the residual error against the SHAM order It canbe seen from Figure 8 that L4 gives the best convergencefollowed by L5 and the slowest convergence is obtainedwhen using L1 The linear operators L1 L2 L3 andL5 converge to the same level of residual error whichis higher than that of L4 Again it can be inferred that
L4 gives superior accuracy compared to the other linearoperators
Table 6 shows the effect of increasing the Reynolds num-ber (Re) on the minimum SHAM orders and correspondingresidual errors in Example 4 It can be observed that theminimumSHAMorders increasewith an increase inRewhenthe SHAM is used with L1 L2 and L3 In the case of L4andL5 it seems possible that an increase in Re has no stronginfluence in the convergence speed of the SHAM It can alsobe noted from Table 6 that the residual error is the smallestwhen L4 is used This is in accord with the trends observedin all the other examples considered in this work Table 7shows the effect of the nondimensional wall dilation rate 120572on the minimum SHAM order and corresponding residualerrors It can be observed from Table 7 that an increasein 120572 results in a corresponding significant increase in theminimum SHAMorder forL1Theminimum SHAMordersare not significantly influenced by 120572 when the SHAM is usedwithL2L3L4 andL5 Furthermore the best accuracy isobtained when usingL4
Maximum residual error 1198641199031 2103119890 minus 09 4065119890 minus 09 1354119890 minus 08 4946119890 minus 13 6441119890 minus 09
2 1198119890 minus 09 1109119890 minus 08 7087119890 minus 10 7009119890 minus 12 9531119890 minus 09
4 1954119890 minus 09 1017119890 minus 08 1604119890 minus 08 3667119890 minus 11 3063119890 minus 09
6 4419119890 minus 09 5935119890 minus 09 1772119890 minus 08 7262119890 minus 11 1388119890 minus 08
8 1930119890 minus 09 2864119890 minus 09 1449119890 minus 08 9716119890 minus 11 1221119890 minus 09
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 7 Residual ℏ-curve for Example 4 using the 20th orderSHAM with Re = 10 and 120572 = 1
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 8 Variation of residual error with SHAM order for Example4 when Re = 10 and 120572 = 1
14 Mathematical Problems in Engineering
6 Conclusion
This study gives a systematic way of choosing initial approx-imations and linear operators that can be employed inthe spectral homotopy analysis method (SHAM) algorithmThe main aim of the study was to determine the linearoperator definition that gives approximate solutions withoptimal convergence and accuracy A comparison betweenthe accuracy and convergence of SHAM algorithms usingcommonly used linear operators and newly proposed linearoperators was conducted Simulations were conducted onselected nonlinear boundary value problems defined onbounded domains One of the more significant findings toemerge from this study is that the linear operator definedby the highest order differential matching which is themost commonly used linear operator for solving boundaryvalue problems defined on bounded domains results inthe slowest converging and least accurate SHAM algorithmThe SHAM algorithm that uses a linear operator definedby the method of linear partition matching was found toperform slightly better than the SHAMwith the highest orderdifferential matching in some cases However in other casesthe methods were found to be comparable It was also shownthat the SHAMwith linear operator defined by themethod ofcomplete differential matching performed much better thanthe SHAMwith linear partitionmapping or the highest orderdifferential matching The study proposed a modification ofthe complete differential method approach which unlike theoriginal approach does not ignore the nonlinear factors of thederivative terms in the definition of the linear operator Thisnew definition of linear operator was found to perform betterthan the original complete differential method approachThe most optimal linear operator was found to be one thatuses the linear partition mapping after a transformation thatexpresses the unknown function as a sum of a new unknownplus some initial guess that satisfies the underlying boundaryconditions
The current findings add substantially to our under-standing of how homotopy analysis based methods canbe modified to yield approximate solutions with optimumaccuracy and convergence Further work needs to be done toinvestigate the application of the linear operators proposed inthis study in solving nonlinear partial differential equationsIt would also be very interesting to extend the current studyto systems of nonlinear differential equations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is based on the research supported in part by theNational Research Foundation of South Africa (Grant no85596)
References
[1] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[2] S J Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer Berlin Germany 2012
[3] S J Liao Advances in Homotopy Analysis Method WorldScientific Singapore 2014
[5] S S Motsa P Sibanda and S Shateyi ldquoA new spectral-homotopy analysismethod for solving a nonlinear second orderBVPrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 9 pp 2293ndash2302 2010
[6] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers and Fluids vol 39 no 7 pp 1219ndash1225 2010
[7] R A Van Gorder and K Vajravelu ldquoOn the selection of auxil-iary functions operators and convergence control parametersin the application of the homotopy analysismethod to nonlineardifferential equations a general approachrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 12 pp4078ndash4089 2009
[8] Y Shan and T Chaolu ldquoA method to select the initial guesssolution auxiliary linear operator and set of basic functions ofhomotopy analysis methodrdquo in Proceedings of the IEEE Inter-national Conference on Intelligent Computing and IntegratedSystems (ICISS rsquo10) pp 410ndash412 Guilin China October 2010
[9] R A van Gorder ldquoStability of auxiliary linear operatorand convergence-control parameter in the homotopy analysismethodrdquo in Advances in Homotopy Analysis Method S J LiaoEd pp 123ndash180 World Scientific Singapore 2014
[10] S M Rassoulinejad-Mousavi and S Abbasbandy ldquoAnalysisof forced convection in a circular tube filled with a darcy-brinkman-forchheimer porous medium using spectral homo-topy analysis methodrdquo Journal of Fluids Engineering Transac-tions of the ASME vol 133 no 10 Article ID 101207 2011
[11] A A Joneidi G Domairry andM Babaelahi ldquoThree analyticalmethods applied to Jeffery-Hamel flowrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 11 pp3423ndash3434 2010
[12] S M Moghimi G Domairry S Soleimani E Ghasemi and HBararnia ldquoApplication of homotopy analysis method to solveMHD Jeffery-Hamel flows in non-parallel wallsrdquo Advances inEngineering Software vol 42 no 3 pp 108ndash113 2011
[13] M Esmaeilpour and D D Ganji ldquoSolution of the Jeffery-Hamel flowproblemby optimal homotopy asymptoticmethodrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3405ndash3411 2010
[14] A Basiri Parsa M M Rashidi O Anwar Beg and S M SadrildquoSemi-computational simulation of magneto-hemodynamicflow in a semi-porous channel using optimal homotopy anddifferential transform methodsrdquo Computers in Biology andMedicine vol 43 no 9 pp 1142ndash1153 2013
[15] Z Ziabakhsh andGDomairry ldquoSolution of the laminar viscousflow in a semi-porous channel in the presence of a uniformmagnetic field by using the homotopy analysis methodrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 4 pp 1284ndash1294 2009
Mathematical Problems in Engineering 15
[16] S SMotsa S Shateyi GMarewo andP Sibanda ldquoAn improvedspectral homotopy analysis method for MHD flow in a semi-porous channelrdquo Numerical Algorithms vol 60 no 3 pp 463ndash481 2012
[17] S Dinarvand and M M Rashidi ldquoA reliable treatment of ahomotopy analysis method for two-dimensional viscous flowin a rectangular domain bounded by twomoving porous wallsrdquoNonlinear Analysis Real World Applications vol 11 no 3 pp1502ndash1512 2010
[18] H Xu Z Lin S Liao JWu and JMajdalani ldquoHomotopy basedsolutions of the Navier-Stokes equations for a porous channelwith orthogonally moving wallsrdquo Physics of Fluids vol 22 no5 Article ID 053601 pp 1ndash18 2010
[19] M M Rashidi E Erfani and B Rostami ldquoOptimal HomotopyAsymptotic Method for solving viscous ow through expandingor contracting gaps with permeable wallsrdquo Transaction on IoTand Cloud Computing vol 2 no 1 pp 85ndash111 2014
[20] Z GMakukula P Sibanda and S S Motsa ldquoA novel numericaltechnique for two-dimensional laminar flow between twomoving porous wallsrdquo Mathematical Problems in Engineeringvol 2010 Article ID 528956 15 pages 2010
[21] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Series in Com-putational Physics Springer New York NY USA 1988
[22] L N Trefethen Spectral Methods in MATLAB SIAM 2000[23] J Majdalani C Zhou and C A Dawson ldquoTwo-dimensional
viscous flow between slowly expanding or contracting wallswith weak permeabilityrdquo Journal of Biomechanics vol 35 no10 pp 1399ndash1403 2002
43 Example 3 Laminar Viscous Flow in a Semiporous Chan-nel Subject to a Transverse Magnetic Field In this section weconsider the problem of laminar viscous flow in a semiporouschannel subject to a transverse magnetic field given by
1199101015840101584010158401015840minusHa211991010158401015840 + Re (119910119910101584010158401015840 minus 119910101584011991010158401015840) = 0 (50)
subject to the boundary conditions
119910 = 0 1199101015840= 0 119909 = 0
119910 = 1 1199101015840= 0 119909 = 1
(51)
This example was previously investigated using the stan-dardHAMwith the highest order differential matching linearoperatorL1(119910) in [14 15] In [16] a blend between the SHAMand a spectral method based quasilinearisation method wasused to solve the same problem In this investigation weconsider the following auxiliary linear operators
Thus the initial approximations corresponding to the SHAMwithL1 andL2 are given by
1199100 (119909) = 31199092minus 21199092
1199100 (119909) =Ha119909 minus 119890Ha119909 + 119890HaminusHa119909 + 119890Ha (Ha119909 minus 1) + 1
119890Ha (Ha minus 2) +Ha + 2
(54)
We note that in this example 1199100 is the second solution in(54)The initial approximations corresponding toL3 andL5are obtained by solving (53) numerically using the spectralmethod Similarly 1199110(119909) is obtained by numerically solving
1199111015840101584010158401015840
0minusHa211991110158401015840
0
+ Re (1199100119911101584010158401015840
0+ 119910101584010158401015840
01199110 minus 119910
1015840
011991110158401015840
0minus 11991010158401015840
01199111015840
0) = 1206013 (119909)
(55)
subject to the boundary conditions
1199110 (0) = 0 1199111015840
0(0) = 0 1199110 (1) = 0 119911
1015840
0(1) = 0
(56)
where
1206013 (119909) = minus (1199101015840101584010158401015840
0minusHa211991010158401015840
0+ Re (1199100119910
101584010158401015840
0minus 1199101015840
011991010158401015840
0)) (57)
The higher order deformation equations are defined as
L119894 [119910119898 (119909) minus 120594119898119910119898minus1 (119909)]
1198774119898minus1 = L4 (119911119898minus1) minus (1 minus 120594119898) 1206013 (119909)
+ Re119898minus1
sum
119899=0
(119911119899119911101584010158401015840
119898minus1minus119899minus 1199111015840
11989911991110158401015840
119898minus1minus119899)
(61)
44 Example 4 Two-Dimensional Viscous Flow in a Rectan-gular Domain Bounded by Two Moving Porous Walls In thissection we consider the problem of two-dimensional viscousflow in a rectangular domain bounded by twomoving porouswalls The governing equations are given in [17 23] as
101584010158401015840+ 311991010158401015840) + Re (119910119910101584010158401015840 minus 119910101584011991010158401015840) = 0 (62)
subject to the boundary conditions
119910 = 0 11991010158401015840= 0 at 119909 = 0
119910 = 1 1199101015840= 0 at 119909 = 1
(63)
where 120572 is the nondimensional wall dilation rate defined tobe positive for expansion and negative for contraction andRe is the permeation Reynolds number defined positive forinjection and negative for suction through the walls Thenonlinear equation (62) was investigated using the standardHAM with the highest order differential matching linearoperatorL1(119910) in [17 18] Rashidi et al [19] used the optimalhomotopy asymptotic method (OHAM) to solve the sameproblem again using L1(119910) as linear operator The SHAMwith transformed linear partition methodL4(119911) was used in[20]
8 Mathematical Problems in Engineering
In this investigation we consider the following auxiliarylinear operators
1198774119898minus1 = L4 (119911119898minus1) minus (1 minus 120594119898) 1206014 (119909)
+ Re119898minus1
sum
119899=0
(119911119899119911101584010158401015840
119898minus1minus119899minus 1199111015840
11989911991110158401015840
119898minus1minus119899)
(72)
5 Results and Discussion
The SHAM algorithm using the proposed auxiliary linearoperators was applied to the several test problems presentedin the last section In this section we present the numeri-cal results that highlight the difference in the accuracy ofthe approximate solutions and convergence of the differentSHAM algorithms All the simulations were conducted using119872 = 20 collocation points The accuracy and convergenceof the SHAM approximate series solution depend on carefulselection of the auxiliary parameter ℏ The residual errorwas used to determine the optimal values of the auxiliaryparameter Using finite terms of the SHAM series we define119870th order approximation at the collocation points 119909119895 for 119895 =0 1 2 119872 as
Y =119870
sum
119896=0
119910119896 (119909119895) =
119870
sum
119896=0
119884119896 (73)
where 119884119896 = [119910119896(1199090)119910119896(1199091) sdot sdot sdot 119910119896(119909119872)]119905 Given that Y is the
SHAM approximate solution at the collocation points theresidual error at the collocation points is defined as
Re 119904 (119884) =N [ (Y ℏ)] (74)
where N is the nonlinear operator defining the governingnonlinear equation (see (4)) We define the infinity norm ofthe residual error as
119864119903 (ℏ) = N[ (Y ℏ)]infin (75)
The optimal ℏ that gives the best accuracy was identified tobe the value of ℏ located at the minimum of the graph of thevariation of the infinity norm against ℏ
Figure 1 gives the typical residual error curves that areobtained using the 20th order SHAM approximate solutionof the Darcy-Brinkman-Forchheimer equation (Example 1)The minimum of the curve gives the best possible residualerror for each auxiliary linear operator used in the SHAMalgorithm A comparison of the minima of the residual errorcurves gives an indication of which auxiliary linear operatoris likely to give better accuracy when used with the optimalauxiliary linear operator in the SHAM algorithm It can beseen from Figure 1 that the best accuracy at the 20th orderSHAM approximation is obtained when using the linearpartition mapping after transformation (L4) The highestorder differential mapping (L1) gives the least accuracyFigure 2 shows the variation of the norm of the residual errorat the optimal value of ℏ as a function of the SHAMorderThisgraph gives an indication of how the SHAM series convergeswith an increase in the number of iterations (terms used inthe series) Rapid convergence is determined by attaining
Maximum Residual error 1198641199031 1599119890 minus 14 4108119890 minus 15 5329119890 minus 15 4441119890 minus 16 8993119890 minus 15
5 3375119890 minus 14 6439119890 minus 15 3442119890 minus 15 1443119890 minus 15 4663119890 minus 15
10 3952119890 minus 14 9770119890 minus 15 3442119890 minus 15 1776119890 minus 15 5218119890 minus 15
15 5529119890 minus 14 3397119890 minus 14 2887119890 minus 15 3553119890 minus 15 3331119890 minus 15
20 1612119890 minus 13 2620119890 minus 14 4885119890 minus 15 3664119890 minus 15 3331119890 minus 15
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 1 Residual ℏ-curve for Example 1 using the 20th orderSHAM with119872 = 1 119904 = 1 and 119865 = 2
the lowest possible residual error after few iterations In thisexample it can be seen from Figure 2 that the highest orderdifferential mapping linear operator L1 is the slowest interms of convergence followed by the complete differentialmappingL3The operator that yields the fastest convergencein this example isL4 which gives full convergence after only10 iterations It can also be seen from Figure 2 that L2L3andL5 all converge to the same level which is slightly lowerthan that of L1 but slightly higher than L4 This resultsuggests that the SHAM algorithm that applies L4 gives thebest accuracy and when usingL1 the accuracy is the lowestin comparison with the other 4 choices of linear operators inthis example
Table 1 gives the minimum number of terms of theSHAM series that are required to attain full convergenceof the SHAM algorithm using the optimal values of theconvergence controlling parameter ℏ The correspondingmaximum residual errors are also tabulated It can be seen
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 2 Variation of residual error with SHAM order for Example1 when119872 = 1 119904 = 1 and 119865 = 2
from Table 1 that the least error is obtained when using L1and the other linear operators give more or less comparableresidual error values The SHAM with L4 requires thefewest number terms of the SHAM series to attain maximumaccuracy The SHAM with L5 comes second followed byL2 L3 and L1 in terms of the minimum SHAM orderrequired to give best accuracy This observation is in accordwith the trends displayed in Figure 2 The table also indicatesthat the number of terms of the SHAM series required for theSHAMapproximation to attainmaximum accuracy increaseswith an increase in the value of the Forchheimer numberparameter 119865 It is worth noting from the governing equation(22) that the parameter119865multiplies the nonlinear componentof the equation Table 2 shows the effect of increasing theporousmedia shape parameter 119904 on the SHAMorder requiredto attain the best accuracy It can be seen from the table thatthe effect of 119904 on convergence and accuracy is similar to that of119865 In particular it can be noted that as 119904 increasesmore SHAM
Maximum residual error 1198641199031 1599119890 minus 14 4108119890 minus 15 5329119890 minus 15 4441119890 minus 16 8993119890 minus 15
5 3375119890 minus 14 6439119890 minus 15 3442119890 minus 15 1443119890 minus 15 4663119890 minus 15
10 3952119890 minus 14 9770119890 minus 15 3442119890 minus 15 1776119890 minus 15 5218119890 minus 15
15 5529119890 minus 14 3397119890 minus 14 2887119890 minus 15 3553119890 minus 15 3331119890 minus 15
20 1612119890 minus 13 2620119890 minus 14 4885119890 minus 15 3664119890 minus 15 3331119890 minus 15
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 3 Residual ℏ-curve for Example 2 using the 20th orderSHAM with Re = 100 and 120572 = minus5∘
series terms are required to attain the maximum accuracy inall linear operators
Figure 3 shows the ℏ-curves for the Jeffery-Hamel prob-lem (Example 2) corresponding to the different SHAMschemes generated using all the linear operators using 20terms of the SHAM approximation series It can be seenfrom Figure 3 that after 20 iterationsL4 would give the bestaccuracy and L1 gives the least accuracy This is in accordwith the observation made earlier in Example 1 It can alsobe observed that the behaviour of the SHAM with L1 isapproximately the same as that of the SHAM with L2 Thiscan be noted from the observation that the ℏ-curve forL1 lieson top of that forL2 In Figure 4 the variation of the residualerror is shown as a function of the SHAM order It can beobserved that in all test linear operators the residual errordecreases with the number of terms used in the SHAM seriesThis demonstrates that the SHAM approximation converges
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 4 Variation of residual error with SHAM order for Example2 when Re = 100 and 120572 = minus5∘
in all cases considered in this example The convergence isfastest for L4 followed by L5 and is slowest when L1 isused in the SHAM algorithm It can also be observed that theresults for L1 L2 L3 and L5 converge to approximatelythe same level while L4 converges to a level lower than thatof the other linear operators Thus in addition to convergingmuch faster L4 gives results that are slightly more accuratethan those obtained using the other linear operators
Table 3 presents the minimum SHAM order required togive maximum accuracy at the the optimal values of theconvergence controlling parameter ℏ and the correspondingresidual errorsThe results are generated for increasing valuesof the Reynolds number (Re) It can be seen from Table 3that the minimum order of the SHAM required graduallyincreases with an increase in Re for both L1 and L2 Thereis no significant increase in the minimum SHAM orderrequired in the case of L3 L4 and L5 The results from
Maximum residual error 1198641199031 8451119890 minus 12 8987119890 minus 12 1074119890 minus 11 4886119890 minus 14 1256119890 minus 11
50 8636119890 minus 12 8887119890 minus 12 1148119890 minus 11 1357119890 minus 12 7468119890 minus 12
100 7930119890 minus 12 8759119890 minus 12 1303119890 minus 11 1886119890 minus 12 1633119890 minus 11
150 7815119890 minus 12 8092119890 minus 12 2546119890 minus 11 2973119890 minus 12 1808119890 minus 11
200 9128119890 minus 12 8421119890 minus 12 3209119890 minus 11 4435119890 minus 12 1176119890 minus 11
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 5 Residual ℏ-curve for Example 3 using the 20th orderSHAM with Re = 10 and Ha = 1
Table 3 also reveal that the required minimum SHAM orderof L1 and L2 is comparable particularly for larger valuesof Re This confirms the observation made in Figures 3 and4 Thus it can be inferred that the behaviour of the SHAMwith higher order differential mapping (L1) is comparablewith that of SHAMwith linear partitionmapping (L2) in thesolution of the Jeffery-Hamel equation
Figure 5 shows the ℏ-curves corresponding to the exam-ple of laminar viscous flow in a semiporous channel subjectto a transverse magnetic field (Example 3) Again it can beseen from Figure 5 that the residual error is lowest whenthe SHAM is used with L4 and largest when used with L1which as was observed in the case of Jeffery-Hamel flowis comparable with L2 The same observation can be madein Figure 6 where the residual error is plotted against theSHAM order From Figure 6 it can also be seen that L4gives the best convergence followed by L5 and the slowest
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 6 Variation of residual error with SHAM order for Example3 when Re = 10 and Ha = 1
convergence is obtained when usingL1 which is comparabletoL2
Table 4 shows effect of increasing the Reynolds number(Re) on the minimum SHAM orders and the correspondingresidual errors It can be observed from Table 4 that theminimumSHAMorders increasewith an increase inRewhenthe SHAM is used with L1 L2 and L3 It can also beseen that there is no significant increase in the minimumSHAM orders when using L4 and L5 Furthermore theresidual error obtained using L4 is lower than that of theother linear operatorsThis is in linewith similar observationsmade earlier
Table 5 shows the effect of increasing the Hartmannnumber (Ha) on the minimum SHAM orders and thecorresponding residual errors in Example 3 In this case itcan be observed that the minimum SHAM orders increasewith an increase in Ha only in the case of L1 In the case of
Maximum Residual error 1198641199032 1484119890 minus 08 1323119890 minus 08 9815119890 minus 09 3391119890 minus 12 2954119890 minus 09
4 8618119890 minus 09 3334119890 minus 09 3888119890 minus 09 5698119890 minus 12 2855119890 minus 09
6 2258119890 minus 09 6625119890 minus 09 1281119890 minus 08 1461119890 minus 11 6068119890 minus 09
8 1172119890 minus 09 2022119890 minus 09 1012119890 minus 08 1690119890 minus 11 7852119890 minus 09
10 8887119890 minus 09 5535119890 minus 09 1261119890 minus 08 5424119890 minus 11 2253119890 minus 09
15 3325119890 minus 09 1870119890 minus 09 5521119890 minus 09 2964119890 minus 10 5871119890 minus 09
the other linear operators theminimum SHAMorders do notincrease with an increase in HaThis means that convergenceof the SHAM series is independent of the value of Ha whenthe SHAM is used with L2 L3 L4 and L5 It can also benoted from Table 5 that the residual error is smallest whenL4 is used If can be inferred that the SHAM is most accuratewhen used withL4
Figure 7 shows the ℏ-curves corresponding to the exam-ple of two-dimensional viscous flow in a rectangular domainbounded by two moving porous walls (Example 4) It canbe noted from Figure 7 that the residual error is the lowestwhen the SHAM is used with L4 and the largest whenused with L1 after 20 iterations Figure 8 illustrates thevariation of the residual error against the SHAM order It canbe seen from Figure 8 that L4 gives the best convergencefollowed by L5 and the slowest convergence is obtainedwhen using L1 The linear operators L1 L2 L3 andL5 converge to the same level of residual error whichis higher than that of L4 Again it can be inferred that
L4 gives superior accuracy compared to the other linearoperators
Table 6 shows the effect of increasing the Reynolds num-ber (Re) on the minimum SHAM orders and correspondingresidual errors in Example 4 It can be observed that theminimumSHAMorders increasewith an increase inRewhenthe SHAM is used with L1 L2 and L3 In the case of L4andL5 it seems possible that an increase in Re has no stronginfluence in the convergence speed of the SHAM It can alsobe noted from Table 6 that the residual error is the smallestwhen L4 is used This is in accord with the trends observedin all the other examples considered in this work Table 7shows the effect of the nondimensional wall dilation rate 120572on the minimum SHAM order and corresponding residualerrors It can be observed from Table 7 that an increasein 120572 results in a corresponding significant increase in theminimum SHAMorder forL1Theminimum SHAMordersare not significantly influenced by 120572 when the SHAM is usedwithL2L3L4 andL5 Furthermore the best accuracy isobtained when usingL4
Maximum residual error 1198641199031 2103119890 minus 09 4065119890 minus 09 1354119890 minus 08 4946119890 minus 13 6441119890 minus 09
2 1198119890 minus 09 1109119890 minus 08 7087119890 minus 10 7009119890 minus 12 9531119890 minus 09
4 1954119890 minus 09 1017119890 minus 08 1604119890 minus 08 3667119890 minus 11 3063119890 minus 09
6 4419119890 minus 09 5935119890 minus 09 1772119890 minus 08 7262119890 minus 11 1388119890 minus 08
8 1930119890 minus 09 2864119890 minus 09 1449119890 minus 08 9716119890 minus 11 1221119890 minus 09
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 7 Residual ℏ-curve for Example 4 using the 20th orderSHAM with Re = 10 and 120572 = 1
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 8 Variation of residual error with SHAM order for Example4 when Re = 10 and 120572 = 1
14 Mathematical Problems in Engineering
6 Conclusion
This study gives a systematic way of choosing initial approx-imations and linear operators that can be employed inthe spectral homotopy analysis method (SHAM) algorithmThe main aim of the study was to determine the linearoperator definition that gives approximate solutions withoptimal convergence and accuracy A comparison betweenthe accuracy and convergence of SHAM algorithms usingcommonly used linear operators and newly proposed linearoperators was conducted Simulations were conducted onselected nonlinear boundary value problems defined onbounded domains One of the more significant findings toemerge from this study is that the linear operator definedby the highest order differential matching which is themost commonly used linear operator for solving boundaryvalue problems defined on bounded domains results inthe slowest converging and least accurate SHAM algorithmThe SHAM algorithm that uses a linear operator definedby the method of linear partition matching was found toperform slightly better than the SHAMwith the highest orderdifferential matching in some cases However in other casesthe methods were found to be comparable It was also shownthat the SHAMwith linear operator defined by themethod ofcomplete differential matching performed much better thanthe SHAMwith linear partitionmapping or the highest orderdifferential matching The study proposed a modification ofthe complete differential method approach which unlike theoriginal approach does not ignore the nonlinear factors of thederivative terms in the definition of the linear operator Thisnew definition of linear operator was found to perform betterthan the original complete differential method approachThe most optimal linear operator was found to be one thatuses the linear partition mapping after a transformation thatexpresses the unknown function as a sum of a new unknownplus some initial guess that satisfies the underlying boundaryconditions
The current findings add substantially to our under-standing of how homotopy analysis based methods canbe modified to yield approximate solutions with optimumaccuracy and convergence Further work needs to be done toinvestigate the application of the linear operators proposed inthis study in solving nonlinear partial differential equationsIt would also be very interesting to extend the current studyto systems of nonlinear differential equations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is based on the research supported in part by theNational Research Foundation of South Africa (Grant no85596)
References
[1] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[2] S J Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer Berlin Germany 2012
[3] S J Liao Advances in Homotopy Analysis Method WorldScientific Singapore 2014
[5] S S Motsa P Sibanda and S Shateyi ldquoA new spectral-homotopy analysismethod for solving a nonlinear second orderBVPrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 9 pp 2293ndash2302 2010
[6] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers and Fluids vol 39 no 7 pp 1219ndash1225 2010
[7] R A Van Gorder and K Vajravelu ldquoOn the selection of auxil-iary functions operators and convergence control parametersin the application of the homotopy analysismethod to nonlineardifferential equations a general approachrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 12 pp4078ndash4089 2009
[8] Y Shan and T Chaolu ldquoA method to select the initial guesssolution auxiliary linear operator and set of basic functions ofhomotopy analysis methodrdquo in Proceedings of the IEEE Inter-national Conference on Intelligent Computing and IntegratedSystems (ICISS rsquo10) pp 410ndash412 Guilin China October 2010
[9] R A van Gorder ldquoStability of auxiliary linear operatorand convergence-control parameter in the homotopy analysismethodrdquo in Advances in Homotopy Analysis Method S J LiaoEd pp 123ndash180 World Scientific Singapore 2014
[10] S M Rassoulinejad-Mousavi and S Abbasbandy ldquoAnalysisof forced convection in a circular tube filled with a darcy-brinkman-forchheimer porous medium using spectral homo-topy analysis methodrdquo Journal of Fluids Engineering Transac-tions of the ASME vol 133 no 10 Article ID 101207 2011
[11] A A Joneidi G Domairry andM Babaelahi ldquoThree analyticalmethods applied to Jeffery-Hamel flowrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 11 pp3423ndash3434 2010
[12] S M Moghimi G Domairry S Soleimani E Ghasemi and HBararnia ldquoApplication of homotopy analysis method to solveMHD Jeffery-Hamel flows in non-parallel wallsrdquo Advances inEngineering Software vol 42 no 3 pp 108ndash113 2011
[13] M Esmaeilpour and D D Ganji ldquoSolution of the Jeffery-Hamel flowproblemby optimal homotopy asymptoticmethodrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3405ndash3411 2010
[14] A Basiri Parsa M M Rashidi O Anwar Beg and S M SadrildquoSemi-computational simulation of magneto-hemodynamicflow in a semi-porous channel using optimal homotopy anddifferential transform methodsrdquo Computers in Biology andMedicine vol 43 no 9 pp 1142ndash1153 2013
[15] Z Ziabakhsh andGDomairry ldquoSolution of the laminar viscousflow in a semi-porous channel in the presence of a uniformmagnetic field by using the homotopy analysis methodrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 4 pp 1284ndash1294 2009
Mathematical Problems in Engineering 15
[16] S SMotsa S Shateyi GMarewo andP Sibanda ldquoAn improvedspectral homotopy analysis method for MHD flow in a semi-porous channelrdquo Numerical Algorithms vol 60 no 3 pp 463ndash481 2012
[17] S Dinarvand and M M Rashidi ldquoA reliable treatment of ahomotopy analysis method for two-dimensional viscous flowin a rectangular domain bounded by twomoving porous wallsrdquoNonlinear Analysis Real World Applications vol 11 no 3 pp1502ndash1512 2010
[18] H Xu Z Lin S Liao JWu and JMajdalani ldquoHomotopy basedsolutions of the Navier-Stokes equations for a porous channelwith orthogonally moving wallsrdquo Physics of Fluids vol 22 no5 Article ID 053601 pp 1ndash18 2010
[19] M M Rashidi E Erfani and B Rostami ldquoOptimal HomotopyAsymptotic Method for solving viscous ow through expandingor contracting gaps with permeable wallsrdquo Transaction on IoTand Cloud Computing vol 2 no 1 pp 85ndash111 2014
[20] Z GMakukula P Sibanda and S S Motsa ldquoA novel numericaltechnique for two-dimensional laminar flow between twomoving porous wallsrdquo Mathematical Problems in Engineeringvol 2010 Article ID 528956 15 pages 2010
[21] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Series in Com-putational Physics Springer New York NY USA 1988
[22] L N Trefethen Spectral Methods in MATLAB SIAM 2000[23] J Majdalani C Zhou and C A Dawson ldquoTwo-dimensional
viscous flow between slowly expanding or contracting wallswith weak permeabilityrdquo Journal of Biomechanics vol 35 no10 pp 1399ndash1403 2002
1198774119898minus1 = L4 (119911119898minus1) minus (1 minus 120594119898) 1206014 (119909)
+ Re119898minus1
sum
119899=0
(119911119899119911101584010158401015840
119898minus1minus119899minus 1199111015840
11989911991110158401015840
119898minus1minus119899)
(72)
5 Results and Discussion
The SHAM algorithm using the proposed auxiliary linearoperators was applied to the several test problems presentedin the last section In this section we present the numeri-cal results that highlight the difference in the accuracy ofthe approximate solutions and convergence of the differentSHAM algorithms All the simulations were conducted using119872 = 20 collocation points The accuracy and convergenceof the SHAM approximate series solution depend on carefulselection of the auxiliary parameter ℏ The residual errorwas used to determine the optimal values of the auxiliaryparameter Using finite terms of the SHAM series we define119870th order approximation at the collocation points 119909119895 for 119895 =0 1 2 119872 as
Y =119870
sum
119896=0
119910119896 (119909119895) =
119870
sum
119896=0
119884119896 (73)
where 119884119896 = [119910119896(1199090)119910119896(1199091) sdot sdot sdot 119910119896(119909119872)]119905 Given that Y is the
SHAM approximate solution at the collocation points theresidual error at the collocation points is defined as
Re 119904 (119884) =N [ (Y ℏ)] (74)
where N is the nonlinear operator defining the governingnonlinear equation (see (4)) We define the infinity norm ofthe residual error as
119864119903 (ℏ) = N[ (Y ℏ)]infin (75)
The optimal ℏ that gives the best accuracy was identified tobe the value of ℏ located at the minimum of the graph of thevariation of the infinity norm against ℏ
Figure 1 gives the typical residual error curves that areobtained using the 20th order SHAM approximate solutionof the Darcy-Brinkman-Forchheimer equation (Example 1)The minimum of the curve gives the best possible residualerror for each auxiliary linear operator used in the SHAMalgorithm A comparison of the minima of the residual errorcurves gives an indication of which auxiliary linear operatoris likely to give better accuracy when used with the optimalauxiliary linear operator in the SHAM algorithm It can beseen from Figure 1 that the best accuracy at the 20th orderSHAM approximation is obtained when using the linearpartition mapping after transformation (L4) The highestorder differential mapping (L1) gives the least accuracyFigure 2 shows the variation of the norm of the residual errorat the optimal value of ℏ as a function of the SHAMorderThisgraph gives an indication of how the SHAM series convergeswith an increase in the number of iterations (terms used inthe series) Rapid convergence is determined by attaining
Maximum Residual error 1198641199031 1599119890 minus 14 4108119890 minus 15 5329119890 minus 15 4441119890 minus 16 8993119890 minus 15
5 3375119890 minus 14 6439119890 minus 15 3442119890 minus 15 1443119890 minus 15 4663119890 minus 15
10 3952119890 minus 14 9770119890 minus 15 3442119890 minus 15 1776119890 minus 15 5218119890 minus 15
15 5529119890 minus 14 3397119890 minus 14 2887119890 minus 15 3553119890 minus 15 3331119890 minus 15
20 1612119890 minus 13 2620119890 minus 14 4885119890 minus 15 3664119890 minus 15 3331119890 minus 15
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 1 Residual ℏ-curve for Example 1 using the 20th orderSHAM with119872 = 1 119904 = 1 and 119865 = 2
the lowest possible residual error after few iterations In thisexample it can be seen from Figure 2 that the highest orderdifferential mapping linear operator L1 is the slowest interms of convergence followed by the complete differentialmappingL3The operator that yields the fastest convergencein this example isL4 which gives full convergence after only10 iterations It can also be seen from Figure 2 that L2L3andL5 all converge to the same level which is slightly lowerthan that of L1 but slightly higher than L4 This resultsuggests that the SHAM algorithm that applies L4 gives thebest accuracy and when usingL1 the accuracy is the lowestin comparison with the other 4 choices of linear operators inthis example
Table 1 gives the minimum number of terms of theSHAM series that are required to attain full convergenceof the SHAM algorithm using the optimal values of theconvergence controlling parameter ℏ The correspondingmaximum residual errors are also tabulated It can be seen
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 2 Variation of residual error with SHAM order for Example1 when119872 = 1 119904 = 1 and 119865 = 2
from Table 1 that the least error is obtained when using L1and the other linear operators give more or less comparableresidual error values The SHAM with L4 requires thefewest number terms of the SHAM series to attain maximumaccuracy The SHAM with L5 comes second followed byL2 L3 and L1 in terms of the minimum SHAM orderrequired to give best accuracy This observation is in accordwith the trends displayed in Figure 2 The table also indicatesthat the number of terms of the SHAM series required for theSHAMapproximation to attainmaximum accuracy increaseswith an increase in the value of the Forchheimer numberparameter 119865 It is worth noting from the governing equation(22) that the parameter119865multiplies the nonlinear componentof the equation Table 2 shows the effect of increasing theporousmedia shape parameter 119904 on the SHAMorder requiredto attain the best accuracy It can be seen from the table thatthe effect of 119904 on convergence and accuracy is similar to that of119865 In particular it can be noted that as 119904 increasesmore SHAM
Maximum residual error 1198641199031 1599119890 minus 14 4108119890 minus 15 5329119890 minus 15 4441119890 minus 16 8993119890 minus 15
5 3375119890 minus 14 6439119890 minus 15 3442119890 minus 15 1443119890 minus 15 4663119890 minus 15
10 3952119890 minus 14 9770119890 minus 15 3442119890 minus 15 1776119890 minus 15 5218119890 minus 15
15 5529119890 minus 14 3397119890 minus 14 2887119890 minus 15 3553119890 minus 15 3331119890 minus 15
20 1612119890 minus 13 2620119890 minus 14 4885119890 minus 15 3664119890 minus 15 3331119890 minus 15
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 3 Residual ℏ-curve for Example 2 using the 20th orderSHAM with Re = 100 and 120572 = minus5∘
series terms are required to attain the maximum accuracy inall linear operators
Figure 3 shows the ℏ-curves for the Jeffery-Hamel prob-lem (Example 2) corresponding to the different SHAMschemes generated using all the linear operators using 20terms of the SHAM approximation series It can be seenfrom Figure 3 that after 20 iterationsL4 would give the bestaccuracy and L1 gives the least accuracy This is in accordwith the observation made earlier in Example 1 It can alsobe observed that the behaviour of the SHAM with L1 isapproximately the same as that of the SHAM with L2 Thiscan be noted from the observation that the ℏ-curve forL1 lieson top of that forL2 In Figure 4 the variation of the residualerror is shown as a function of the SHAM order It can beobserved that in all test linear operators the residual errordecreases with the number of terms used in the SHAM seriesThis demonstrates that the SHAM approximation converges
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 4 Variation of residual error with SHAM order for Example2 when Re = 100 and 120572 = minus5∘
in all cases considered in this example The convergence isfastest for L4 followed by L5 and is slowest when L1 isused in the SHAM algorithm It can also be observed that theresults for L1 L2 L3 and L5 converge to approximatelythe same level while L4 converges to a level lower than thatof the other linear operators Thus in addition to convergingmuch faster L4 gives results that are slightly more accuratethan those obtained using the other linear operators
Table 3 presents the minimum SHAM order required togive maximum accuracy at the the optimal values of theconvergence controlling parameter ℏ and the correspondingresidual errorsThe results are generated for increasing valuesof the Reynolds number (Re) It can be seen from Table 3that the minimum order of the SHAM required graduallyincreases with an increase in Re for both L1 and L2 Thereis no significant increase in the minimum SHAM orderrequired in the case of L3 L4 and L5 The results from
Maximum residual error 1198641199031 8451119890 minus 12 8987119890 minus 12 1074119890 minus 11 4886119890 minus 14 1256119890 minus 11
50 8636119890 minus 12 8887119890 minus 12 1148119890 minus 11 1357119890 minus 12 7468119890 minus 12
100 7930119890 minus 12 8759119890 minus 12 1303119890 minus 11 1886119890 minus 12 1633119890 minus 11
150 7815119890 minus 12 8092119890 minus 12 2546119890 minus 11 2973119890 minus 12 1808119890 minus 11
200 9128119890 minus 12 8421119890 minus 12 3209119890 minus 11 4435119890 minus 12 1176119890 minus 11
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 5 Residual ℏ-curve for Example 3 using the 20th orderSHAM with Re = 10 and Ha = 1
Table 3 also reveal that the required minimum SHAM orderof L1 and L2 is comparable particularly for larger valuesof Re This confirms the observation made in Figures 3 and4 Thus it can be inferred that the behaviour of the SHAMwith higher order differential mapping (L1) is comparablewith that of SHAMwith linear partitionmapping (L2) in thesolution of the Jeffery-Hamel equation
Figure 5 shows the ℏ-curves corresponding to the exam-ple of laminar viscous flow in a semiporous channel subjectto a transverse magnetic field (Example 3) Again it can beseen from Figure 5 that the residual error is lowest whenthe SHAM is used with L4 and largest when used with L1which as was observed in the case of Jeffery-Hamel flowis comparable with L2 The same observation can be madein Figure 6 where the residual error is plotted against theSHAM order From Figure 6 it can also be seen that L4gives the best convergence followed by L5 and the slowest
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 6 Variation of residual error with SHAM order for Example3 when Re = 10 and Ha = 1
convergence is obtained when usingL1 which is comparabletoL2
Table 4 shows effect of increasing the Reynolds number(Re) on the minimum SHAM orders and the correspondingresidual errors It can be observed from Table 4 that theminimumSHAMorders increasewith an increase inRewhenthe SHAM is used with L1 L2 and L3 It can also beseen that there is no significant increase in the minimumSHAM orders when using L4 and L5 Furthermore theresidual error obtained using L4 is lower than that of theother linear operatorsThis is in linewith similar observationsmade earlier
Table 5 shows the effect of increasing the Hartmannnumber (Ha) on the minimum SHAM orders and thecorresponding residual errors in Example 3 In this case itcan be observed that the minimum SHAM orders increasewith an increase in Ha only in the case of L1 In the case of
Maximum Residual error 1198641199032 1484119890 minus 08 1323119890 minus 08 9815119890 minus 09 3391119890 minus 12 2954119890 minus 09
4 8618119890 minus 09 3334119890 minus 09 3888119890 minus 09 5698119890 minus 12 2855119890 minus 09
6 2258119890 minus 09 6625119890 minus 09 1281119890 minus 08 1461119890 minus 11 6068119890 minus 09
8 1172119890 minus 09 2022119890 minus 09 1012119890 minus 08 1690119890 minus 11 7852119890 minus 09
10 8887119890 minus 09 5535119890 minus 09 1261119890 minus 08 5424119890 minus 11 2253119890 minus 09
15 3325119890 minus 09 1870119890 minus 09 5521119890 minus 09 2964119890 minus 10 5871119890 minus 09
the other linear operators theminimum SHAMorders do notincrease with an increase in HaThis means that convergenceof the SHAM series is independent of the value of Ha whenthe SHAM is used with L2 L3 L4 and L5 It can also benoted from Table 5 that the residual error is smallest whenL4 is used If can be inferred that the SHAM is most accuratewhen used withL4
Figure 7 shows the ℏ-curves corresponding to the exam-ple of two-dimensional viscous flow in a rectangular domainbounded by two moving porous walls (Example 4) It canbe noted from Figure 7 that the residual error is the lowestwhen the SHAM is used with L4 and the largest whenused with L1 after 20 iterations Figure 8 illustrates thevariation of the residual error against the SHAM order It canbe seen from Figure 8 that L4 gives the best convergencefollowed by L5 and the slowest convergence is obtainedwhen using L1 The linear operators L1 L2 L3 andL5 converge to the same level of residual error whichis higher than that of L4 Again it can be inferred that
L4 gives superior accuracy compared to the other linearoperators
Table 6 shows the effect of increasing the Reynolds num-ber (Re) on the minimum SHAM orders and correspondingresidual errors in Example 4 It can be observed that theminimumSHAMorders increasewith an increase inRewhenthe SHAM is used with L1 L2 and L3 In the case of L4andL5 it seems possible that an increase in Re has no stronginfluence in the convergence speed of the SHAM It can alsobe noted from Table 6 that the residual error is the smallestwhen L4 is used This is in accord with the trends observedin all the other examples considered in this work Table 7shows the effect of the nondimensional wall dilation rate 120572on the minimum SHAM order and corresponding residualerrors It can be observed from Table 7 that an increasein 120572 results in a corresponding significant increase in theminimum SHAMorder forL1Theminimum SHAMordersare not significantly influenced by 120572 when the SHAM is usedwithL2L3L4 andL5 Furthermore the best accuracy isobtained when usingL4
Maximum residual error 1198641199031 2103119890 minus 09 4065119890 minus 09 1354119890 minus 08 4946119890 minus 13 6441119890 minus 09
2 1198119890 minus 09 1109119890 minus 08 7087119890 minus 10 7009119890 minus 12 9531119890 minus 09
4 1954119890 minus 09 1017119890 minus 08 1604119890 minus 08 3667119890 minus 11 3063119890 minus 09
6 4419119890 minus 09 5935119890 minus 09 1772119890 minus 08 7262119890 minus 11 1388119890 minus 08
8 1930119890 minus 09 2864119890 minus 09 1449119890 minus 08 9716119890 minus 11 1221119890 minus 09
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 7 Residual ℏ-curve for Example 4 using the 20th orderSHAM with Re = 10 and 120572 = 1
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 8 Variation of residual error with SHAM order for Example4 when Re = 10 and 120572 = 1
14 Mathematical Problems in Engineering
6 Conclusion
This study gives a systematic way of choosing initial approx-imations and linear operators that can be employed inthe spectral homotopy analysis method (SHAM) algorithmThe main aim of the study was to determine the linearoperator definition that gives approximate solutions withoptimal convergence and accuracy A comparison betweenthe accuracy and convergence of SHAM algorithms usingcommonly used linear operators and newly proposed linearoperators was conducted Simulations were conducted onselected nonlinear boundary value problems defined onbounded domains One of the more significant findings toemerge from this study is that the linear operator definedby the highest order differential matching which is themost commonly used linear operator for solving boundaryvalue problems defined on bounded domains results inthe slowest converging and least accurate SHAM algorithmThe SHAM algorithm that uses a linear operator definedby the method of linear partition matching was found toperform slightly better than the SHAMwith the highest orderdifferential matching in some cases However in other casesthe methods were found to be comparable It was also shownthat the SHAMwith linear operator defined by themethod ofcomplete differential matching performed much better thanthe SHAMwith linear partitionmapping or the highest orderdifferential matching The study proposed a modification ofthe complete differential method approach which unlike theoriginal approach does not ignore the nonlinear factors of thederivative terms in the definition of the linear operator Thisnew definition of linear operator was found to perform betterthan the original complete differential method approachThe most optimal linear operator was found to be one thatuses the linear partition mapping after a transformation thatexpresses the unknown function as a sum of a new unknownplus some initial guess that satisfies the underlying boundaryconditions
The current findings add substantially to our under-standing of how homotopy analysis based methods canbe modified to yield approximate solutions with optimumaccuracy and convergence Further work needs to be done toinvestigate the application of the linear operators proposed inthis study in solving nonlinear partial differential equationsIt would also be very interesting to extend the current studyto systems of nonlinear differential equations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is based on the research supported in part by theNational Research Foundation of South Africa (Grant no85596)
References
[1] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[2] S J Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer Berlin Germany 2012
[3] S J Liao Advances in Homotopy Analysis Method WorldScientific Singapore 2014
[5] S S Motsa P Sibanda and S Shateyi ldquoA new spectral-homotopy analysismethod for solving a nonlinear second orderBVPrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 9 pp 2293ndash2302 2010
[6] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers and Fluids vol 39 no 7 pp 1219ndash1225 2010
[7] R A Van Gorder and K Vajravelu ldquoOn the selection of auxil-iary functions operators and convergence control parametersin the application of the homotopy analysismethod to nonlineardifferential equations a general approachrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 12 pp4078ndash4089 2009
[8] Y Shan and T Chaolu ldquoA method to select the initial guesssolution auxiliary linear operator and set of basic functions ofhomotopy analysis methodrdquo in Proceedings of the IEEE Inter-national Conference on Intelligent Computing and IntegratedSystems (ICISS rsquo10) pp 410ndash412 Guilin China October 2010
[9] R A van Gorder ldquoStability of auxiliary linear operatorand convergence-control parameter in the homotopy analysismethodrdquo in Advances in Homotopy Analysis Method S J LiaoEd pp 123ndash180 World Scientific Singapore 2014
[10] S M Rassoulinejad-Mousavi and S Abbasbandy ldquoAnalysisof forced convection in a circular tube filled with a darcy-brinkman-forchheimer porous medium using spectral homo-topy analysis methodrdquo Journal of Fluids Engineering Transac-tions of the ASME vol 133 no 10 Article ID 101207 2011
[11] A A Joneidi G Domairry andM Babaelahi ldquoThree analyticalmethods applied to Jeffery-Hamel flowrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 11 pp3423ndash3434 2010
[12] S M Moghimi G Domairry S Soleimani E Ghasemi and HBararnia ldquoApplication of homotopy analysis method to solveMHD Jeffery-Hamel flows in non-parallel wallsrdquo Advances inEngineering Software vol 42 no 3 pp 108ndash113 2011
[13] M Esmaeilpour and D D Ganji ldquoSolution of the Jeffery-Hamel flowproblemby optimal homotopy asymptoticmethodrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3405ndash3411 2010
[14] A Basiri Parsa M M Rashidi O Anwar Beg and S M SadrildquoSemi-computational simulation of magneto-hemodynamicflow in a semi-porous channel using optimal homotopy anddifferential transform methodsrdquo Computers in Biology andMedicine vol 43 no 9 pp 1142ndash1153 2013
[15] Z Ziabakhsh andGDomairry ldquoSolution of the laminar viscousflow in a semi-porous channel in the presence of a uniformmagnetic field by using the homotopy analysis methodrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 4 pp 1284ndash1294 2009
Mathematical Problems in Engineering 15
[16] S SMotsa S Shateyi GMarewo andP Sibanda ldquoAn improvedspectral homotopy analysis method for MHD flow in a semi-porous channelrdquo Numerical Algorithms vol 60 no 3 pp 463ndash481 2012
[17] S Dinarvand and M M Rashidi ldquoA reliable treatment of ahomotopy analysis method for two-dimensional viscous flowin a rectangular domain bounded by twomoving porous wallsrdquoNonlinear Analysis Real World Applications vol 11 no 3 pp1502ndash1512 2010
[18] H Xu Z Lin S Liao JWu and JMajdalani ldquoHomotopy basedsolutions of the Navier-Stokes equations for a porous channelwith orthogonally moving wallsrdquo Physics of Fluids vol 22 no5 Article ID 053601 pp 1ndash18 2010
[19] M M Rashidi E Erfani and B Rostami ldquoOptimal HomotopyAsymptotic Method for solving viscous ow through expandingor contracting gaps with permeable wallsrdquo Transaction on IoTand Cloud Computing vol 2 no 1 pp 85ndash111 2014
[20] Z GMakukula P Sibanda and S S Motsa ldquoA novel numericaltechnique for two-dimensional laminar flow between twomoving porous wallsrdquo Mathematical Problems in Engineeringvol 2010 Article ID 528956 15 pages 2010
[21] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Series in Com-putational Physics Springer New York NY USA 1988
[22] L N Trefethen Spectral Methods in MATLAB SIAM 2000[23] J Majdalani C Zhou and C A Dawson ldquoTwo-dimensional
viscous flow between slowly expanding or contracting wallswith weak permeabilityrdquo Journal of Biomechanics vol 35 no10 pp 1399ndash1403 2002
Maximum Residual error 1198641199031 1599119890 minus 14 4108119890 minus 15 5329119890 minus 15 4441119890 minus 16 8993119890 minus 15
5 3375119890 minus 14 6439119890 minus 15 3442119890 minus 15 1443119890 minus 15 4663119890 minus 15
10 3952119890 minus 14 9770119890 minus 15 3442119890 minus 15 1776119890 minus 15 5218119890 minus 15
15 5529119890 minus 14 3397119890 minus 14 2887119890 minus 15 3553119890 minus 15 3331119890 minus 15
20 1612119890 minus 13 2620119890 minus 14 4885119890 minus 15 3664119890 minus 15 3331119890 minus 15
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 1 Residual ℏ-curve for Example 1 using the 20th orderSHAM with119872 = 1 119904 = 1 and 119865 = 2
the lowest possible residual error after few iterations In thisexample it can be seen from Figure 2 that the highest orderdifferential mapping linear operator L1 is the slowest interms of convergence followed by the complete differentialmappingL3The operator that yields the fastest convergencein this example isL4 which gives full convergence after only10 iterations It can also be seen from Figure 2 that L2L3andL5 all converge to the same level which is slightly lowerthan that of L1 but slightly higher than L4 This resultsuggests that the SHAM algorithm that applies L4 gives thebest accuracy and when usingL1 the accuracy is the lowestin comparison with the other 4 choices of linear operators inthis example
Table 1 gives the minimum number of terms of theSHAM series that are required to attain full convergenceof the SHAM algorithm using the optimal values of theconvergence controlling parameter ℏ The correspondingmaximum residual errors are also tabulated It can be seen
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 2 Variation of residual error with SHAM order for Example1 when119872 = 1 119904 = 1 and 119865 = 2
from Table 1 that the least error is obtained when using L1and the other linear operators give more or less comparableresidual error values The SHAM with L4 requires thefewest number terms of the SHAM series to attain maximumaccuracy The SHAM with L5 comes second followed byL2 L3 and L1 in terms of the minimum SHAM orderrequired to give best accuracy This observation is in accordwith the trends displayed in Figure 2 The table also indicatesthat the number of terms of the SHAM series required for theSHAMapproximation to attainmaximum accuracy increaseswith an increase in the value of the Forchheimer numberparameter 119865 It is worth noting from the governing equation(22) that the parameter119865multiplies the nonlinear componentof the equation Table 2 shows the effect of increasing theporousmedia shape parameter 119904 on the SHAMorder requiredto attain the best accuracy It can be seen from the table thatthe effect of 119904 on convergence and accuracy is similar to that of119865 In particular it can be noted that as 119904 increasesmore SHAM
Maximum residual error 1198641199031 1599119890 minus 14 4108119890 minus 15 5329119890 minus 15 4441119890 minus 16 8993119890 minus 15
5 3375119890 minus 14 6439119890 minus 15 3442119890 minus 15 1443119890 minus 15 4663119890 minus 15
10 3952119890 minus 14 9770119890 minus 15 3442119890 minus 15 1776119890 minus 15 5218119890 minus 15
15 5529119890 minus 14 3397119890 minus 14 2887119890 minus 15 3553119890 minus 15 3331119890 minus 15
20 1612119890 minus 13 2620119890 minus 14 4885119890 minus 15 3664119890 minus 15 3331119890 minus 15
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 3 Residual ℏ-curve for Example 2 using the 20th orderSHAM with Re = 100 and 120572 = minus5∘
series terms are required to attain the maximum accuracy inall linear operators
Figure 3 shows the ℏ-curves for the Jeffery-Hamel prob-lem (Example 2) corresponding to the different SHAMschemes generated using all the linear operators using 20terms of the SHAM approximation series It can be seenfrom Figure 3 that after 20 iterationsL4 would give the bestaccuracy and L1 gives the least accuracy This is in accordwith the observation made earlier in Example 1 It can alsobe observed that the behaviour of the SHAM with L1 isapproximately the same as that of the SHAM with L2 Thiscan be noted from the observation that the ℏ-curve forL1 lieson top of that forL2 In Figure 4 the variation of the residualerror is shown as a function of the SHAM order It can beobserved that in all test linear operators the residual errordecreases with the number of terms used in the SHAM seriesThis demonstrates that the SHAM approximation converges
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 4 Variation of residual error with SHAM order for Example2 when Re = 100 and 120572 = minus5∘
in all cases considered in this example The convergence isfastest for L4 followed by L5 and is slowest when L1 isused in the SHAM algorithm It can also be observed that theresults for L1 L2 L3 and L5 converge to approximatelythe same level while L4 converges to a level lower than thatof the other linear operators Thus in addition to convergingmuch faster L4 gives results that are slightly more accuratethan those obtained using the other linear operators
Table 3 presents the minimum SHAM order required togive maximum accuracy at the the optimal values of theconvergence controlling parameter ℏ and the correspondingresidual errorsThe results are generated for increasing valuesof the Reynolds number (Re) It can be seen from Table 3that the minimum order of the SHAM required graduallyincreases with an increase in Re for both L1 and L2 Thereis no significant increase in the minimum SHAM orderrequired in the case of L3 L4 and L5 The results from
Maximum residual error 1198641199031 8451119890 minus 12 8987119890 minus 12 1074119890 minus 11 4886119890 minus 14 1256119890 minus 11
50 8636119890 minus 12 8887119890 minus 12 1148119890 minus 11 1357119890 minus 12 7468119890 minus 12
100 7930119890 minus 12 8759119890 minus 12 1303119890 minus 11 1886119890 minus 12 1633119890 minus 11
150 7815119890 minus 12 8092119890 minus 12 2546119890 minus 11 2973119890 minus 12 1808119890 minus 11
200 9128119890 minus 12 8421119890 minus 12 3209119890 minus 11 4435119890 minus 12 1176119890 minus 11
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 5 Residual ℏ-curve for Example 3 using the 20th orderSHAM with Re = 10 and Ha = 1
Table 3 also reveal that the required minimum SHAM orderof L1 and L2 is comparable particularly for larger valuesof Re This confirms the observation made in Figures 3 and4 Thus it can be inferred that the behaviour of the SHAMwith higher order differential mapping (L1) is comparablewith that of SHAMwith linear partitionmapping (L2) in thesolution of the Jeffery-Hamel equation
Figure 5 shows the ℏ-curves corresponding to the exam-ple of laminar viscous flow in a semiporous channel subjectto a transverse magnetic field (Example 3) Again it can beseen from Figure 5 that the residual error is lowest whenthe SHAM is used with L4 and largest when used with L1which as was observed in the case of Jeffery-Hamel flowis comparable with L2 The same observation can be madein Figure 6 where the residual error is plotted against theSHAM order From Figure 6 it can also be seen that L4gives the best convergence followed by L5 and the slowest
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 6 Variation of residual error with SHAM order for Example3 when Re = 10 and Ha = 1
convergence is obtained when usingL1 which is comparabletoL2
Table 4 shows effect of increasing the Reynolds number(Re) on the minimum SHAM orders and the correspondingresidual errors It can be observed from Table 4 that theminimumSHAMorders increasewith an increase inRewhenthe SHAM is used with L1 L2 and L3 It can also beseen that there is no significant increase in the minimumSHAM orders when using L4 and L5 Furthermore theresidual error obtained using L4 is lower than that of theother linear operatorsThis is in linewith similar observationsmade earlier
Table 5 shows the effect of increasing the Hartmannnumber (Ha) on the minimum SHAM orders and thecorresponding residual errors in Example 3 In this case itcan be observed that the minimum SHAM orders increasewith an increase in Ha only in the case of L1 In the case of
Maximum Residual error 1198641199032 1484119890 minus 08 1323119890 minus 08 9815119890 minus 09 3391119890 minus 12 2954119890 minus 09
4 8618119890 minus 09 3334119890 minus 09 3888119890 minus 09 5698119890 minus 12 2855119890 minus 09
6 2258119890 minus 09 6625119890 minus 09 1281119890 minus 08 1461119890 minus 11 6068119890 minus 09
8 1172119890 minus 09 2022119890 minus 09 1012119890 minus 08 1690119890 minus 11 7852119890 minus 09
10 8887119890 minus 09 5535119890 minus 09 1261119890 minus 08 5424119890 minus 11 2253119890 minus 09
15 3325119890 minus 09 1870119890 minus 09 5521119890 minus 09 2964119890 minus 10 5871119890 minus 09
the other linear operators theminimum SHAMorders do notincrease with an increase in HaThis means that convergenceof the SHAM series is independent of the value of Ha whenthe SHAM is used with L2 L3 L4 and L5 It can also benoted from Table 5 that the residual error is smallest whenL4 is used If can be inferred that the SHAM is most accuratewhen used withL4
Figure 7 shows the ℏ-curves corresponding to the exam-ple of two-dimensional viscous flow in a rectangular domainbounded by two moving porous walls (Example 4) It canbe noted from Figure 7 that the residual error is the lowestwhen the SHAM is used with L4 and the largest whenused with L1 after 20 iterations Figure 8 illustrates thevariation of the residual error against the SHAM order It canbe seen from Figure 8 that L4 gives the best convergencefollowed by L5 and the slowest convergence is obtainedwhen using L1 The linear operators L1 L2 L3 andL5 converge to the same level of residual error whichis higher than that of L4 Again it can be inferred that
L4 gives superior accuracy compared to the other linearoperators
Table 6 shows the effect of increasing the Reynolds num-ber (Re) on the minimum SHAM orders and correspondingresidual errors in Example 4 It can be observed that theminimumSHAMorders increasewith an increase inRewhenthe SHAM is used with L1 L2 and L3 In the case of L4andL5 it seems possible that an increase in Re has no stronginfluence in the convergence speed of the SHAM It can alsobe noted from Table 6 that the residual error is the smallestwhen L4 is used This is in accord with the trends observedin all the other examples considered in this work Table 7shows the effect of the nondimensional wall dilation rate 120572on the minimum SHAM order and corresponding residualerrors It can be observed from Table 7 that an increasein 120572 results in a corresponding significant increase in theminimum SHAMorder forL1Theminimum SHAMordersare not significantly influenced by 120572 when the SHAM is usedwithL2L3L4 andL5 Furthermore the best accuracy isobtained when usingL4
Maximum residual error 1198641199031 2103119890 minus 09 4065119890 minus 09 1354119890 minus 08 4946119890 minus 13 6441119890 minus 09
2 1198119890 minus 09 1109119890 minus 08 7087119890 minus 10 7009119890 minus 12 9531119890 minus 09
4 1954119890 minus 09 1017119890 minus 08 1604119890 minus 08 3667119890 minus 11 3063119890 minus 09
6 4419119890 minus 09 5935119890 minus 09 1772119890 minus 08 7262119890 minus 11 1388119890 minus 08
8 1930119890 minus 09 2864119890 minus 09 1449119890 minus 08 9716119890 minus 11 1221119890 minus 09
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 7 Residual ℏ-curve for Example 4 using the 20th orderSHAM with Re = 10 and 120572 = 1
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 8 Variation of residual error with SHAM order for Example4 when Re = 10 and 120572 = 1
14 Mathematical Problems in Engineering
6 Conclusion
This study gives a systematic way of choosing initial approx-imations and linear operators that can be employed inthe spectral homotopy analysis method (SHAM) algorithmThe main aim of the study was to determine the linearoperator definition that gives approximate solutions withoptimal convergence and accuracy A comparison betweenthe accuracy and convergence of SHAM algorithms usingcommonly used linear operators and newly proposed linearoperators was conducted Simulations were conducted onselected nonlinear boundary value problems defined onbounded domains One of the more significant findings toemerge from this study is that the linear operator definedby the highest order differential matching which is themost commonly used linear operator for solving boundaryvalue problems defined on bounded domains results inthe slowest converging and least accurate SHAM algorithmThe SHAM algorithm that uses a linear operator definedby the method of linear partition matching was found toperform slightly better than the SHAMwith the highest orderdifferential matching in some cases However in other casesthe methods were found to be comparable It was also shownthat the SHAMwith linear operator defined by themethod ofcomplete differential matching performed much better thanthe SHAMwith linear partitionmapping or the highest orderdifferential matching The study proposed a modification ofthe complete differential method approach which unlike theoriginal approach does not ignore the nonlinear factors of thederivative terms in the definition of the linear operator Thisnew definition of linear operator was found to perform betterthan the original complete differential method approachThe most optimal linear operator was found to be one thatuses the linear partition mapping after a transformation thatexpresses the unknown function as a sum of a new unknownplus some initial guess that satisfies the underlying boundaryconditions
The current findings add substantially to our under-standing of how homotopy analysis based methods canbe modified to yield approximate solutions with optimumaccuracy and convergence Further work needs to be done toinvestigate the application of the linear operators proposed inthis study in solving nonlinear partial differential equationsIt would also be very interesting to extend the current studyto systems of nonlinear differential equations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is based on the research supported in part by theNational Research Foundation of South Africa (Grant no85596)
References
[1] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[2] S J Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer Berlin Germany 2012
[3] S J Liao Advances in Homotopy Analysis Method WorldScientific Singapore 2014
[5] S S Motsa P Sibanda and S Shateyi ldquoA new spectral-homotopy analysismethod for solving a nonlinear second orderBVPrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 9 pp 2293ndash2302 2010
[6] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers and Fluids vol 39 no 7 pp 1219ndash1225 2010
[7] R A Van Gorder and K Vajravelu ldquoOn the selection of auxil-iary functions operators and convergence control parametersin the application of the homotopy analysismethod to nonlineardifferential equations a general approachrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 12 pp4078ndash4089 2009
[8] Y Shan and T Chaolu ldquoA method to select the initial guesssolution auxiliary linear operator and set of basic functions ofhomotopy analysis methodrdquo in Proceedings of the IEEE Inter-national Conference on Intelligent Computing and IntegratedSystems (ICISS rsquo10) pp 410ndash412 Guilin China October 2010
[9] R A van Gorder ldquoStability of auxiliary linear operatorand convergence-control parameter in the homotopy analysismethodrdquo in Advances in Homotopy Analysis Method S J LiaoEd pp 123ndash180 World Scientific Singapore 2014
[10] S M Rassoulinejad-Mousavi and S Abbasbandy ldquoAnalysisof forced convection in a circular tube filled with a darcy-brinkman-forchheimer porous medium using spectral homo-topy analysis methodrdquo Journal of Fluids Engineering Transac-tions of the ASME vol 133 no 10 Article ID 101207 2011
[11] A A Joneidi G Domairry andM Babaelahi ldquoThree analyticalmethods applied to Jeffery-Hamel flowrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 11 pp3423ndash3434 2010
[12] S M Moghimi G Domairry S Soleimani E Ghasemi and HBararnia ldquoApplication of homotopy analysis method to solveMHD Jeffery-Hamel flows in non-parallel wallsrdquo Advances inEngineering Software vol 42 no 3 pp 108ndash113 2011
[13] M Esmaeilpour and D D Ganji ldquoSolution of the Jeffery-Hamel flowproblemby optimal homotopy asymptoticmethodrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3405ndash3411 2010
[14] A Basiri Parsa M M Rashidi O Anwar Beg and S M SadrildquoSemi-computational simulation of magneto-hemodynamicflow in a semi-porous channel using optimal homotopy anddifferential transform methodsrdquo Computers in Biology andMedicine vol 43 no 9 pp 1142ndash1153 2013
[15] Z Ziabakhsh andGDomairry ldquoSolution of the laminar viscousflow in a semi-porous channel in the presence of a uniformmagnetic field by using the homotopy analysis methodrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 4 pp 1284ndash1294 2009
Mathematical Problems in Engineering 15
[16] S SMotsa S Shateyi GMarewo andP Sibanda ldquoAn improvedspectral homotopy analysis method for MHD flow in a semi-porous channelrdquo Numerical Algorithms vol 60 no 3 pp 463ndash481 2012
[17] S Dinarvand and M M Rashidi ldquoA reliable treatment of ahomotopy analysis method for two-dimensional viscous flowin a rectangular domain bounded by twomoving porous wallsrdquoNonlinear Analysis Real World Applications vol 11 no 3 pp1502ndash1512 2010
[18] H Xu Z Lin S Liao JWu and JMajdalani ldquoHomotopy basedsolutions of the Navier-Stokes equations for a porous channelwith orthogonally moving wallsrdquo Physics of Fluids vol 22 no5 Article ID 053601 pp 1ndash18 2010
[19] M M Rashidi E Erfani and B Rostami ldquoOptimal HomotopyAsymptotic Method for solving viscous ow through expandingor contracting gaps with permeable wallsrdquo Transaction on IoTand Cloud Computing vol 2 no 1 pp 85ndash111 2014
[20] Z GMakukula P Sibanda and S S Motsa ldquoA novel numericaltechnique for two-dimensional laminar flow between twomoving porous wallsrdquo Mathematical Problems in Engineeringvol 2010 Article ID 528956 15 pages 2010
[21] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Series in Com-putational Physics Springer New York NY USA 1988
[22] L N Trefethen Spectral Methods in MATLAB SIAM 2000[23] J Majdalani C Zhou and C A Dawson ldquoTwo-dimensional
viscous flow between slowly expanding or contracting wallswith weak permeabilityrdquo Journal of Biomechanics vol 35 no10 pp 1399ndash1403 2002
Maximum residual error 1198641199031 1599119890 minus 14 4108119890 minus 15 5329119890 minus 15 4441119890 minus 16 8993119890 minus 15
5 3375119890 minus 14 6439119890 minus 15 3442119890 minus 15 1443119890 minus 15 4663119890 minus 15
10 3952119890 minus 14 9770119890 minus 15 3442119890 minus 15 1776119890 minus 15 5218119890 minus 15
15 5529119890 minus 14 3397119890 minus 14 2887119890 minus 15 3553119890 minus 15 3331119890 minus 15
20 1612119890 minus 13 2620119890 minus 14 4885119890 minus 15 3664119890 minus 15 3331119890 minus 15
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 3 Residual ℏ-curve for Example 2 using the 20th orderSHAM with Re = 100 and 120572 = minus5∘
series terms are required to attain the maximum accuracy inall linear operators
Figure 3 shows the ℏ-curves for the Jeffery-Hamel prob-lem (Example 2) corresponding to the different SHAMschemes generated using all the linear operators using 20terms of the SHAM approximation series It can be seenfrom Figure 3 that after 20 iterationsL4 would give the bestaccuracy and L1 gives the least accuracy This is in accordwith the observation made earlier in Example 1 It can alsobe observed that the behaviour of the SHAM with L1 isapproximately the same as that of the SHAM with L2 Thiscan be noted from the observation that the ℏ-curve forL1 lieson top of that forL2 In Figure 4 the variation of the residualerror is shown as a function of the SHAM order It can beobserved that in all test linear operators the residual errordecreases with the number of terms used in the SHAM seriesThis demonstrates that the SHAM approximation converges
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 4 Variation of residual error with SHAM order for Example2 when Re = 100 and 120572 = minus5∘
in all cases considered in this example The convergence isfastest for L4 followed by L5 and is slowest when L1 isused in the SHAM algorithm It can also be observed that theresults for L1 L2 L3 and L5 converge to approximatelythe same level while L4 converges to a level lower than thatof the other linear operators Thus in addition to convergingmuch faster L4 gives results that are slightly more accuratethan those obtained using the other linear operators
Table 3 presents the minimum SHAM order required togive maximum accuracy at the the optimal values of theconvergence controlling parameter ℏ and the correspondingresidual errorsThe results are generated for increasing valuesof the Reynolds number (Re) It can be seen from Table 3that the minimum order of the SHAM required graduallyincreases with an increase in Re for both L1 and L2 Thereis no significant increase in the minimum SHAM orderrequired in the case of L3 L4 and L5 The results from
Maximum residual error 1198641199031 8451119890 minus 12 8987119890 minus 12 1074119890 minus 11 4886119890 minus 14 1256119890 minus 11
50 8636119890 minus 12 8887119890 minus 12 1148119890 minus 11 1357119890 minus 12 7468119890 minus 12
100 7930119890 minus 12 8759119890 minus 12 1303119890 minus 11 1886119890 minus 12 1633119890 minus 11
150 7815119890 minus 12 8092119890 minus 12 2546119890 minus 11 2973119890 minus 12 1808119890 minus 11
200 9128119890 minus 12 8421119890 minus 12 3209119890 minus 11 4435119890 minus 12 1176119890 minus 11
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 5 Residual ℏ-curve for Example 3 using the 20th orderSHAM with Re = 10 and Ha = 1
Table 3 also reveal that the required minimum SHAM orderof L1 and L2 is comparable particularly for larger valuesof Re This confirms the observation made in Figures 3 and4 Thus it can be inferred that the behaviour of the SHAMwith higher order differential mapping (L1) is comparablewith that of SHAMwith linear partitionmapping (L2) in thesolution of the Jeffery-Hamel equation
Figure 5 shows the ℏ-curves corresponding to the exam-ple of laminar viscous flow in a semiporous channel subjectto a transverse magnetic field (Example 3) Again it can beseen from Figure 5 that the residual error is lowest whenthe SHAM is used with L4 and largest when used with L1which as was observed in the case of Jeffery-Hamel flowis comparable with L2 The same observation can be madein Figure 6 where the residual error is plotted against theSHAM order From Figure 6 it can also be seen that L4gives the best convergence followed by L5 and the slowest
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 6 Variation of residual error with SHAM order for Example3 when Re = 10 and Ha = 1
convergence is obtained when usingL1 which is comparabletoL2
Table 4 shows effect of increasing the Reynolds number(Re) on the minimum SHAM orders and the correspondingresidual errors It can be observed from Table 4 that theminimumSHAMorders increasewith an increase inRewhenthe SHAM is used with L1 L2 and L3 It can also beseen that there is no significant increase in the minimumSHAM orders when using L4 and L5 Furthermore theresidual error obtained using L4 is lower than that of theother linear operatorsThis is in linewith similar observationsmade earlier
Table 5 shows the effect of increasing the Hartmannnumber (Ha) on the minimum SHAM orders and thecorresponding residual errors in Example 3 In this case itcan be observed that the minimum SHAM orders increasewith an increase in Ha only in the case of L1 In the case of
Maximum Residual error 1198641199032 1484119890 minus 08 1323119890 minus 08 9815119890 minus 09 3391119890 minus 12 2954119890 minus 09
4 8618119890 minus 09 3334119890 minus 09 3888119890 minus 09 5698119890 minus 12 2855119890 minus 09
6 2258119890 minus 09 6625119890 minus 09 1281119890 minus 08 1461119890 minus 11 6068119890 minus 09
8 1172119890 minus 09 2022119890 minus 09 1012119890 minus 08 1690119890 minus 11 7852119890 minus 09
10 8887119890 minus 09 5535119890 minus 09 1261119890 minus 08 5424119890 minus 11 2253119890 minus 09
15 3325119890 minus 09 1870119890 minus 09 5521119890 minus 09 2964119890 minus 10 5871119890 minus 09
the other linear operators theminimum SHAMorders do notincrease with an increase in HaThis means that convergenceof the SHAM series is independent of the value of Ha whenthe SHAM is used with L2 L3 L4 and L5 It can also benoted from Table 5 that the residual error is smallest whenL4 is used If can be inferred that the SHAM is most accuratewhen used withL4
Figure 7 shows the ℏ-curves corresponding to the exam-ple of two-dimensional viscous flow in a rectangular domainbounded by two moving porous walls (Example 4) It canbe noted from Figure 7 that the residual error is the lowestwhen the SHAM is used with L4 and the largest whenused with L1 after 20 iterations Figure 8 illustrates thevariation of the residual error against the SHAM order It canbe seen from Figure 8 that L4 gives the best convergencefollowed by L5 and the slowest convergence is obtainedwhen using L1 The linear operators L1 L2 L3 andL5 converge to the same level of residual error whichis higher than that of L4 Again it can be inferred that
L4 gives superior accuracy compared to the other linearoperators
Table 6 shows the effect of increasing the Reynolds num-ber (Re) on the minimum SHAM orders and correspondingresidual errors in Example 4 It can be observed that theminimumSHAMorders increasewith an increase inRewhenthe SHAM is used with L1 L2 and L3 In the case of L4andL5 it seems possible that an increase in Re has no stronginfluence in the convergence speed of the SHAM It can alsobe noted from Table 6 that the residual error is the smallestwhen L4 is used This is in accord with the trends observedin all the other examples considered in this work Table 7shows the effect of the nondimensional wall dilation rate 120572on the minimum SHAM order and corresponding residualerrors It can be observed from Table 7 that an increasein 120572 results in a corresponding significant increase in theminimum SHAMorder forL1Theminimum SHAMordersare not significantly influenced by 120572 when the SHAM is usedwithL2L3L4 andL5 Furthermore the best accuracy isobtained when usingL4
Maximum residual error 1198641199031 2103119890 minus 09 4065119890 minus 09 1354119890 minus 08 4946119890 minus 13 6441119890 minus 09
2 1198119890 minus 09 1109119890 minus 08 7087119890 minus 10 7009119890 minus 12 9531119890 minus 09
4 1954119890 minus 09 1017119890 minus 08 1604119890 minus 08 3667119890 minus 11 3063119890 minus 09
6 4419119890 minus 09 5935119890 minus 09 1772119890 minus 08 7262119890 minus 11 1388119890 minus 08
8 1930119890 minus 09 2864119890 minus 09 1449119890 minus 08 9716119890 minus 11 1221119890 minus 09
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 7 Residual ℏ-curve for Example 4 using the 20th orderSHAM with Re = 10 and 120572 = 1
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 8 Variation of residual error with SHAM order for Example4 when Re = 10 and 120572 = 1
14 Mathematical Problems in Engineering
6 Conclusion
This study gives a systematic way of choosing initial approx-imations and linear operators that can be employed inthe spectral homotopy analysis method (SHAM) algorithmThe main aim of the study was to determine the linearoperator definition that gives approximate solutions withoptimal convergence and accuracy A comparison betweenthe accuracy and convergence of SHAM algorithms usingcommonly used linear operators and newly proposed linearoperators was conducted Simulations were conducted onselected nonlinear boundary value problems defined onbounded domains One of the more significant findings toemerge from this study is that the linear operator definedby the highest order differential matching which is themost commonly used linear operator for solving boundaryvalue problems defined on bounded domains results inthe slowest converging and least accurate SHAM algorithmThe SHAM algorithm that uses a linear operator definedby the method of linear partition matching was found toperform slightly better than the SHAMwith the highest orderdifferential matching in some cases However in other casesthe methods were found to be comparable It was also shownthat the SHAMwith linear operator defined by themethod ofcomplete differential matching performed much better thanthe SHAMwith linear partitionmapping or the highest orderdifferential matching The study proposed a modification ofthe complete differential method approach which unlike theoriginal approach does not ignore the nonlinear factors of thederivative terms in the definition of the linear operator Thisnew definition of linear operator was found to perform betterthan the original complete differential method approachThe most optimal linear operator was found to be one thatuses the linear partition mapping after a transformation thatexpresses the unknown function as a sum of a new unknownplus some initial guess that satisfies the underlying boundaryconditions
The current findings add substantially to our under-standing of how homotopy analysis based methods canbe modified to yield approximate solutions with optimumaccuracy and convergence Further work needs to be done toinvestigate the application of the linear operators proposed inthis study in solving nonlinear partial differential equationsIt would also be very interesting to extend the current studyto systems of nonlinear differential equations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is based on the research supported in part by theNational Research Foundation of South Africa (Grant no85596)
References
[1] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[2] S J Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer Berlin Germany 2012
[3] S J Liao Advances in Homotopy Analysis Method WorldScientific Singapore 2014
[5] S S Motsa P Sibanda and S Shateyi ldquoA new spectral-homotopy analysismethod for solving a nonlinear second orderBVPrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 9 pp 2293ndash2302 2010
[6] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers and Fluids vol 39 no 7 pp 1219ndash1225 2010
[7] R A Van Gorder and K Vajravelu ldquoOn the selection of auxil-iary functions operators and convergence control parametersin the application of the homotopy analysismethod to nonlineardifferential equations a general approachrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 12 pp4078ndash4089 2009
[8] Y Shan and T Chaolu ldquoA method to select the initial guesssolution auxiliary linear operator and set of basic functions ofhomotopy analysis methodrdquo in Proceedings of the IEEE Inter-national Conference on Intelligent Computing and IntegratedSystems (ICISS rsquo10) pp 410ndash412 Guilin China October 2010
[9] R A van Gorder ldquoStability of auxiliary linear operatorand convergence-control parameter in the homotopy analysismethodrdquo in Advances in Homotopy Analysis Method S J LiaoEd pp 123ndash180 World Scientific Singapore 2014
[10] S M Rassoulinejad-Mousavi and S Abbasbandy ldquoAnalysisof forced convection in a circular tube filled with a darcy-brinkman-forchheimer porous medium using spectral homo-topy analysis methodrdquo Journal of Fluids Engineering Transac-tions of the ASME vol 133 no 10 Article ID 101207 2011
[11] A A Joneidi G Domairry andM Babaelahi ldquoThree analyticalmethods applied to Jeffery-Hamel flowrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 11 pp3423ndash3434 2010
[12] S M Moghimi G Domairry S Soleimani E Ghasemi and HBararnia ldquoApplication of homotopy analysis method to solveMHD Jeffery-Hamel flows in non-parallel wallsrdquo Advances inEngineering Software vol 42 no 3 pp 108ndash113 2011
[13] M Esmaeilpour and D D Ganji ldquoSolution of the Jeffery-Hamel flowproblemby optimal homotopy asymptoticmethodrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3405ndash3411 2010
[14] A Basiri Parsa M M Rashidi O Anwar Beg and S M SadrildquoSemi-computational simulation of magneto-hemodynamicflow in a semi-porous channel using optimal homotopy anddifferential transform methodsrdquo Computers in Biology andMedicine vol 43 no 9 pp 1142ndash1153 2013
[15] Z Ziabakhsh andGDomairry ldquoSolution of the laminar viscousflow in a semi-porous channel in the presence of a uniformmagnetic field by using the homotopy analysis methodrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 4 pp 1284ndash1294 2009
Mathematical Problems in Engineering 15
[16] S SMotsa S Shateyi GMarewo andP Sibanda ldquoAn improvedspectral homotopy analysis method for MHD flow in a semi-porous channelrdquo Numerical Algorithms vol 60 no 3 pp 463ndash481 2012
[17] S Dinarvand and M M Rashidi ldquoA reliable treatment of ahomotopy analysis method for two-dimensional viscous flowin a rectangular domain bounded by twomoving porous wallsrdquoNonlinear Analysis Real World Applications vol 11 no 3 pp1502ndash1512 2010
[18] H Xu Z Lin S Liao JWu and JMajdalani ldquoHomotopy basedsolutions of the Navier-Stokes equations for a porous channelwith orthogonally moving wallsrdquo Physics of Fluids vol 22 no5 Article ID 053601 pp 1ndash18 2010
[19] M M Rashidi E Erfani and B Rostami ldquoOptimal HomotopyAsymptotic Method for solving viscous ow through expandingor contracting gaps with permeable wallsrdquo Transaction on IoTand Cloud Computing vol 2 no 1 pp 85ndash111 2014
[20] Z GMakukula P Sibanda and S S Motsa ldquoA novel numericaltechnique for two-dimensional laminar flow between twomoving porous wallsrdquo Mathematical Problems in Engineeringvol 2010 Article ID 528956 15 pages 2010
[21] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Series in Com-putational Physics Springer New York NY USA 1988
[22] L N Trefethen Spectral Methods in MATLAB SIAM 2000[23] J Majdalani C Zhou and C A Dawson ldquoTwo-dimensional
viscous flow between slowly expanding or contracting wallswith weak permeabilityrdquo Journal of Biomechanics vol 35 no10 pp 1399ndash1403 2002
Maximum residual error 1198641199031 8451119890 minus 12 8987119890 minus 12 1074119890 minus 11 4886119890 minus 14 1256119890 minus 11
50 8636119890 minus 12 8887119890 minus 12 1148119890 minus 11 1357119890 minus 12 7468119890 minus 12
100 7930119890 minus 12 8759119890 minus 12 1303119890 minus 11 1886119890 minus 12 1633119890 minus 11
150 7815119890 minus 12 8092119890 minus 12 2546119890 minus 11 2973119890 minus 12 1808119890 minus 11
200 9128119890 minus 12 8421119890 minus 12 3209119890 minus 11 4435119890 minus 12 1176119890 minus 11
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 5 Residual ℏ-curve for Example 3 using the 20th orderSHAM with Re = 10 and Ha = 1
Table 3 also reveal that the required minimum SHAM orderof L1 and L2 is comparable particularly for larger valuesof Re This confirms the observation made in Figures 3 and4 Thus it can be inferred that the behaviour of the SHAMwith higher order differential mapping (L1) is comparablewith that of SHAMwith linear partitionmapping (L2) in thesolution of the Jeffery-Hamel equation
Figure 5 shows the ℏ-curves corresponding to the exam-ple of laminar viscous flow in a semiporous channel subjectto a transverse magnetic field (Example 3) Again it can beseen from Figure 5 that the residual error is lowest whenthe SHAM is used with L4 and largest when used with L1which as was observed in the case of Jeffery-Hamel flowis comparable with L2 The same observation can be madein Figure 6 where the residual error is plotted against theSHAM order From Figure 6 it can also be seen that L4gives the best convergence followed by L5 and the slowest
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 6 Variation of residual error with SHAM order for Example3 when Re = 10 and Ha = 1
convergence is obtained when usingL1 which is comparabletoL2
Table 4 shows effect of increasing the Reynolds number(Re) on the minimum SHAM orders and the correspondingresidual errors It can be observed from Table 4 that theminimumSHAMorders increasewith an increase inRewhenthe SHAM is used with L1 L2 and L3 It can also beseen that there is no significant increase in the minimumSHAM orders when using L4 and L5 Furthermore theresidual error obtained using L4 is lower than that of theother linear operatorsThis is in linewith similar observationsmade earlier
Table 5 shows the effect of increasing the Hartmannnumber (Ha) on the minimum SHAM orders and thecorresponding residual errors in Example 3 In this case itcan be observed that the minimum SHAM orders increasewith an increase in Ha only in the case of L1 In the case of
Maximum Residual error 1198641199032 1484119890 minus 08 1323119890 minus 08 9815119890 minus 09 3391119890 minus 12 2954119890 minus 09
4 8618119890 minus 09 3334119890 minus 09 3888119890 minus 09 5698119890 minus 12 2855119890 minus 09
6 2258119890 minus 09 6625119890 minus 09 1281119890 minus 08 1461119890 minus 11 6068119890 minus 09
8 1172119890 minus 09 2022119890 minus 09 1012119890 minus 08 1690119890 minus 11 7852119890 minus 09
10 8887119890 minus 09 5535119890 minus 09 1261119890 minus 08 5424119890 minus 11 2253119890 minus 09
15 3325119890 minus 09 1870119890 minus 09 5521119890 minus 09 2964119890 minus 10 5871119890 minus 09
the other linear operators theminimum SHAMorders do notincrease with an increase in HaThis means that convergenceof the SHAM series is independent of the value of Ha whenthe SHAM is used with L2 L3 L4 and L5 It can also benoted from Table 5 that the residual error is smallest whenL4 is used If can be inferred that the SHAM is most accuratewhen used withL4
Figure 7 shows the ℏ-curves corresponding to the exam-ple of two-dimensional viscous flow in a rectangular domainbounded by two moving porous walls (Example 4) It canbe noted from Figure 7 that the residual error is the lowestwhen the SHAM is used with L4 and the largest whenused with L1 after 20 iterations Figure 8 illustrates thevariation of the residual error against the SHAM order It canbe seen from Figure 8 that L4 gives the best convergencefollowed by L5 and the slowest convergence is obtainedwhen using L1 The linear operators L1 L2 L3 andL5 converge to the same level of residual error whichis higher than that of L4 Again it can be inferred that
L4 gives superior accuracy compared to the other linearoperators
Table 6 shows the effect of increasing the Reynolds num-ber (Re) on the minimum SHAM orders and correspondingresidual errors in Example 4 It can be observed that theminimumSHAMorders increasewith an increase inRewhenthe SHAM is used with L1 L2 and L3 In the case of L4andL5 it seems possible that an increase in Re has no stronginfluence in the convergence speed of the SHAM It can alsobe noted from Table 6 that the residual error is the smallestwhen L4 is used This is in accord with the trends observedin all the other examples considered in this work Table 7shows the effect of the nondimensional wall dilation rate 120572on the minimum SHAM order and corresponding residualerrors It can be observed from Table 7 that an increasein 120572 results in a corresponding significant increase in theminimum SHAMorder forL1Theminimum SHAMordersare not significantly influenced by 120572 when the SHAM is usedwithL2L3L4 andL5 Furthermore the best accuracy isobtained when usingL4
Maximum residual error 1198641199031 2103119890 minus 09 4065119890 minus 09 1354119890 minus 08 4946119890 minus 13 6441119890 minus 09
2 1198119890 minus 09 1109119890 minus 08 7087119890 minus 10 7009119890 minus 12 9531119890 minus 09
4 1954119890 minus 09 1017119890 minus 08 1604119890 minus 08 3667119890 minus 11 3063119890 minus 09
6 4419119890 minus 09 5935119890 minus 09 1772119890 minus 08 7262119890 minus 11 1388119890 minus 08
8 1930119890 minus 09 2864119890 minus 09 1449119890 minus 08 9716119890 minus 11 1221119890 minus 09
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 7 Residual ℏ-curve for Example 4 using the 20th orderSHAM with Re = 10 and 120572 = 1
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 8 Variation of residual error with SHAM order for Example4 when Re = 10 and 120572 = 1
14 Mathematical Problems in Engineering
6 Conclusion
This study gives a systematic way of choosing initial approx-imations and linear operators that can be employed inthe spectral homotopy analysis method (SHAM) algorithmThe main aim of the study was to determine the linearoperator definition that gives approximate solutions withoptimal convergence and accuracy A comparison betweenthe accuracy and convergence of SHAM algorithms usingcommonly used linear operators and newly proposed linearoperators was conducted Simulations were conducted onselected nonlinear boundary value problems defined onbounded domains One of the more significant findings toemerge from this study is that the linear operator definedby the highest order differential matching which is themost commonly used linear operator for solving boundaryvalue problems defined on bounded domains results inthe slowest converging and least accurate SHAM algorithmThe SHAM algorithm that uses a linear operator definedby the method of linear partition matching was found toperform slightly better than the SHAMwith the highest orderdifferential matching in some cases However in other casesthe methods were found to be comparable It was also shownthat the SHAMwith linear operator defined by themethod ofcomplete differential matching performed much better thanthe SHAMwith linear partitionmapping or the highest orderdifferential matching The study proposed a modification ofthe complete differential method approach which unlike theoriginal approach does not ignore the nonlinear factors of thederivative terms in the definition of the linear operator Thisnew definition of linear operator was found to perform betterthan the original complete differential method approachThe most optimal linear operator was found to be one thatuses the linear partition mapping after a transformation thatexpresses the unknown function as a sum of a new unknownplus some initial guess that satisfies the underlying boundaryconditions
The current findings add substantially to our under-standing of how homotopy analysis based methods canbe modified to yield approximate solutions with optimumaccuracy and convergence Further work needs to be done toinvestigate the application of the linear operators proposed inthis study in solving nonlinear partial differential equationsIt would also be very interesting to extend the current studyto systems of nonlinear differential equations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is based on the research supported in part by theNational Research Foundation of South Africa (Grant no85596)
References
[1] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[2] S J Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer Berlin Germany 2012
[3] S J Liao Advances in Homotopy Analysis Method WorldScientific Singapore 2014
[5] S S Motsa P Sibanda and S Shateyi ldquoA new spectral-homotopy analysismethod for solving a nonlinear second orderBVPrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 9 pp 2293ndash2302 2010
[6] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers and Fluids vol 39 no 7 pp 1219ndash1225 2010
[7] R A Van Gorder and K Vajravelu ldquoOn the selection of auxil-iary functions operators and convergence control parametersin the application of the homotopy analysismethod to nonlineardifferential equations a general approachrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 12 pp4078ndash4089 2009
[8] Y Shan and T Chaolu ldquoA method to select the initial guesssolution auxiliary linear operator and set of basic functions ofhomotopy analysis methodrdquo in Proceedings of the IEEE Inter-national Conference on Intelligent Computing and IntegratedSystems (ICISS rsquo10) pp 410ndash412 Guilin China October 2010
[9] R A van Gorder ldquoStability of auxiliary linear operatorand convergence-control parameter in the homotopy analysismethodrdquo in Advances in Homotopy Analysis Method S J LiaoEd pp 123ndash180 World Scientific Singapore 2014
[10] S M Rassoulinejad-Mousavi and S Abbasbandy ldquoAnalysisof forced convection in a circular tube filled with a darcy-brinkman-forchheimer porous medium using spectral homo-topy analysis methodrdquo Journal of Fluids Engineering Transac-tions of the ASME vol 133 no 10 Article ID 101207 2011
[11] A A Joneidi G Domairry andM Babaelahi ldquoThree analyticalmethods applied to Jeffery-Hamel flowrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 11 pp3423ndash3434 2010
[12] S M Moghimi G Domairry S Soleimani E Ghasemi and HBararnia ldquoApplication of homotopy analysis method to solveMHD Jeffery-Hamel flows in non-parallel wallsrdquo Advances inEngineering Software vol 42 no 3 pp 108ndash113 2011
[13] M Esmaeilpour and D D Ganji ldquoSolution of the Jeffery-Hamel flowproblemby optimal homotopy asymptoticmethodrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3405ndash3411 2010
[14] A Basiri Parsa M M Rashidi O Anwar Beg and S M SadrildquoSemi-computational simulation of magneto-hemodynamicflow in a semi-porous channel using optimal homotopy anddifferential transform methodsrdquo Computers in Biology andMedicine vol 43 no 9 pp 1142ndash1153 2013
[15] Z Ziabakhsh andGDomairry ldquoSolution of the laminar viscousflow in a semi-porous channel in the presence of a uniformmagnetic field by using the homotopy analysis methodrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 4 pp 1284ndash1294 2009
Mathematical Problems in Engineering 15
[16] S SMotsa S Shateyi GMarewo andP Sibanda ldquoAn improvedspectral homotopy analysis method for MHD flow in a semi-porous channelrdquo Numerical Algorithms vol 60 no 3 pp 463ndash481 2012
[17] S Dinarvand and M M Rashidi ldquoA reliable treatment of ahomotopy analysis method for two-dimensional viscous flowin a rectangular domain bounded by twomoving porous wallsrdquoNonlinear Analysis Real World Applications vol 11 no 3 pp1502ndash1512 2010
[18] H Xu Z Lin S Liao JWu and JMajdalani ldquoHomotopy basedsolutions of the Navier-Stokes equations for a porous channelwith orthogonally moving wallsrdquo Physics of Fluids vol 22 no5 Article ID 053601 pp 1ndash18 2010
[19] M M Rashidi E Erfani and B Rostami ldquoOptimal HomotopyAsymptotic Method for solving viscous ow through expandingor contracting gaps with permeable wallsrdquo Transaction on IoTand Cloud Computing vol 2 no 1 pp 85ndash111 2014
[20] Z GMakukula P Sibanda and S S Motsa ldquoA novel numericaltechnique for two-dimensional laminar flow between twomoving porous wallsrdquo Mathematical Problems in Engineeringvol 2010 Article ID 528956 15 pages 2010
[21] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Series in Com-putational Physics Springer New York NY USA 1988
[22] L N Trefethen Spectral Methods in MATLAB SIAM 2000[23] J Majdalani C Zhou and C A Dawson ldquoTwo-dimensional
viscous flow between slowly expanding or contracting wallswith weak permeabilityrdquo Journal of Biomechanics vol 35 no10 pp 1399ndash1403 2002
Maximum Residual error 1198641199032 1484119890 minus 08 1323119890 minus 08 9815119890 minus 09 3391119890 minus 12 2954119890 minus 09
4 8618119890 minus 09 3334119890 minus 09 3888119890 minus 09 5698119890 minus 12 2855119890 minus 09
6 2258119890 minus 09 6625119890 minus 09 1281119890 minus 08 1461119890 minus 11 6068119890 minus 09
8 1172119890 minus 09 2022119890 minus 09 1012119890 minus 08 1690119890 minus 11 7852119890 minus 09
10 8887119890 minus 09 5535119890 minus 09 1261119890 minus 08 5424119890 minus 11 2253119890 minus 09
15 3325119890 minus 09 1870119890 minus 09 5521119890 minus 09 2964119890 minus 10 5871119890 minus 09
the other linear operators theminimum SHAMorders do notincrease with an increase in HaThis means that convergenceof the SHAM series is independent of the value of Ha whenthe SHAM is used with L2 L3 L4 and L5 It can also benoted from Table 5 that the residual error is smallest whenL4 is used If can be inferred that the SHAM is most accuratewhen used withL4
Figure 7 shows the ℏ-curves corresponding to the exam-ple of two-dimensional viscous flow in a rectangular domainbounded by two moving porous walls (Example 4) It canbe noted from Figure 7 that the residual error is the lowestwhen the SHAM is used with L4 and the largest whenused with L1 after 20 iterations Figure 8 illustrates thevariation of the residual error against the SHAM order It canbe seen from Figure 8 that L4 gives the best convergencefollowed by L5 and the slowest convergence is obtainedwhen using L1 The linear operators L1 L2 L3 andL5 converge to the same level of residual error whichis higher than that of L4 Again it can be inferred that
L4 gives superior accuracy compared to the other linearoperators
Table 6 shows the effect of increasing the Reynolds num-ber (Re) on the minimum SHAM orders and correspondingresidual errors in Example 4 It can be observed that theminimumSHAMorders increasewith an increase inRewhenthe SHAM is used with L1 L2 and L3 In the case of L4andL5 it seems possible that an increase in Re has no stronginfluence in the convergence speed of the SHAM It can alsobe noted from Table 6 that the residual error is the smallestwhen L4 is used This is in accord with the trends observedin all the other examples considered in this work Table 7shows the effect of the nondimensional wall dilation rate 120572on the minimum SHAM order and corresponding residualerrors It can be observed from Table 7 that an increasein 120572 results in a corresponding significant increase in theminimum SHAMorder forL1Theminimum SHAMordersare not significantly influenced by 120572 when the SHAM is usedwithL2L3L4 andL5 Furthermore the best accuracy isobtained when usingL4
Maximum residual error 1198641199031 2103119890 minus 09 4065119890 minus 09 1354119890 minus 08 4946119890 minus 13 6441119890 minus 09
2 1198119890 minus 09 1109119890 minus 08 7087119890 minus 10 7009119890 minus 12 9531119890 minus 09
4 1954119890 minus 09 1017119890 minus 08 1604119890 minus 08 3667119890 minus 11 3063119890 minus 09
6 4419119890 minus 09 5935119890 minus 09 1772119890 minus 08 7262119890 minus 11 1388119890 minus 08
8 1930119890 minus 09 2864119890 minus 09 1449119890 minus 08 9716119890 minus 11 1221119890 minus 09
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 7 Residual ℏ-curve for Example 4 using the 20th orderSHAM with Re = 10 and 120572 = 1
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 8 Variation of residual error with SHAM order for Example4 when Re = 10 and 120572 = 1
14 Mathematical Problems in Engineering
6 Conclusion
This study gives a systematic way of choosing initial approx-imations and linear operators that can be employed inthe spectral homotopy analysis method (SHAM) algorithmThe main aim of the study was to determine the linearoperator definition that gives approximate solutions withoptimal convergence and accuracy A comparison betweenthe accuracy and convergence of SHAM algorithms usingcommonly used linear operators and newly proposed linearoperators was conducted Simulations were conducted onselected nonlinear boundary value problems defined onbounded domains One of the more significant findings toemerge from this study is that the linear operator definedby the highest order differential matching which is themost commonly used linear operator for solving boundaryvalue problems defined on bounded domains results inthe slowest converging and least accurate SHAM algorithmThe SHAM algorithm that uses a linear operator definedby the method of linear partition matching was found toperform slightly better than the SHAMwith the highest orderdifferential matching in some cases However in other casesthe methods were found to be comparable It was also shownthat the SHAMwith linear operator defined by themethod ofcomplete differential matching performed much better thanthe SHAMwith linear partitionmapping or the highest orderdifferential matching The study proposed a modification ofthe complete differential method approach which unlike theoriginal approach does not ignore the nonlinear factors of thederivative terms in the definition of the linear operator Thisnew definition of linear operator was found to perform betterthan the original complete differential method approachThe most optimal linear operator was found to be one thatuses the linear partition mapping after a transformation thatexpresses the unknown function as a sum of a new unknownplus some initial guess that satisfies the underlying boundaryconditions
The current findings add substantially to our under-standing of how homotopy analysis based methods canbe modified to yield approximate solutions with optimumaccuracy and convergence Further work needs to be done toinvestigate the application of the linear operators proposed inthis study in solving nonlinear partial differential equationsIt would also be very interesting to extend the current studyto systems of nonlinear differential equations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is based on the research supported in part by theNational Research Foundation of South Africa (Grant no85596)
References
[1] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[2] S J Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer Berlin Germany 2012
[3] S J Liao Advances in Homotopy Analysis Method WorldScientific Singapore 2014
[5] S S Motsa P Sibanda and S Shateyi ldquoA new spectral-homotopy analysismethod for solving a nonlinear second orderBVPrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 9 pp 2293ndash2302 2010
[6] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers and Fluids vol 39 no 7 pp 1219ndash1225 2010
[7] R A Van Gorder and K Vajravelu ldquoOn the selection of auxil-iary functions operators and convergence control parametersin the application of the homotopy analysismethod to nonlineardifferential equations a general approachrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 12 pp4078ndash4089 2009
[8] Y Shan and T Chaolu ldquoA method to select the initial guesssolution auxiliary linear operator and set of basic functions ofhomotopy analysis methodrdquo in Proceedings of the IEEE Inter-national Conference on Intelligent Computing and IntegratedSystems (ICISS rsquo10) pp 410ndash412 Guilin China October 2010
[9] R A van Gorder ldquoStability of auxiliary linear operatorand convergence-control parameter in the homotopy analysismethodrdquo in Advances in Homotopy Analysis Method S J LiaoEd pp 123ndash180 World Scientific Singapore 2014
[10] S M Rassoulinejad-Mousavi and S Abbasbandy ldquoAnalysisof forced convection in a circular tube filled with a darcy-brinkman-forchheimer porous medium using spectral homo-topy analysis methodrdquo Journal of Fluids Engineering Transac-tions of the ASME vol 133 no 10 Article ID 101207 2011
[11] A A Joneidi G Domairry andM Babaelahi ldquoThree analyticalmethods applied to Jeffery-Hamel flowrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 11 pp3423ndash3434 2010
[12] S M Moghimi G Domairry S Soleimani E Ghasemi and HBararnia ldquoApplication of homotopy analysis method to solveMHD Jeffery-Hamel flows in non-parallel wallsrdquo Advances inEngineering Software vol 42 no 3 pp 108ndash113 2011
[13] M Esmaeilpour and D D Ganji ldquoSolution of the Jeffery-Hamel flowproblemby optimal homotopy asymptoticmethodrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3405ndash3411 2010
[14] A Basiri Parsa M M Rashidi O Anwar Beg and S M SadrildquoSemi-computational simulation of magneto-hemodynamicflow in a semi-porous channel using optimal homotopy anddifferential transform methodsrdquo Computers in Biology andMedicine vol 43 no 9 pp 1142ndash1153 2013
[15] Z Ziabakhsh andGDomairry ldquoSolution of the laminar viscousflow in a semi-porous channel in the presence of a uniformmagnetic field by using the homotopy analysis methodrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 4 pp 1284ndash1294 2009
Mathematical Problems in Engineering 15
[16] S SMotsa S Shateyi GMarewo andP Sibanda ldquoAn improvedspectral homotopy analysis method for MHD flow in a semi-porous channelrdquo Numerical Algorithms vol 60 no 3 pp 463ndash481 2012
[17] S Dinarvand and M M Rashidi ldquoA reliable treatment of ahomotopy analysis method for two-dimensional viscous flowin a rectangular domain bounded by twomoving porous wallsrdquoNonlinear Analysis Real World Applications vol 11 no 3 pp1502ndash1512 2010
[18] H Xu Z Lin S Liao JWu and JMajdalani ldquoHomotopy basedsolutions of the Navier-Stokes equations for a porous channelwith orthogonally moving wallsrdquo Physics of Fluids vol 22 no5 Article ID 053601 pp 1ndash18 2010
[19] M M Rashidi E Erfani and B Rostami ldquoOptimal HomotopyAsymptotic Method for solving viscous ow through expandingor contracting gaps with permeable wallsrdquo Transaction on IoTand Cloud Computing vol 2 no 1 pp 85ndash111 2014
[20] Z GMakukula P Sibanda and S S Motsa ldquoA novel numericaltechnique for two-dimensional laminar flow between twomoving porous wallsrdquo Mathematical Problems in Engineeringvol 2010 Article ID 528956 15 pages 2010
[21] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Series in Com-putational Physics Springer New York NY USA 1988
[22] L N Trefethen Spectral Methods in MATLAB SIAM 2000[23] J Majdalani C Zhou and C A Dawson ldquoTwo-dimensional
viscous flow between slowly expanding or contracting wallswith weak permeabilityrdquo Journal of Biomechanics vol 35 no10 pp 1399ndash1403 2002
Maximum residual error 1198641199031 2103119890 minus 09 4065119890 minus 09 1354119890 minus 08 4946119890 minus 13 6441119890 minus 09
2 1198119890 minus 09 1109119890 minus 08 7087119890 minus 10 7009119890 minus 12 9531119890 minus 09
4 1954119890 minus 09 1017119890 minus 08 1604119890 minus 08 3667119890 minus 11 3063119890 minus 09
6 4419119890 minus 09 5935119890 minus 09 1772119890 minus 08 7262119890 minus 11 1388119890 minus 08
8 1930119890 minus 09 2864119890 minus 09 1449119890 minus 08 9716119890 minus 11 1221119890 minus 09
0
ℏ
minus2 minus15 minus1 minus05
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 7 Residual ℏ-curve for Example 4 using the 20th orderSHAM with Re = 10 and 120572 = 1
0 10 20 30 40 50 60
Order
Er
100
10minus5
10minus10
10minus15
ℒ2
ℒ3
ℒ4
ℒ5
ℒ1
Figure 8 Variation of residual error with SHAM order for Example4 when Re = 10 and 120572 = 1
14 Mathematical Problems in Engineering
6 Conclusion
This study gives a systematic way of choosing initial approx-imations and linear operators that can be employed inthe spectral homotopy analysis method (SHAM) algorithmThe main aim of the study was to determine the linearoperator definition that gives approximate solutions withoptimal convergence and accuracy A comparison betweenthe accuracy and convergence of SHAM algorithms usingcommonly used linear operators and newly proposed linearoperators was conducted Simulations were conducted onselected nonlinear boundary value problems defined onbounded domains One of the more significant findings toemerge from this study is that the linear operator definedby the highest order differential matching which is themost commonly used linear operator for solving boundaryvalue problems defined on bounded domains results inthe slowest converging and least accurate SHAM algorithmThe SHAM algorithm that uses a linear operator definedby the method of linear partition matching was found toperform slightly better than the SHAMwith the highest orderdifferential matching in some cases However in other casesthe methods were found to be comparable It was also shownthat the SHAMwith linear operator defined by themethod ofcomplete differential matching performed much better thanthe SHAMwith linear partitionmapping or the highest orderdifferential matching The study proposed a modification ofthe complete differential method approach which unlike theoriginal approach does not ignore the nonlinear factors of thederivative terms in the definition of the linear operator Thisnew definition of linear operator was found to perform betterthan the original complete differential method approachThe most optimal linear operator was found to be one thatuses the linear partition mapping after a transformation thatexpresses the unknown function as a sum of a new unknownplus some initial guess that satisfies the underlying boundaryconditions
The current findings add substantially to our under-standing of how homotopy analysis based methods canbe modified to yield approximate solutions with optimumaccuracy and convergence Further work needs to be done toinvestigate the application of the linear operators proposed inthis study in solving nonlinear partial differential equationsIt would also be very interesting to extend the current studyto systems of nonlinear differential equations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is based on the research supported in part by theNational Research Foundation of South Africa (Grant no85596)
References
[1] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[2] S J Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer Berlin Germany 2012
[3] S J Liao Advances in Homotopy Analysis Method WorldScientific Singapore 2014
[5] S S Motsa P Sibanda and S Shateyi ldquoA new spectral-homotopy analysismethod for solving a nonlinear second orderBVPrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 9 pp 2293ndash2302 2010
[6] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers and Fluids vol 39 no 7 pp 1219ndash1225 2010
[7] R A Van Gorder and K Vajravelu ldquoOn the selection of auxil-iary functions operators and convergence control parametersin the application of the homotopy analysismethod to nonlineardifferential equations a general approachrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 12 pp4078ndash4089 2009
[8] Y Shan and T Chaolu ldquoA method to select the initial guesssolution auxiliary linear operator and set of basic functions ofhomotopy analysis methodrdquo in Proceedings of the IEEE Inter-national Conference on Intelligent Computing and IntegratedSystems (ICISS rsquo10) pp 410ndash412 Guilin China October 2010
[9] R A van Gorder ldquoStability of auxiliary linear operatorand convergence-control parameter in the homotopy analysismethodrdquo in Advances in Homotopy Analysis Method S J LiaoEd pp 123ndash180 World Scientific Singapore 2014
[10] S M Rassoulinejad-Mousavi and S Abbasbandy ldquoAnalysisof forced convection in a circular tube filled with a darcy-brinkman-forchheimer porous medium using spectral homo-topy analysis methodrdquo Journal of Fluids Engineering Transac-tions of the ASME vol 133 no 10 Article ID 101207 2011
[11] A A Joneidi G Domairry andM Babaelahi ldquoThree analyticalmethods applied to Jeffery-Hamel flowrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 11 pp3423ndash3434 2010
[12] S M Moghimi G Domairry S Soleimani E Ghasemi and HBararnia ldquoApplication of homotopy analysis method to solveMHD Jeffery-Hamel flows in non-parallel wallsrdquo Advances inEngineering Software vol 42 no 3 pp 108ndash113 2011
[13] M Esmaeilpour and D D Ganji ldquoSolution of the Jeffery-Hamel flowproblemby optimal homotopy asymptoticmethodrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3405ndash3411 2010
[14] A Basiri Parsa M M Rashidi O Anwar Beg and S M SadrildquoSemi-computational simulation of magneto-hemodynamicflow in a semi-porous channel using optimal homotopy anddifferential transform methodsrdquo Computers in Biology andMedicine vol 43 no 9 pp 1142ndash1153 2013
[15] Z Ziabakhsh andGDomairry ldquoSolution of the laminar viscousflow in a semi-porous channel in the presence of a uniformmagnetic field by using the homotopy analysis methodrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 4 pp 1284ndash1294 2009
Mathematical Problems in Engineering 15
[16] S SMotsa S Shateyi GMarewo andP Sibanda ldquoAn improvedspectral homotopy analysis method for MHD flow in a semi-porous channelrdquo Numerical Algorithms vol 60 no 3 pp 463ndash481 2012
[17] S Dinarvand and M M Rashidi ldquoA reliable treatment of ahomotopy analysis method for two-dimensional viscous flowin a rectangular domain bounded by twomoving porous wallsrdquoNonlinear Analysis Real World Applications vol 11 no 3 pp1502ndash1512 2010
[18] H Xu Z Lin S Liao JWu and JMajdalani ldquoHomotopy basedsolutions of the Navier-Stokes equations for a porous channelwith orthogonally moving wallsrdquo Physics of Fluids vol 22 no5 Article ID 053601 pp 1ndash18 2010
[19] M M Rashidi E Erfani and B Rostami ldquoOptimal HomotopyAsymptotic Method for solving viscous ow through expandingor contracting gaps with permeable wallsrdquo Transaction on IoTand Cloud Computing vol 2 no 1 pp 85ndash111 2014
[20] Z GMakukula P Sibanda and S S Motsa ldquoA novel numericaltechnique for two-dimensional laminar flow between twomoving porous wallsrdquo Mathematical Problems in Engineeringvol 2010 Article ID 528956 15 pages 2010
[21] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Series in Com-putational Physics Springer New York NY USA 1988
[22] L N Trefethen Spectral Methods in MATLAB SIAM 2000[23] J Majdalani C Zhou and C A Dawson ldquoTwo-dimensional
viscous flow between slowly expanding or contracting wallswith weak permeabilityrdquo Journal of Biomechanics vol 35 no10 pp 1399ndash1403 2002
This study gives a systematic way of choosing initial approx-imations and linear operators that can be employed inthe spectral homotopy analysis method (SHAM) algorithmThe main aim of the study was to determine the linearoperator definition that gives approximate solutions withoptimal convergence and accuracy A comparison betweenthe accuracy and convergence of SHAM algorithms usingcommonly used linear operators and newly proposed linearoperators was conducted Simulations were conducted onselected nonlinear boundary value problems defined onbounded domains One of the more significant findings toemerge from this study is that the linear operator definedby the highest order differential matching which is themost commonly used linear operator for solving boundaryvalue problems defined on bounded domains results inthe slowest converging and least accurate SHAM algorithmThe SHAM algorithm that uses a linear operator definedby the method of linear partition matching was found toperform slightly better than the SHAMwith the highest orderdifferential matching in some cases However in other casesthe methods were found to be comparable It was also shownthat the SHAMwith linear operator defined by themethod ofcomplete differential matching performed much better thanthe SHAMwith linear partitionmapping or the highest orderdifferential matching The study proposed a modification ofthe complete differential method approach which unlike theoriginal approach does not ignore the nonlinear factors of thederivative terms in the definition of the linear operator Thisnew definition of linear operator was found to perform betterthan the original complete differential method approachThe most optimal linear operator was found to be one thatuses the linear partition mapping after a transformation thatexpresses the unknown function as a sum of a new unknownplus some initial guess that satisfies the underlying boundaryconditions
The current findings add substantially to our under-standing of how homotopy analysis based methods canbe modified to yield approximate solutions with optimumaccuracy and convergence Further work needs to be done toinvestigate the application of the linear operators proposed inthis study in solving nonlinear partial differential equationsIt would also be very interesting to extend the current studyto systems of nonlinear differential equations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is based on the research supported in part by theNational Research Foundation of South Africa (Grant no85596)
References
[1] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[2] S J Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer Berlin Germany 2012
[3] S J Liao Advances in Homotopy Analysis Method WorldScientific Singapore 2014
[5] S S Motsa P Sibanda and S Shateyi ldquoA new spectral-homotopy analysismethod for solving a nonlinear second orderBVPrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 9 pp 2293ndash2302 2010
[6] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers and Fluids vol 39 no 7 pp 1219ndash1225 2010
[7] R A Van Gorder and K Vajravelu ldquoOn the selection of auxil-iary functions operators and convergence control parametersin the application of the homotopy analysismethod to nonlineardifferential equations a general approachrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 12 pp4078ndash4089 2009
[8] Y Shan and T Chaolu ldquoA method to select the initial guesssolution auxiliary linear operator and set of basic functions ofhomotopy analysis methodrdquo in Proceedings of the IEEE Inter-national Conference on Intelligent Computing and IntegratedSystems (ICISS rsquo10) pp 410ndash412 Guilin China October 2010
[9] R A van Gorder ldquoStability of auxiliary linear operatorand convergence-control parameter in the homotopy analysismethodrdquo in Advances in Homotopy Analysis Method S J LiaoEd pp 123ndash180 World Scientific Singapore 2014
[10] S M Rassoulinejad-Mousavi and S Abbasbandy ldquoAnalysisof forced convection in a circular tube filled with a darcy-brinkman-forchheimer porous medium using spectral homo-topy analysis methodrdquo Journal of Fluids Engineering Transac-tions of the ASME vol 133 no 10 Article ID 101207 2011
[11] A A Joneidi G Domairry andM Babaelahi ldquoThree analyticalmethods applied to Jeffery-Hamel flowrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 11 pp3423ndash3434 2010
[12] S M Moghimi G Domairry S Soleimani E Ghasemi and HBararnia ldquoApplication of homotopy analysis method to solveMHD Jeffery-Hamel flows in non-parallel wallsrdquo Advances inEngineering Software vol 42 no 3 pp 108ndash113 2011
[13] M Esmaeilpour and D D Ganji ldquoSolution of the Jeffery-Hamel flowproblemby optimal homotopy asymptoticmethodrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3405ndash3411 2010
[14] A Basiri Parsa M M Rashidi O Anwar Beg and S M SadrildquoSemi-computational simulation of magneto-hemodynamicflow in a semi-porous channel using optimal homotopy anddifferential transform methodsrdquo Computers in Biology andMedicine vol 43 no 9 pp 1142ndash1153 2013
[15] Z Ziabakhsh andGDomairry ldquoSolution of the laminar viscousflow in a semi-porous channel in the presence of a uniformmagnetic field by using the homotopy analysis methodrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 4 pp 1284ndash1294 2009
Mathematical Problems in Engineering 15
[16] S SMotsa S Shateyi GMarewo andP Sibanda ldquoAn improvedspectral homotopy analysis method for MHD flow in a semi-porous channelrdquo Numerical Algorithms vol 60 no 3 pp 463ndash481 2012
[17] S Dinarvand and M M Rashidi ldquoA reliable treatment of ahomotopy analysis method for two-dimensional viscous flowin a rectangular domain bounded by twomoving porous wallsrdquoNonlinear Analysis Real World Applications vol 11 no 3 pp1502ndash1512 2010
[18] H Xu Z Lin S Liao JWu and JMajdalani ldquoHomotopy basedsolutions of the Navier-Stokes equations for a porous channelwith orthogonally moving wallsrdquo Physics of Fluids vol 22 no5 Article ID 053601 pp 1ndash18 2010
[19] M M Rashidi E Erfani and B Rostami ldquoOptimal HomotopyAsymptotic Method for solving viscous ow through expandingor contracting gaps with permeable wallsrdquo Transaction on IoTand Cloud Computing vol 2 no 1 pp 85ndash111 2014
[20] Z GMakukula P Sibanda and S S Motsa ldquoA novel numericaltechnique for two-dimensional laminar flow between twomoving porous wallsrdquo Mathematical Problems in Engineeringvol 2010 Article ID 528956 15 pages 2010
[21] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Series in Com-putational Physics Springer New York NY USA 1988
[22] L N Trefethen Spectral Methods in MATLAB SIAM 2000[23] J Majdalani C Zhou and C A Dawson ldquoTwo-dimensional
viscous flow between slowly expanding or contracting wallswith weak permeabilityrdquo Journal of Biomechanics vol 35 no10 pp 1399ndash1403 2002
[16] S SMotsa S Shateyi GMarewo andP Sibanda ldquoAn improvedspectral homotopy analysis method for MHD flow in a semi-porous channelrdquo Numerical Algorithms vol 60 no 3 pp 463ndash481 2012
[17] S Dinarvand and M M Rashidi ldquoA reliable treatment of ahomotopy analysis method for two-dimensional viscous flowin a rectangular domain bounded by twomoving porous wallsrdquoNonlinear Analysis Real World Applications vol 11 no 3 pp1502ndash1512 2010
[18] H Xu Z Lin S Liao JWu and JMajdalani ldquoHomotopy basedsolutions of the Navier-Stokes equations for a porous channelwith orthogonally moving wallsrdquo Physics of Fluids vol 22 no5 Article ID 053601 pp 1ndash18 2010
[19] M M Rashidi E Erfani and B Rostami ldquoOptimal HomotopyAsymptotic Method for solving viscous ow through expandingor contracting gaps with permeable wallsrdquo Transaction on IoTand Cloud Computing vol 2 no 1 pp 85ndash111 2014
[20] Z GMakukula P Sibanda and S S Motsa ldquoA novel numericaltechnique for two-dimensional laminar flow between twomoving porous wallsrdquo Mathematical Problems in Engineeringvol 2010 Article ID 528956 15 pages 2010
[21] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Series in Com-putational Physics Springer New York NY USA 1988
[22] L N Trefethen Spectral Methods in MATLAB SIAM 2000[23] J Majdalani C Zhou and C A Dawson ldquoTwo-dimensional
viscous flow between slowly expanding or contracting wallswith weak permeabilityrdquo Journal of Biomechanics vol 35 no10 pp 1399ndash1403 2002
