Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 879195 9 pageshttpdxdoiorg1011552013879195
Research ArticleOn Extending the Quasilinearization Method toHigher Order Convergent Hybrid Schemes Using the SpectralHomotopy Analysis Method
Sandile S Motsa and Precious Sibanda
School of Mathematics Statistics amp Computer Science University of KwaZulu-Natal Private Bag X01 ScottsvillePietermaritzburg 3209 South Africa
Correspondence should be addressed to Precious Sibanda sibandapukznacza
Received 22 January 2013 Revised 22 March 2013 Accepted 5 April 2013
Academic Editor Saeid Abbasbandy
Copyright copy 2013 S S Motsa and P Sibanda This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We propose a sequence of highly accurate higher order convergent iterative schemes by embedding the quasilinearization algorithmwithin a spectral collocation method The iterative schemes are simple to use and significantly reduce the time and number ofiterations required to find solutions of highly nonlinear boundary value problems to any arbitrary level of accuracy The accuracyand convergence properties of the proposed algorithms are tested numerically by solving three Falkner-Skan type boundary layerflow problems and comparing the results to the most accurate results currently available in the literature We show for instancethat precision of up to 29 significant figures can be attained with no more than 5 iterations of each algorithm
1 Introduction
The quasilinearization method (QLM) was originally devel-oped by Bellman and Kalaba [1] as a generalization of theNewton-Raphsonmethod to provide lower and upper boundsolutions of nonlinear differential equationsThe attraction ofquasilinearization is that the algorithm is easy to understandand themethod generally converges rapidly if the initial guessis close to the true solution
Bellman and Kalaba [1] established that the method con-verges quadratically However the original proof of quadraticconvergence was subject to restrictive conditions of smallstep size and convexity or concavity of nonlinear functionsMaleknejad and Najafi [2] These conditions were subse-quently relaxed and the method generalized to be applicableto a wider class of problems see for instance papers byMandelzweig and his coworkers [3ndash6] and Lakshmikantham[7 8] Parand et al [9] used the quasilinearization methodto solve Volterrarsquos model for population growth in a closedsystem Other uses of the quasilinearization method includeapplication to reaction diffusion equations Jiang and Vatsala
[10] and to Volterra integro-differential equations Ahmad[11] Pandit [12] and Ramos [13]
An often noted disadvantage of quasilinearization is theinstability of the method whenever a poor initial guess ischosen Tuffuor and Labadie [14] To improve the accuracyand convergence of the quasilinearization method for allinitial guesses we propose in this paper to embed the QLMalgorithm within the spectral homotopy analysis method(SHAM) to obtain a sequence of integration schemes witharbitrary higher order convergence
The spectral homotopy analysis method was introducedby Motsa et al [15 16] to address some limitations of thestandard homotopy analysis method of Liao [17 18] by forexample improving the rate of convergence and extendingthe region of validity of solutions The SHAM has been usedto solve nonlinear equations that arise in the study of fluidflow problems and other areas of science and engineeringSibanda et al [19] and Motsa and Sibanda [20]
Abbasbandy [21] and Chun [22] proposed and studiedseveral methods for nonlinear equations with higher orderconvergence by using the Adomian decomposition technique
2 Journal of Applied Mathematics
[23ndash25] Higher order Newton-like iteration formulae for thecomputation of the solutions of nonlinear equationswere alsoderived in Chun [26] using the homotopy analysis methodand by [27] using the homotopy perturbationmethod In thispaper we extend the ideas used for the solution of nonlinearequations to obtain higher order iteration schemes for solvingnonlinear boundary value problems We propose an exten-sion of the quasilinearization method by using the spectralhomotopy analysis method within the QLM algorithm toobtain a sequence of highly accurate and convergent higherorder iterative schemes for solving boundary value problemsFor illustration purposes we have presented three QLM-SHAMhybrid iteration schemes that are used to solve the Bla-sius and Falkner-Skan equationsThe results are compared tothemost accurate skin friction coefficients currently availablein the literature by Boyd [28] and Ganapol [29] Ganapol [29]used an algorithm based on a Maclaurin series with Wynn-epsilon convergence acceleration and analytical continuationto obtain highly accurate skin friction coefficients for theBlasius and Falkner-Skan boundary layer flows The schemesderived in this paper however require neither convergenceacceleration nor analytical continuation to remain steady andaccurate for up to 29 digits of precision In addition thepresent schemes are highly efficient with 29-digit precisionachieved with five or fewer iterations as compared with atleast 104 iterations of Ganapolrsquos algorithm
The structure of this paper is as follows Section 2gives a general framework for the derivation of the hybridquasilinearization-SHAMschemes for the solution of nonlin-ear differential equations Section 3 illustrates the applicationof the three schemes derived in this paper to the solution ofBlasius and Falkner-Skan equations In Section 4 the resultsare presented and comparison made with the most accurateskin friction results for Blasius and Falkner-Skan equationscurrently available in the literature
2 Derivation of the Iterative Schemes
In this section we present a framework for the derivationof general QLM-SHAM iterative schemes for solving one-dimensional nonlinear differential equations We consider ageneral 119899-order nonlinear ordinary differential equation ofthe form
where 120595(119909) is a known function of the independent variable119909 and 119910(119909) is an unknown function The functions 119871 and119865 represent the linear and nonlinear components of thegoverning equation respectively We assume that (1) is to besolved for 119909 isin [119886 119887] subject to the boundary conditions
where 119861119886 and 119861119887 are linear operators
Following [22 27] we assume that the true solution of(1) is 119910120572(119909) and that 119910120574(119909) is an initial approximation thatis sufficiently close to 119910120572(119909) After expanding 119865 using Taylorseries up to first order about 119910120574 119910
where for compactness (sdot sdot sdot ) denotes (119910120574 119910(1)120574 119910
(119899)120574 )
Note that adding (3) and (4) gives (1) Equation (3) can berewritten in the form
L1 [119910 119910(1) 119910
(119899)] +G1 [119910 119910
(1) 119910
(119899)]
= Φ (119910120574 119910(1)
120574 119910(119899)
120574 )
(5)
where
L1 [119910 119910(1) 119910
(119899)]
= 119871 [119910 119910(1) 119910
(119899)] +
119899
sum
119904=0
119910(119904) 120597119865
120597119910(119904)(119910120574 119910
(1)
120574 119910(119899)
120574 )
(6)
Φ(119910120574 119910(1)
120574 119910(119899)
120574 ) =
119899
sum
119904=0
119910(119904)
120574
120597119865
120597119910(119904)(119910120574 119910
(1)
120574 119910(119899)
120574 )
minus 119865 [119910120574 119910(1)
120574 119910(119899)
120574 ] + 120595 (119909)
(7)
G1 [119910 119910(1) 119910
(119899)] = 119866 [119910 119910
(1) 119910
(119899)] (8)
HereG1 is a nonlinear function that is decomposed using thespectral homotopy analysis method [15 16] We define thefollowing zeroth-order deformation equations
(1 minus 119902)L1 [119884 (119909 119902) minus 1199100 (119909)]= 119902ℎ N1 [119884 (119909 119902)]minus Φ (119910120574)
(9)
where 119902 isin [0 1] is an embedding parameter 119884(119909 119902) areunknown functions ℎ is the convergence controlling param-eter and the ldquobarrdquo has been introduced for convenienceto denote the associated function and its 119899 derivatives Forexample
By differentiating the zeroth-order equations (11) 119898 timeswith respect to 119902 setting 119902 = 0 and finally dividing theresulting equations by 119898 (see eg [17 30ndash32]) we obtainthe following119898th order deformation equations
L1 [119910119898 (119909) minus (120594119898 + ℎ) 119910119898minus1 (119909)]
After obtaining solutions for (12) the approximate solutionfor 119910(119909) is determined as the series solution
119910 (119909) =
infin
sum
119898=0
119910119898 (119909) (14)
The SHAM solution is said to be of order 119872 if the previousseries is truncated at119898 = 119872 that is if
119910 (119909) =
119872
sum
119898=0
119910119898 (119909) (15)
The initial approximation 1199100 required for solving thesequence of linear higher order deformation equations (12)is chosen as the solution that results from solving the linearpart of (5) subject to the given boundary conditions (2)Thatis we solve
L1 [1199100 119910(1)
0 119910(119899)
0 ] = Φ (119910120574 119910(1)
120574 119910(119899)
120574 ) (16)
We note that with L1 as defined in (6) (16) cannot besolved exactly by means of analytical techniques Numericalmethods such as finite differences finite element methodand spectral method can be used to solve equations of theform (16) Thus if only the initial approximation is used toapproximate the solution 119910(119909) of the governing nonlineardifferential equation (1) that is if 119910(119909) asymp 1199100(119909) the (119903 + 1)thapproximation of (1) is a solution of
L1 [119910119903+1 119910(1)
119903+1 119910(119899)
119903+1] = Φ (119910120574 119910(1)
120574 119910(119899)
120574 ) (17)
which on using the definitions (6) and (7) can be written as
119871 [119910119903+1 119910(1)
119903+1 119910(119899)
119903+1]
+
119899
sum
119904=0
(119910(119904)
119903+1 minus 119910(119904)
119903 )120597119865
120597119910(119904)(119910119903 119910
(1)
119903 119910(119899)
119903 )
+ 119865 [119910119903 119910(1)
119903 119910(119899)
119903 ] = 0
(18)
We note that the iterative scheme (18) is in fact the quasilin-earization method of Bellman and Kalaba [1] For119872 = 1 wehave
In general for any119872 gt 1 we have119910 (119909) = 1199100 (119909) + 1199101 (119909) + sdot sdot sdot + 119910119872 (119909) (28)
where 119910119872(119909) is obtained as a solution ofL1 [119910119872] = (1 + ℎ)L1 [119910119872minus1] + ℎ119877119872minus1 [1199100 1199101 119910119872minus1]
(29)Thus a general scheme when the SHAM is truncated at order119872 (where119872 ge 1) hereinafter referred to as scheme-119872 canbe obtained asL1 [119910119903+1 119910
Using (35)ndash(37) the first three iterative schemes correspond-ing to119872 = 0 1 2may now be defined as follows
31 Scheme-0 In this scheme we set
L1 (119910119903+1 1199101015840
119903+1 11991010158401015840
119903+1 119910101584010158401015840
119903+1) = Φ (119910119903 1199101015840
119903 11991010158401015840
119903 ) (39)
subject to
119910119903+1 (0) = 1199101015840
119903+1 (0) = 0 1199101015840
119903+1 (infin) = 1 (40)
It is worth noting that Scheme-0 is in fact equivalent to theoriginal QLM algorithm see Mandelzweig and Tabakin [5]and Mandelzweig [6]
32 Scheme-1 For this scheme we set
L1 (119910119903+1 1199101015840
119903+1 11991010158401015840
119903+1 119910101584010158401015840
119903+1)
= Φ (119910119903 1199101015840
119903 11991010158401015840
119903 ) + ℎL1 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1 119910101584010158401015840
0119903+1)
+ ℎ1198770 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1)
(41)
subject to
119910119903+1 (0) = 1199101015840
119903+1 (0) = 0 1199101015840
119903+1 (infin) = 1 (42)
where
1198770 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1)
= 1205731199100119903+111991010158401015840
0119903+1 minus 120573111991010158402
0119903+1 + 1205731 + 212057311199101015840
1199031199101015840
0119903+1
minus 120573 (11991011990311991010158401015840
0119903+1 + 11991010158401015840
119903 1199100119903+1)
(43)
and 1199100119903+1 is the solution of
L1 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1 119910101584010158401015840
0119903+1) = Φ (119910119903 1199101015840
119903 11991010158401015840
119903 ) (44)
subject to
1199100 119903+1 (0) = 1199101015840
0119903+1 (0) = 0 1199101015840
0119903+1 (infin) = 1 (45)
33 Scheme-2 The complexity of the defining equationsincreases with the order of the scheme For Scheme-2 we have
L1 (119910119903+1 1199101015840
119903+1 11991010158401015840
119903+1 119910101584010158401015840
119903+1)
= Φ (119910119903 1199101015840
119903 11991010158401015840
119903 ) + ℎL1 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1 119910101584010158401015840
0119903+1)
+ (1 + ℎ)L1 (1199101119903+1 1199101015840
1119903+1 11991010158401015840
1119903+1 119910101584010158401015840
1119903+1)
+ ℎ1198770 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1)
+ ℎ1198771 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1 1199101119903+1 1199101015840
1119903+1 11991010158401015840
1119903+1)
(46)
Journal of Applied Mathematics 5
subject to
119910119903+1 (0) = 1199101015840
119903+1 (0) = 0 1199101015840
119903+1 (infin) = 1 (47)
where
1198771 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1 1199101119903+1 1199101015840
1119903+1 11991010158401015840
1119903+1)
= 120573 (1199100119903+111991010158401015840
1119903+1 + 1199101119903+111991010158401015840
0119903+1) minus 212057311199101015840
0119903+11199101119903+1
+ 212057311199101015840
1199031199101015840
1119903+1 minus 120573 (11991011990311991010158401015840
1119903+1 + 11991010158401015840
119903 1199101119903+1)
(48)
and 1199101119903+1 is the solution of
L1 (1199101119903+1 1199101015840
1119903+1 11991010158401015840
1119903+1 119910101584010158401015840
1119903+1)
= ℎL1 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1 119910101584010158401015840
0119903+1)
+ ℎ1198770 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1)
(49)
with
1199101119903+1 (0) = 1199101015840
1119903+1 (0) = 0 1199101015840
1119903+1 (infin) = 0 (50)
Equations (39) (41) and (46) describing the three solutionschemes can be solved numerically using standard meth-ods such as finite difference finite elements and splinecollocation methods In this study we use the Chebyshevspectral collocation method to solve the iteration schemes(see [33ndash36]) To allow for numerical implementation ofthe pseudospectral method the physical region [0infin) istruncated to [0 119871] where 119871 is chosen to be sufficiently largeThe truncated region is further transformed to the space[minus1 1] using the transformation
120585 =
2
119871infin
120578 minus 1 (51)
As with any other numerical approximation method somesort of discretization is introduced in the interval [minus1 1] Wechoose the Gauss-Lobatto collocation points to define thenodes in [minus1 1] as
120585119895 = cos(120587119895
119873
) 119895 = 0 1 119873 (52)
where (119873 + 1) is the number of collocation points Theessence of the Chebyshev spectral collocation method isthe idea of introducing a differentiation matrix 119863 Thedifferentiation matrix maps a vector of the function valuesY = [119910(1205850) 119910(120585119873)]
119879 at the collocation points to a vectorY1015840 defined as
Starting from a suitable initial guess 1199100(120578) the iterationscheme (60) can be used to iteratively give approximatesolutions of the governing equation (32) for Scheme-0 Theapplication of the pseudospectral method for Scheme-1 andScheme-2 can be done in a similar manner The initialapproximation used in all the algorithms is
The number of collocation points used in all the resultspresented here is119873 = 200 with 119871infin = 20
4 Results and Discussion
In this section we present solutions of the Falkner-Skan equa-tion (32) using the QLM-SHAM hybrid iteration schemesNumerical simulations were conducted for the followingspecial classes of the F-S equations
To assess the accuracy and performance of our schemesthe numerical results were compared to the recently reportedresults of Ganapol [29] To date these results are the most
accurate results for the Blasius and Falkner-Skan class ofequations Ganapol [29] reported highly accurate resultsbetween 10 and 30 decimal places using a robust algorithmbased onMaclaurin series with convergence acceleration andanalytical continuation techniques
The comparison between the present findings and theresults in the literature is made for the skin friction which isproportional to 11989110158401015840(0) Table 1 shows a comparison betweenthe computed skin friction values of the Blasius equationusing the three QLM-SHAM iteration schemes The resultsare comparedwith the results reported inGanapol [29] whichare accurate to 29 decimal places We observe that all theiteration schemes rapidly converge to the results of [29] toall 29 decimal places Full convergence is achieved after 6iterations when using Scheme-0 4 iterations when usingScheme-1 and after 3 iterations when using Scheme-2 Itis worth noting that the results of [29] were achieved after104 decimal places Prior to Ganapol [29] the most accurateBlasius skin friction results had been published to 17 decimalplaces by Boyd [28] as 11989110158401015840(0) = 033205733621519630 Thisresult was obtained after 5 iterations using Scheme-0 and3 iterations for both Schemes-1 and -2 The value reportedafter 52 iterations in [29] is 11989110158401015840(0) = 03320573362151965It is clear that the proposed iteration schemes convergesignificantly faster than the method of [29] That the results
8 Journal of Applied Mathematics
of Boyd [28] and Ganapol [29] were obtained only after a fewiterations validates both the higher order convergence and theaccuracy of the present solution methods
Table 2 shows the computed values of 11989110158401015840(0) using
Schemes-0 and -1 for the Pohlhausen flow (120573 = 0 1205731 = 1)For this particular flow the exact value of 11989110158401015840(0) is knownto be 2radic3 The iteration Scheme-0 matches the exact resultafter only 5 iterations and Scheme-1 converges after only 3iterationsThemethod used in Ganapol [29] converged to theexact result after 104 iterations This again demonstrates thesuperior convergence of the present method
Table 3 gives the numerical simulations of the skin fric-tion results for theHomann flowWe observe that the 29-digitresults reported in [29] are achieved in 5 iterations 4 itera-tions and 3 iterations for Scheme-0 -1 and -2 respectivelyThis result indicates that adding an additional level in theQLM-SHAM schemewould further significantly increase theconvergence of the iteration scheme We further note fromTables 1ndash3 that all three schemes converge significantly muchfaster than the spectral homotopy analysismethod on its own
5 Conclusion
In this study we presented three hybrid QLM-SHAM itera-tion schemes for the solution of Falkner-Skan type boundarylayer equations We have shown through numerical exper-imentation that the proposed numerical schemes signifi-cantly enhance the convergence rate of the quasilinearizationmethod By comparison with the most accurate solutionsof the Falkner-Skan equations currently available in theliterature we have shown that the schemes are highly accurateand efficient in terms of the number of iterations requiredto determine the solution to the required level of accuracyThe schemes presented provide robust tools for the efficientsolution of nonlinear equations by offering superior accuracyto many existing methods In addition the approach usedin deriving these schemes provides a suitable framework forextension to higher level schemes by adding more terms ofthe SHAM component of the method
References
[1] R E Bellman and R E Kalaba Quasilinearization and Non-linear Boundary-Value Problems Elsevier New York NY USA1965
[2] K Maleknejad and E Najafi ldquoNumerical solution of nonlinearVolterra integral equations using the idea of quasilinearizationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 1 pp 93ndash100 2011
[3] R Krivec and V B Mandelzweig ldquoNumerical investigation ofquasilinearization method in quantum mechanicsrdquo ComputerPhysics Communications vol 138 pp 69ndash79 2001
[4] V BMandelzweig ldquoQuasilinearizationmethod and its verifica-tion on exactly solvablemodels in quantummechanicsrdquo Journalof Mathematical Physics vol 40 no 12 pp 6266ndash6291 1999
[5] V B Mandelzweig and F Tabakin ldquoQuasilinearizationapproach to nonlinear problems in physics with application tononlinear ODEsrdquo Computer Physics Communications vol 141no 2 pp 268ndash281 2001
[6] V B Mandelzweig ldquoQuasilinearization method nonperturba-tive approach to physical problemsrdquo Physics of Atomic Nucleivol 68 no 7 pp 1227ndash1258 2005
[7] V Lakshmikantham S Leela and S Sivasundaram ldquoExtensionsof the method of quasilinearizationrdquo Journal of OptimizationTheory and Applications vol 87 no 2 pp 379ndash401 1995
[8] V Lakshmikantham ldquoFurther improvement of generalizedquasilinearization methodrdquo Nonlinear Analysis Theory Meth-ods amp Applications vol 27 no 2 pp 223ndash227 1996
[9] K Parand M Ghasemi S Rezazadeh A Peiravi A Ghor-banpour and A Tavakoli Golpaygani ldquoQuasilinearizationapproach for solving Volterrarsquos population modelrdquo Applied andComputational Mathematics vol 9 no 1 pp 95ndash103 2010
[10] J Jiang and A S Vatsala ldquoThe quasilinearization method in thesystem of reaction diffusion equationsrdquo Applied Mathematicsand Computation vol 97 no 2-3 pp 223ndash235 1998
[11] B Ahmad ldquoA quasilinearization method for a class of integro-differential equations with mixed nonlinearitiesrdquo NonlinearAnalysis Real World Applications vol 7 no 5 pp 997ndash10042006
[12] S G Pandit ldquoQuadratically converging iterative schemes fornonlinear Volterra integral equations and an applicationrdquo Jour-nal of Applied Mathematics and Stochastic Analysis vol 10 no2 pp 169ndash178 1997
[13] J I Ramos ldquoPiecewise-quasilinearization techniques for singu-larly perturbed Volterra integro-differential equationsrdquo AppliedMathematics and Computation vol 188 no 2 pp 1221ndash12332007
[14] S Tuffuor and J W Labadie ldquoA nonlinear time variant rainfall-runoff model for augmenting monthly datardquo Water ResourcesResearch vol 10 pp 1161ndash1166 1974
[15] 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
[16] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers amp Fluids vol 39 no 7 pp 1219ndash1225 2010
[17] S J Liao Beyond Perturbation Introduction to HomotopyAnalysis Method Chapman amp HallCRC Press 2003
[18] S Li and S-J Liao ldquoAn analytic approach to solve multiplesolutions of a strongly nonlinear problemrdquoAppliedMathematicsand Computation vol 169 no 2 pp 854ndash865 2005
[19] P Sibanda S Motsa and Z Makukula ldquoA spectral-homotopyanalysis method for heat transfer flow of a third grade fluidbetween parallel platesrdquo International Journal of NumericalMethods for Heat amp Fluid Flow vol 22 no 1 pp 4ndash23 2012
[20] S S Motsa and P Sibanda ldquoOn the solution of MHD flowover a nonlinear stretching sheet by an efficient semi-analyticaltechniquerdquo International Journal for Numerical Methods inFluids vol 68 no 12 pp 1524ndash1537 2012
[21] S Abbasbandy ldquoImproving Newton-Raphson method for non-linear equations bymodifiedAdomiandecompositionmethodrdquoApplied Mathematics and Computation vol 145 no 2-3 pp887ndash893 2003
[22] C Chun ldquoIterative methods improving Newtonrsquos method bythe decomposition methodrdquo Computers amp Mathematics withApplications vol 50 no 10ndash12 pp 1559ndash1568 2005
[23] G Adomian ldquoA review of the decomposition method andsome recent results for nonlinear equationsrdquoMathematical andComputer Modelling vol 13 no 7 pp 17ndash43 1990
Journal of Applied Mathematics 9
[24] G Adomian and R Rach ldquoNoise terms in decompositionsolution seriesrdquo Computers amp Mathematics with Applicationsvol 24 no 11 pp 61ndash64 1992
[25] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer Academic Publishers Group BostonMass USA 1994
[26] C Chun ldquoConstruction of Newton-like iteration methods forsolving nonlinear equationsrdquoNumerische Mathematik vol 104no 3 pp 297ndash315 2006
[27] M A Noor ldquoIterative methods for nonlinear equations usinghomotopy perturbation techniquerdquo Applied Mathematics ampInformation Sciences vol 4 no 2 pp 227ndash235 2010
[28] J P Boyd ldquoThe Blasius function computations before com-puters the value of tricks undergraduate projects and openresearch problemsrdquo SIAM Review vol 50 no 4 pp 791ndash8042008
[29] B D Ganapol ldquoHighly accurate solutions of the BlasiusandFalkner-Skan boundary layer equations via convergence accel-erationrdquo httparxivorgabs10063888
[30] A S Bataineh M S M Noorani and I Hashim ldquoSeriessolution of the multispecies Lotka-Volterra equations by meansof the homotopy analysis methodrdquo Differential Equations andNonlinear Mechanics vol 2008 Article ID 816787 14 pages2008
[31] A S Bataineh M S M Noorani and I Hashim ldquoHomotopyanalysis method for singular IVPs of Emden-Fowler typerdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 4 pp 1121ndash1131 2009
[32] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[33] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Berlin Germany1988
[34] W S Don and A Solomonoff ldquoAccuracy and speed in com-puting the Chebyshev collocation derivativerdquo SIAM Journal onScientific Computing vol 16 no 6 pp 1253ndash1268 1995
[35] L N Trefethen Spectral Methods in MATLAB SIAM 2000[36] J A CWeideman and S C Reddy ldquoAMATLAB differentiation
matrix suiterdquoACMTransactions onMathematical Software vol26 no 4 pp 465ndash519 2000
[23ndash25] Higher order Newton-like iteration formulae for thecomputation of the solutions of nonlinear equationswere alsoderived in Chun [26] using the homotopy analysis methodand by [27] using the homotopy perturbationmethod In thispaper we extend the ideas used for the solution of nonlinearequations to obtain higher order iteration schemes for solvingnonlinear boundary value problems We propose an exten-sion of the quasilinearization method by using the spectralhomotopy analysis method within the QLM algorithm toobtain a sequence of highly accurate and convergent higherorder iterative schemes for solving boundary value problemsFor illustration purposes we have presented three QLM-SHAMhybrid iteration schemes that are used to solve the Bla-sius and Falkner-Skan equationsThe results are compared tothemost accurate skin friction coefficients currently availablein the literature by Boyd [28] and Ganapol [29] Ganapol [29]used an algorithm based on a Maclaurin series with Wynn-epsilon convergence acceleration and analytical continuationto obtain highly accurate skin friction coefficients for theBlasius and Falkner-Skan boundary layer flows The schemesderived in this paper however require neither convergenceacceleration nor analytical continuation to remain steady andaccurate for up to 29 digits of precision In addition thepresent schemes are highly efficient with 29-digit precisionachieved with five or fewer iterations as compared with atleast 104 iterations of Ganapolrsquos algorithm
The structure of this paper is as follows Section 2gives a general framework for the derivation of the hybridquasilinearization-SHAMschemes for the solution of nonlin-ear differential equations Section 3 illustrates the applicationof the three schemes derived in this paper to the solution ofBlasius and Falkner-Skan equations In Section 4 the resultsare presented and comparison made with the most accurateskin friction results for Blasius and Falkner-Skan equationscurrently available in the literature
2 Derivation of the Iterative Schemes
In this section we present a framework for the derivationof general QLM-SHAM iterative schemes for solving one-dimensional nonlinear differential equations We consider ageneral 119899-order nonlinear ordinary differential equation ofthe form
where 120595(119909) is a known function of the independent variable119909 and 119910(119909) is an unknown function The functions 119871 and119865 represent the linear and nonlinear components of thegoverning equation respectively We assume that (1) is to besolved for 119909 isin [119886 119887] subject to the boundary conditions
where 119861119886 and 119861119887 are linear operators
Following [22 27] we assume that the true solution of(1) is 119910120572(119909) and that 119910120574(119909) is an initial approximation thatis sufficiently close to 119910120572(119909) After expanding 119865 using Taylorseries up to first order about 119910120574 119910
where for compactness (sdot sdot sdot ) denotes (119910120574 119910(1)120574 119910
(119899)120574 )
Note that adding (3) and (4) gives (1) Equation (3) can berewritten in the form
L1 [119910 119910(1) 119910
(119899)] +G1 [119910 119910
(1) 119910
(119899)]
= Φ (119910120574 119910(1)
120574 119910(119899)
120574 )
(5)
where
L1 [119910 119910(1) 119910
(119899)]
= 119871 [119910 119910(1) 119910
(119899)] +
119899
sum
119904=0
119910(119904) 120597119865
120597119910(119904)(119910120574 119910
(1)
120574 119910(119899)
120574 )
(6)
Φ(119910120574 119910(1)
120574 119910(119899)
120574 ) =
119899
sum
119904=0
119910(119904)
120574
120597119865
120597119910(119904)(119910120574 119910
(1)
120574 119910(119899)
120574 )
minus 119865 [119910120574 119910(1)
120574 119910(119899)
120574 ] + 120595 (119909)
(7)
G1 [119910 119910(1) 119910
(119899)] = 119866 [119910 119910
(1) 119910
(119899)] (8)
HereG1 is a nonlinear function that is decomposed using thespectral homotopy analysis method [15 16] We define thefollowing zeroth-order deformation equations
(1 minus 119902)L1 [119884 (119909 119902) minus 1199100 (119909)]= 119902ℎ N1 [119884 (119909 119902)]minus Φ (119910120574)
(9)
where 119902 isin [0 1] is an embedding parameter 119884(119909 119902) areunknown functions ℎ is the convergence controlling param-eter and the ldquobarrdquo has been introduced for convenienceto denote the associated function and its 119899 derivatives Forexample
By differentiating the zeroth-order equations (11) 119898 timeswith respect to 119902 setting 119902 = 0 and finally dividing theresulting equations by 119898 (see eg [17 30ndash32]) we obtainthe following119898th order deformation equations
L1 [119910119898 (119909) minus (120594119898 + ℎ) 119910119898minus1 (119909)]
After obtaining solutions for (12) the approximate solutionfor 119910(119909) is determined as the series solution
119910 (119909) =
infin
sum
119898=0
119910119898 (119909) (14)
The SHAM solution is said to be of order 119872 if the previousseries is truncated at119898 = 119872 that is if
119910 (119909) =
119872
sum
119898=0
119910119898 (119909) (15)
The initial approximation 1199100 required for solving thesequence of linear higher order deformation equations (12)is chosen as the solution that results from solving the linearpart of (5) subject to the given boundary conditions (2)Thatis we solve
L1 [1199100 119910(1)
0 119910(119899)
0 ] = Φ (119910120574 119910(1)
120574 119910(119899)
120574 ) (16)
We note that with L1 as defined in (6) (16) cannot besolved exactly by means of analytical techniques Numericalmethods such as finite differences finite element methodand spectral method can be used to solve equations of theform (16) Thus if only the initial approximation is used toapproximate the solution 119910(119909) of the governing nonlineardifferential equation (1) that is if 119910(119909) asymp 1199100(119909) the (119903 + 1)thapproximation of (1) is a solution of
L1 [119910119903+1 119910(1)
119903+1 119910(119899)
119903+1] = Φ (119910120574 119910(1)
120574 119910(119899)
120574 ) (17)
which on using the definitions (6) and (7) can be written as
119871 [119910119903+1 119910(1)
119903+1 119910(119899)
119903+1]
+
119899
sum
119904=0
(119910(119904)
119903+1 minus 119910(119904)
119903 )120597119865
120597119910(119904)(119910119903 119910
(1)
119903 119910(119899)
119903 )
+ 119865 [119910119903 119910(1)
119903 119910(119899)
119903 ] = 0
(18)
We note that the iterative scheme (18) is in fact the quasilin-earization method of Bellman and Kalaba [1] For119872 = 1 wehave
In general for any119872 gt 1 we have119910 (119909) = 1199100 (119909) + 1199101 (119909) + sdot sdot sdot + 119910119872 (119909) (28)
where 119910119872(119909) is obtained as a solution ofL1 [119910119872] = (1 + ℎ)L1 [119910119872minus1] + ℎ119877119872minus1 [1199100 1199101 119910119872minus1]
(29)Thus a general scheme when the SHAM is truncated at order119872 (where119872 ge 1) hereinafter referred to as scheme-119872 canbe obtained asL1 [119910119903+1 119910
Using (35)ndash(37) the first three iterative schemes correspond-ing to119872 = 0 1 2may now be defined as follows
31 Scheme-0 In this scheme we set
L1 (119910119903+1 1199101015840
119903+1 11991010158401015840
119903+1 119910101584010158401015840
119903+1) = Φ (119910119903 1199101015840
119903 11991010158401015840
119903 ) (39)
subject to
119910119903+1 (0) = 1199101015840
119903+1 (0) = 0 1199101015840
119903+1 (infin) = 1 (40)
It is worth noting that Scheme-0 is in fact equivalent to theoriginal QLM algorithm see Mandelzweig and Tabakin [5]and Mandelzweig [6]
32 Scheme-1 For this scheme we set
L1 (119910119903+1 1199101015840
119903+1 11991010158401015840
119903+1 119910101584010158401015840
119903+1)
= Φ (119910119903 1199101015840
119903 11991010158401015840
119903 ) + ℎL1 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1 119910101584010158401015840
0119903+1)
+ ℎ1198770 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1)
(41)
subject to
119910119903+1 (0) = 1199101015840
119903+1 (0) = 0 1199101015840
119903+1 (infin) = 1 (42)
where
1198770 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1)
= 1205731199100119903+111991010158401015840
0119903+1 minus 120573111991010158402
0119903+1 + 1205731 + 212057311199101015840
1199031199101015840
0119903+1
minus 120573 (11991011990311991010158401015840
0119903+1 + 11991010158401015840
119903 1199100119903+1)
(43)
and 1199100119903+1 is the solution of
L1 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1 119910101584010158401015840
0119903+1) = Φ (119910119903 1199101015840
119903 11991010158401015840
119903 ) (44)
subject to
1199100 119903+1 (0) = 1199101015840
0119903+1 (0) = 0 1199101015840
0119903+1 (infin) = 1 (45)
33 Scheme-2 The complexity of the defining equationsincreases with the order of the scheme For Scheme-2 we have
L1 (119910119903+1 1199101015840
119903+1 11991010158401015840
119903+1 119910101584010158401015840
119903+1)
= Φ (119910119903 1199101015840
119903 11991010158401015840
119903 ) + ℎL1 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1 119910101584010158401015840
0119903+1)
+ (1 + ℎ)L1 (1199101119903+1 1199101015840
1119903+1 11991010158401015840
1119903+1 119910101584010158401015840
1119903+1)
+ ℎ1198770 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1)
+ ℎ1198771 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1 1199101119903+1 1199101015840
1119903+1 11991010158401015840
1119903+1)
(46)
Journal of Applied Mathematics 5
subject to
119910119903+1 (0) = 1199101015840
119903+1 (0) = 0 1199101015840
119903+1 (infin) = 1 (47)
where
1198771 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1 1199101119903+1 1199101015840
1119903+1 11991010158401015840
1119903+1)
= 120573 (1199100119903+111991010158401015840
1119903+1 + 1199101119903+111991010158401015840
0119903+1) minus 212057311199101015840
0119903+11199101119903+1
+ 212057311199101015840
1199031199101015840
1119903+1 minus 120573 (11991011990311991010158401015840
1119903+1 + 11991010158401015840
119903 1199101119903+1)
(48)
and 1199101119903+1 is the solution of
L1 (1199101119903+1 1199101015840
1119903+1 11991010158401015840
1119903+1 119910101584010158401015840
1119903+1)
= ℎL1 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1 119910101584010158401015840
0119903+1)
+ ℎ1198770 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1)
(49)
with
1199101119903+1 (0) = 1199101015840
1119903+1 (0) = 0 1199101015840
1119903+1 (infin) = 0 (50)
Equations (39) (41) and (46) describing the three solutionschemes can be solved numerically using standard meth-ods such as finite difference finite elements and splinecollocation methods In this study we use the Chebyshevspectral collocation method to solve the iteration schemes(see [33ndash36]) To allow for numerical implementation ofthe pseudospectral method the physical region [0infin) istruncated to [0 119871] where 119871 is chosen to be sufficiently largeThe truncated region is further transformed to the space[minus1 1] using the transformation
120585 =
2
119871infin
120578 minus 1 (51)
As with any other numerical approximation method somesort of discretization is introduced in the interval [minus1 1] Wechoose the Gauss-Lobatto collocation points to define thenodes in [minus1 1] as
120585119895 = cos(120587119895
119873
) 119895 = 0 1 119873 (52)
where (119873 + 1) is the number of collocation points Theessence of the Chebyshev spectral collocation method isthe idea of introducing a differentiation matrix 119863 Thedifferentiation matrix maps a vector of the function valuesY = [119910(1205850) 119910(120585119873)]
119879 at the collocation points to a vectorY1015840 defined as
Starting from a suitable initial guess 1199100(120578) the iterationscheme (60) can be used to iteratively give approximatesolutions of the governing equation (32) for Scheme-0 Theapplication of the pseudospectral method for Scheme-1 andScheme-2 can be done in a similar manner The initialapproximation used in all the algorithms is
The number of collocation points used in all the resultspresented here is119873 = 200 with 119871infin = 20
4 Results and Discussion
In this section we present solutions of the Falkner-Skan equa-tion (32) using the QLM-SHAM hybrid iteration schemesNumerical simulations were conducted for the followingspecial classes of the F-S equations
To assess the accuracy and performance of our schemesthe numerical results were compared to the recently reportedresults of Ganapol [29] To date these results are the most
accurate results for the Blasius and Falkner-Skan class ofequations Ganapol [29] reported highly accurate resultsbetween 10 and 30 decimal places using a robust algorithmbased onMaclaurin series with convergence acceleration andanalytical continuation techniques
The comparison between the present findings and theresults in the literature is made for the skin friction which isproportional to 11989110158401015840(0) Table 1 shows a comparison betweenthe computed skin friction values of the Blasius equationusing the three QLM-SHAM iteration schemes The resultsare comparedwith the results reported inGanapol [29] whichare accurate to 29 decimal places We observe that all theiteration schemes rapidly converge to the results of [29] toall 29 decimal places Full convergence is achieved after 6iterations when using Scheme-0 4 iterations when usingScheme-1 and after 3 iterations when using Scheme-2 Itis worth noting that the results of [29] were achieved after104 decimal places Prior to Ganapol [29] the most accurateBlasius skin friction results had been published to 17 decimalplaces by Boyd [28] as 11989110158401015840(0) = 033205733621519630 Thisresult was obtained after 5 iterations using Scheme-0 and3 iterations for both Schemes-1 and -2 The value reportedafter 52 iterations in [29] is 11989110158401015840(0) = 03320573362151965It is clear that the proposed iteration schemes convergesignificantly faster than the method of [29] That the results
8 Journal of Applied Mathematics
of Boyd [28] and Ganapol [29] were obtained only after a fewiterations validates both the higher order convergence and theaccuracy of the present solution methods
Table 2 shows the computed values of 11989110158401015840(0) using
Schemes-0 and -1 for the Pohlhausen flow (120573 = 0 1205731 = 1)For this particular flow the exact value of 11989110158401015840(0) is knownto be 2radic3 The iteration Scheme-0 matches the exact resultafter only 5 iterations and Scheme-1 converges after only 3iterationsThemethod used in Ganapol [29] converged to theexact result after 104 iterations This again demonstrates thesuperior convergence of the present method
Table 3 gives the numerical simulations of the skin fric-tion results for theHomann flowWe observe that the 29-digitresults reported in [29] are achieved in 5 iterations 4 itera-tions and 3 iterations for Scheme-0 -1 and -2 respectivelyThis result indicates that adding an additional level in theQLM-SHAM schemewould further significantly increase theconvergence of the iteration scheme We further note fromTables 1ndash3 that all three schemes converge significantly muchfaster than the spectral homotopy analysismethod on its own
5 Conclusion
In this study we presented three hybrid QLM-SHAM itera-tion schemes for the solution of Falkner-Skan type boundarylayer equations We have shown through numerical exper-imentation that the proposed numerical schemes signifi-cantly enhance the convergence rate of the quasilinearizationmethod By comparison with the most accurate solutionsof the Falkner-Skan equations currently available in theliterature we have shown that the schemes are highly accurateand efficient in terms of the number of iterations requiredto determine the solution to the required level of accuracyThe schemes presented provide robust tools for the efficientsolution of nonlinear equations by offering superior accuracyto many existing methods In addition the approach usedin deriving these schemes provides a suitable framework forextension to higher level schemes by adding more terms ofthe SHAM component of the method
References
[1] R E Bellman and R E Kalaba Quasilinearization and Non-linear Boundary-Value Problems Elsevier New York NY USA1965
[2] K Maleknejad and E Najafi ldquoNumerical solution of nonlinearVolterra integral equations using the idea of quasilinearizationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 1 pp 93ndash100 2011
[3] R Krivec and V B Mandelzweig ldquoNumerical investigation ofquasilinearization method in quantum mechanicsrdquo ComputerPhysics Communications vol 138 pp 69ndash79 2001
[4] V BMandelzweig ldquoQuasilinearizationmethod and its verifica-tion on exactly solvablemodels in quantummechanicsrdquo Journalof Mathematical Physics vol 40 no 12 pp 6266ndash6291 1999
[5] V B Mandelzweig and F Tabakin ldquoQuasilinearizationapproach to nonlinear problems in physics with application tononlinear ODEsrdquo Computer Physics Communications vol 141no 2 pp 268ndash281 2001
[6] V B Mandelzweig ldquoQuasilinearization method nonperturba-tive approach to physical problemsrdquo Physics of Atomic Nucleivol 68 no 7 pp 1227ndash1258 2005
[7] V Lakshmikantham S Leela and S Sivasundaram ldquoExtensionsof the method of quasilinearizationrdquo Journal of OptimizationTheory and Applications vol 87 no 2 pp 379ndash401 1995
[8] V Lakshmikantham ldquoFurther improvement of generalizedquasilinearization methodrdquo Nonlinear Analysis Theory Meth-ods amp Applications vol 27 no 2 pp 223ndash227 1996
[9] K Parand M Ghasemi S Rezazadeh A Peiravi A Ghor-banpour and A Tavakoli Golpaygani ldquoQuasilinearizationapproach for solving Volterrarsquos population modelrdquo Applied andComputational Mathematics vol 9 no 1 pp 95ndash103 2010
[10] J Jiang and A S Vatsala ldquoThe quasilinearization method in thesystem of reaction diffusion equationsrdquo Applied Mathematicsand Computation vol 97 no 2-3 pp 223ndash235 1998
[11] B Ahmad ldquoA quasilinearization method for a class of integro-differential equations with mixed nonlinearitiesrdquo NonlinearAnalysis Real World Applications vol 7 no 5 pp 997ndash10042006
[12] S G Pandit ldquoQuadratically converging iterative schemes fornonlinear Volterra integral equations and an applicationrdquo Jour-nal of Applied Mathematics and Stochastic Analysis vol 10 no2 pp 169ndash178 1997
[13] J I Ramos ldquoPiecewise-quasilinearization techniques for singu-larly perturbed Volterra integro-differential equationsrdquo AppliedMathematics and Computation vol 188 no 2 pp 1221ndash12332007
[14] S Tuffuor and J W Labadie ldquoA nonlinear time variant rainfall-runoff model for augmenting monthly datardquo Water ResourcesResearch vol 10 pp 1161ndash1166 1974
[15] 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
[16] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers amp Fluids vol 39 no 7 pp 1219ndash1225 2010
[17] S J Liao Beyond Perturbation Introduction to HomotopyAnalysis Method Chapman amp HallCRC Press 2003
[18] S Li and S-J Liao ldquoAn analytic approach to solve multiplesolutions of a strongly nonlinear problemrdquoAppliedMathematicsand Computation vol 169 no 2 pp 854ndash865 2005
[19] P Sibanda S Motsa and Z Makukula ldquoA spectral-homotopyanalysis method for heat transfer flow of a third grade fluidbetween parallel platesrdquo International Journal of NumericalMethods for Heat amp Fluid Flow vol 22 no 1 pp 4ndash23 2012
[20] S S Motsa and P Sibanda ldquoOn the solution of MHD flowover a nonlinear stretching sheet by an efficient semi-analyticaltechniquerdquo International Journal for Numerical Methods inFluids vol 68 no 12 pp 1524ndash1537 2012
[21] S Abbasbandy ldquoImproving Newton-Raphson method for non-linear equations bymodifiedAdomiandecompositionmethodrdquoApplied Mathematics and Computation vol 145 no 2-3 pp887ndash893 2003
[22] C Chun ldquoIterative methods improving Newtonrsquos method bythe decomposition methodrdquo Computers amp Mathematics withApplications vol 50 no 10ndash12 pp 1559ndash1568 2005
[23] G Adomian ldquoA review of the decomposition method andsome recent results for nonlinear equationsrdquoMathematical andComputer Modelling vol 13 no 7 pp 17ndash43 1990
Journal of Applied Mathematics 9
[24] G Adomian and R Rach ldquoNoise terms in decompositionsolution seriesrdquo Computers amp Mathematics with Applicationsvol 24 no 11 pp 61ndash64 1992
[25] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer Academic Publishers Group BostonMass USA 1994
[26] C Chun ldquoConstruction of Newton-like iteration methods forsolving nonlinear equationsrdquoNumerische Mathematik vol 104no 3 pp 297ndash315 2006
[27] M A Noor ldquoIterative methods for nonlinear equations usinghomotopy perturbation techniquerdquo Applied Mathematics ampInformation Sciences vol 4 no 2 pp 227ndash235 2010
[28] J P Boyd ldquoThe Blasius function computations before com-puters the value of tricks undergraduate projects and openresearch problemsrdquo SIAM Review vol 50 no 4 pp 791ndash8042008
[29] B D Ganapol ldquoHighly accurate solutions of the BlasiusandFalkner-Skan boundary layer equations via convergence accel-erationrdquo httparxivorgabs10063888
[30] A S Bataineh M S M Noorani and I Hashim ldquoSeriessolution of the multispecies Lotka-Volterra equations by meansof the homotopy analysis methodrdquo Differential Equations andNonlinear Mechanics vol 2008 Article ID 816787 14 pages2008
[31] A S Bataineh M S M Noorani and I Hashim ldquoHomotopyanalysis method for singular IVPs of Emden-Fowler typerdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 4 pp 1121ndash1131 2009
[32] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[33] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Berlin Germany1988
[34] W S Don and A Solomonoff ldquoAccuracy and speed in com-puting the Chebyshev collocation derivativerdquo SIAM Journal onScientific Computing vol 16 no 6 pp 1253ndash1268 1995
[35] L N Trefethen Spectral Methods in MATLAB SIAM 2000[36] J A CWeideman and S C Reddy ldquoAMATLAB differentiation
matrix suiterdquoACMTransactions onMathematical Software vol26 no 4 pp 465ndash519 2000
By differentiating the zeroth-order equations (11) 119898 timeswith respect to 119902 setting 119902 = 0 and finally dividing theresulting equations by 119898 (see eg [17 30ndash32]) we obtainthe following119898th order deformation equations
L1 [119910119898 (119909) minus (120594119898 + ℎ) 119910119898minus1 (119909)]
After obtaining solutions for (12) the approximate solutionfor 119910(119909) is determined as the series solution
119910 (119909) =
infin
sum
119898=0
119910119898 (119909) (14)
The SHAM solution is said to be of order 119872 if the previousseries is truncated at119898 = 119872 that is if
119910 (119909) =
119872
sum
119898=0
119910119898 (119909) (15)
The initial approximation 1199100 required for solving thesequence of linear higher order deformation equations (12)is chosen as the solution that results from solving the linearpart of (5) subject to the given boundary conditions (2)Thatis we solve
L1 [1199100 119910(1)
0 119910(119899)
0 ] = Φ (119910120574 119910(1)
120574 119910(119899)
120574 ) (16)
We note that with L1 as defined in (6) (16) cannot besolved exactly by means of analytical techniques Numericalmethods such as finite differences finite element methodand spectral method can be used to solve equations of theform (16) Thus if only the initial approximation is used toapproximate the solution 119910(119909) of the governing nonlineardifferential equation (1) that is if 119910(119909) asymp 1199100(119909) the (119903 + 1)thapproximation of (1) is a solution of
L1 [119910119903+1 119910(1)
119903+1 119910(119899)
119903+1] = Φ (119910120574 119910(1)
120574 119910(119899)
120574 ) (17)
which on using the definitions (6) and (7) can be written as
119871 [119910119903+1 119910(1)
119903+1 119910(119899)
119903+1]
+
119899
sum
119904=0
(119910(119904)
119903+1 minus 119910(119904)
119903 )120597119865
120597119910(119904)(119910119903 119910
(1)
119903 119910(119899)
119903 )
+ 119865 [119910119903 119910(1)
119903 119910(119899)
119903 ] = 0
(18)
We note that the iterative scheme (18) is in fact the quasilin-earization method of Bellman and Kalaba [1] For119872 = 1 wehave
In general for any119872 gt 1 we have119910 (119909) = 1199100 (119909) + 1199101 (119909) + sdot sdot sdot + 119910119872 (119909) (28)
where 119910119872(119909) is obtained as a solution ofL1 [119910119872] = (1 + ℎ)L1 [119910119872minus1] + ℎ119877119872minus1 [1199100 1199101 119910119872minus1]
(29)Thus a general scheme when the SHAM is truncated at order119872 (where119872 ge 1) hereinafter referred to as scheme-119872 canbe obtained asL1 [119910119903+1 119910
Using (35)ndash(37) the first three iterative schemes correspond-ing to119872 = 0 1 2may now be defined as follows
31 Scheme-0 In this scheme we set
L1 (119910119903+1 1199101015840
119903+1 11991010158401015840
119903+1 119910101584010158401015840
119903+1) = Φ (119910119903 1199101015840
119903 11991010158401015840
119903 ) (39)
subject to
119910119903+1 (0) = 1199101015840
119903+1 (0) = 0 1199101015840
119903+1 (infin) = 1 (40)
It is worth noting that Scheme-0 is in fact equivalent to theoriginal QLM algorithm see Mandelzweig and Tabakin [5]and Mandelzweig [6]
32 Scheme-1 For this scheme we set
L1 (119910119903+1 1199101015840
119903+1 11991010158401015840
119903+1 119910101584010158401015840
119903+1)
= Φ (119910119903 1199101015840
119903 11991010158401015840
119903 ) + ℎL1 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1 119910101584010158401015840
0119903+1)
+ ℎ1198770 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1)
(41)
subject to
119910119903+1 (0) = 1199101015840
119903+1 (0) = 0 1199101015840
119903+1 (infin) = 1 (42)
where
1198770 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1)
= 1205731199100119903+111991010158401015840
0119903+1 minus 120573111991010158402
0119903+1 + 1205731 + 212057311199101015840
1199031199101015840
0119903+1
minus 120573 (11991011990311991010158401015840
0119903+1 + 11991010158401015840
119903 1199100119903+1)
(43)
and 1199100119903+1 is the solution of
L1 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1 119910101584010158401015840
0119903+1) = Φ (119910119903 1199101015840
119903 11991010158401015840
119903 ) (44)
subject to
1199100 119903+1 (0) = 1199101015840
0119903+1 (0) = 0 1199101015840
0119903+1 (infin) = 1 (45)
33 Scheme-2 The complexity of the defining equationsincreases with the order of the scheme For Scheme-2 we have
L1 (119910119903+1 1199101015840
119903+1 11991010158401015840
119903+1 119910101584010158401015840
119903+1)
= Φ (119910119903 1199101015840
119903 11991010158401015840
119903 ) + ℎL1 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1 119910101584010158401015840
0119903+1)
+ (1 + ℎ)L1 (1199101119903+1 1199101015840
1119903+1 11991010158401015840
1119903+1 119910101584010158401015840
1119903+1)
+ ℎ1198770 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1)
+ ℎ1198771 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1 1199101119903+1 1199101015840
1119903+1 11991010158401015840
1119903+1)
(46)
Journal of Applied Mathematics 5
subject to
119910119903+1 (0) = 1199101015840
119903+1 (0) = 0 1199101015840
119903+1 (infin) = 1 (47)
where
1198771 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1 1199101119903+1 1199101015840
1119903+1 11991010158401015840
1119903+1)
= 120573 (1199100119903+111991010158401015840
1119903+1 + 1199101119903+111991010158401015840
0119903+1) minus 212057311199101015840
0119903+11199101119903+1
+ 212057311199101015840
1199031199101015840
1119903+1 minus 120573 (11991011990311991010158401015840
1119903+1 + 11991010158401015840
119903 1199101119903+1)
(48)
and 1199101119903+1 is the solution of
L1 (1199101119903+1 1199101015840
1119903+1 11991010158401015840
1119903+1 119910101584010158401015840
1119903+1)
= ℎL1 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1 119910101584010158401015840
0119903+1)
+ ℎ1198770 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1)
(49)
with
1199101119903+1 (0) = 1199101015840
1119903+1 (0) = 0 1199101015840
1119903+1 (infin) = 0 (50)
Equations (39) (41) and (46) describing the three solutionschemes can be solved numerically using standard meth-ods such as finite difference finite elements and splinecollocation methods In this study we use the Chebyshevspectral collocation method to solve the iteration schemes(see [33ndash36]) To allow for numerical implementation ofthe pseudospectral method the physical region [0infin) istruncated to [0 119871] where 119871 is chosen to be sufficiently largeThe truncated region is further transformed to the space[minus1 1] using the transformation
120585 =
2
119871infin
120578 minus 1 (51)
As with any other numerical approximation method somesort of discretization is introduced in the interval [minus1 1] Wechoose the Gauss-Lobatto collocation points to define thenodes in [minus1 1] as
120585119895 = cos(120587119895
119873
) 119895 = 0 1 119873 (52)
where (119873 + 1) is the number of collocation points Theessence of the Chebyshev spectral collocation method isthe idea of introducing a differentiation matrix 119863 Thedifferentiation matrix maps a vector of the function valuesY = [119910(1205850) 119910(120585119873)]
119879 at the collocation points to a vectorY1015840 defined as
Starting from a suitable initial guess 1199100(120578) the iterationscheme (60) can be used to iteratively give approximatesolutions of the governing equation (32) for Scheme-0 Theapplication of the pseudospectral method for Scheme-1 andScheme-2 can be done in a similar manner The initialapproximation used in all the algorithms is
The number of collocation points used in all the resultspresented here is119873 = 200 with 119871infin = 20
4 Results and Discussion
In this section we present solutions of the Falkner-Skan equa-tion (32) using the QLM-SHAM hybrid iteration schemesNumerical simulations were conducted for the followingspecial classes of the F-S equations
To assess the accuracy and performance of our schemesthe numerical results were compared to the recently reportedresults of Ganapol [29] To date these results are the most
accurate results for the Blasius and Falkner-Skan class ofequations Ganapol [29] reported highly accurate resultsbetween 10 and 30 decimal places using a robust algorithmbased onMaclaurin series with convergence acceleration andanalytical continuation techniques
The comparison between the present findings and theresults in the literature is made for the skin friction which isproportional to 11989110158401015840(0) Table 1 shows a comparison betweenthe computed skin friction values of the Blasius equationusing the three QLM-SHAM iteration schemes The resultsare comparedwith the results reported inGanapol [29] whichare accurate to 29 decimal places We observe that all theiteration schemes rapidly converge to the results of [29] toall 29 decimal places Full convergence is achieved after 6iterations when using Scheme-0 4 iterations when usingScheme-1 and after 3 iterations when using Scheme-2 Itis worth noting that the results of [29] were achieved after104 decimal places Prior to Ganapol [29] the most accurateBlasius skin friction results had been published to 17 decimalplaces by Boyd [28] as 11989110158401015840(0) = 033205733621519630 Thisresult was obtained after 5 iterations using Scheme-0 and3 iterations for both Schemes-1 and -2 The value reportedafter 52 iterations in [29] is 11989110158401015840(0) = 03320573362151965It is clear that the proposed iteration schemes convergesignificantly faster than the method of [29] That the results
8 Journal of Applied Mathematics
of Boyd [28] and Ganapol [29] were obtained only after a fewiterations validates both the higher order convergence and theaccuracy of the present solution methods
Table 2 shows the computed values of 11989110158401015840(0) using
Schemes-0 and -1 for the Pohlhausen flow (120573 = 0 1205731 = 1)For this particular flow the exact value of 11989110158401015840(0) is knownto be 2radic3 The iteration Scheme-0 matches the exact resultafter only 5 iterations and Scheme-1 converges after only 3iterationsThemethod used in Ganapol [29] converged to theexact result after 104 iterations This again demonstrates thesuperior convergence of the present method
Table 3 gives the numerical simulations of the skin fric-tion results for theHomann flowWe observe that the 29-digitresults reported in [29] are achieved in 5 iterations 4 itera-tions and 3 iterations for Scheme-0 -1 and -2 respectivelyThis result indicates that adding an additional level in theQLM-SHAM schemewould further significantly increase theconvergence of the iteration scheme We further note fromTables 1ndash3 that all three schemes converge significantly muchfaster than the spectral homotopy analysismethod on its own
5 Conclusion
In this study we presented three hybrid QLM-SHAM itera-tion schemes for the solution of Falkner-Skan type boundarylayer equations We have shown through numerical exper-imentation that the proposed numerical schemes signifi-cantly enhance the convergence rate of the quasilinearizationmethod By comparison with the most accurate solutionsof the Falkner-Skan equations currently available in theliterature we have shown that the schemes are highly accurateand efficient in terms of the number of iterations requiredto determine the solution to the required level of accuracyThe schemes presented provide robust tools for the efficientsolution of nonlinear equations by offering superior accuracyto many existing methods In addition the approach usedin deriving these schemes provides a suitable framework forextension to higher level schemes by adding more terms ofthe SHAM component of the method
References
[1] R E Bellman and R E Kalaba Quasilinearization and Non-linear Boundary-Value Problems Elsevier New York NY USA1965
[2] K Maleknejad and E Najafi ldquoNumerical solution of nonlinearVolterra integral equations using the idea of quasilinearizationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 1 pp 93ndash100 2011
[3] R Krivec and V B Mandelzweig ldquoNumerical investigation ofquasilinearization method in quantum mechanicsrdquo ComputerPhysics Communications vol 138 pp 69ndash79 2001
[4] V BMandelzweig ldquoQuasilinearizationmethod and its verifica-tion on exactly solvablemodels in quantummechanicsrdquo Journalof Mathematical Physics vol 40 no 12 pp 6266ndash6291 1999
[5] V B Mandelzweig and F Tabakin ldquoQuasilinearizationapproach to nonlinear problems in physics with application tononlinear ODEsrdquo Computer Physics Communications vol 141no 2 pp 268ndash281 2001
[6] V B Mandelzweig ldquoQuasilinearization method nonperturba-tive approach to physical problemsrdquo Physics of Atomic Nucleivol 68 no 7 pp 1227ndash1258 2005
[7] V Lakshmikantham S Leela and S Sivasundaram ldquoExtensionsof the method of quasilinearizationrdquo Journal of OptimizationTheory and Applications vol 87 no 2 pp 379ndash401 1995
[8] V Lakshmikantham ldquoFurther improvement of generalizedquasilinearization methodrdquo Nonlinear Analysis Theory Meth-ods amp Applications vol 27 no 2 pp 223ndash227 1996
[9] K Parand M Ghasemi S Rezazadeh A Peiravi A Ghor-banpour and A Tavakoli Golpaygani ldquoQuasilinearizationapproach for solving Volterrarsquos population modelrdquo Applied andComputational Mathematics vol 9 no 1 pp 95ndash103 2010
[10] J Jiang and A S Vatsala ldquoThe quasilinearization method in thesystem of reaction diffusion equationsrdquo Applied Mathematicsand Computation vol 97 no 2-3 pp 223ndash235 1998
[11] B Ahmad ldquoA quasilinearization method for a class of integro-differential equations with mixed nonlinearitiesrdquo NonlinearAnalysis Real World Applications vol 7 no 5 pp 997ndash10042006
[12] S G Pandit ldquoQuadratically converging iterative schemes fornonlinear Volterra integral equations and an applicationrdquo Jour-nal of Applied Mathematics and Stochastic Analysis vol 10 no2 pp 169ndash178 1997
[13] J I Ramos ldquoPiecewise-quasilinearization techniques for singu-larly perturbed Volterra integro-differential equationsrdquo AppliedMathematics and Computation vol 188 no 2 pp 1221ndash12332007
[14] S Tuffuor and J W Labadie ldquoA nonlinear time variant rainfall-runoff model for augmenting monthly datardquo Water ResourcesResearch vol 10 pp 1161ndash1166 1974
[15] 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
[16] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers amp Fluids vol 39 no 7 pp 1219ndash1225 2010
[17] S J Liao Beyond Perturbation Introduction to HomotopyAnalysis Method Chapman amp HallCRC Press 2003
[18] S Li and S-J Liao ldquoAn analytic approach to solve multiplesolutions of a strongly nonlinear problemrdquoAppliedMathematicsand Computation vol 169 no 2 pp 854ndash865 2005
[19] P Sibanda S Motsa and Z Makukula ldquoA spectral-homotopyanalysis method for heat transfer flow of a third grade fluidbetween parallel platesrdquo International Journal of NumericalMethods for Heat amp Fluid Flow vol 22 no 1 pp 4ndash23 2012
[20] S S Motsa and P Sibanda ldquoOn the solution of MHD flowover a nonlinear stretching sheet by an efficient semi-analyticaltechniquerdquo International Journal for Numerical Methods inFluids vol 68 no 12 pp 1524ndash1537 2012
[21] S Abbasbandy ldquoImproving Newton-Raphson method for non-linear equations bymodifiedAdomiandecompositionmethodrdquoApplied Mathematics and Computation vol 145 no 2-3 pp887ndash893 2003
[22] C Chun ldquoIterative methods improving Newtonrsquos method bythe decomposition methodrdquo Computers amp Mathematics withApplications vol 50 no 10ndash12 pp 1559ndash1568 2005
[23] G Adomian ldquoA review of the decomposition method andsome recent results for nonlinear equationsrdquoMathematical andComputer Modelling vol 13 no 7 pp 17ndash43 1990
Journal of Applied Mathematics 9
[24] G Adomian and R Rach ldquoNoise terms in decompositionsolution seriesrdquo Computers amp Mathematics with Applicationsvol 24 no 11 pp 61ndash64 1992
[25] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer Academic Publishers Group BostonMass USA 1994
[26] C Chun ldquoConstruction of Newton-like iteration methods forsolving nonlinear equationsrdquoNumerische Mathematik vol 104no 3 pp 297ndash315 2006
[27] M A Noor ldquoIterative methods for nonlinear equations usinghomotopy perturbation techniquerdquo Applied Mathematics ampInformation Sciences vol 4 no 2 pp 227ndash235 2010
[28] J P Boyd ldquoThe Blasius function computations before com-puters the value of tricks undergraduate projects and openresearch problemsrdquo SIAM Review vol 50 no 4 pp 791ndash8042008
[29] B D Ganapol ldquoHighly accurate solutions of the BlasiusandFalkner-Skan boundary layer equations via convergence accel-erationrdquo httparxivorgabs10063888
[30] A S Bataineh M S M Noorani and I Hashim ldquoSeriessolution of the multispecies Lotka-Volterra equations by meansof the homotopy analysis methodrdquo Differential Equations andNonlinear Mechanics vol 2008 Article ID 816787 14 pages2008
[31] A S Bataineh M S M Noorani and I Hashim ldquoHomotopyanalysis method for singular IVPs of Emden-Fowler typerdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 4 pp 1121ndash1131 2009
[32] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[33] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Berlin Germany1988
[34] W S Don and A Solomonoff ldquoAccuracy and speed in com-puting the Chebyshev collocation derivativerdquo SIAM Journal onScientific Computing vol 16 no 6 pp 1253ndash1268 1995
[35] L N Trefethen Spectral Methods in MATLAB SIAM 2000[36] J A CWeideman and S C Reddy ldquoAMATLAB differentiation
matrix suiterdquoACMTransactions onMathematical Software vol26 no 4 pp 465ndash519 2000
In general for any119872 gt 1 we have119910 (119909) = 1199100 (119909) + 1199101 (119909) + sdot sdot sdot + 119910119872 (119909) (28)
where 119910119872(119909) is obtained as a solution ofL1 [119910119872] = (1 + ℎ)L1 [119910119872minus1] + ℎ119877119872minus1 [1199100 1199101 119910119872minus1]
(29)Thus a general scheme when the SHAM is truncated at order119872 (where119872 ge 1) hereinafter referred to as scheme-119872 canbe obtained asL1 [119910119903+1 119910
Using (35)ndash(37) the first three iterative schemes correspond-ing to119872 = 0 1 2may now be defined as follows
31 Scheme-0 In this scheme we set
L1 (119910119903+1 1199101015840
119903+1 11991010158401015840
119903+1 119910101584010158401015840
119903+1) = Φ (119910119903 1199101015840
119903 11991010158401015840
119903 ) (39)
subject to
119910119903+1 (0) = 1199101015840
119903+1 (0) = 0 1199101015840
119903+1 (infin) = 1 (40)
It is worth noting that Scheme-0 is in fact equivalent to theoriginal QLM algorithm see Mandelzweig and Tabakin [5]and Mandelzweig [6]
32 Scheme-1 For this scheme we set
L1 (119910119903+1 1199101015840
119903+1 11991010158401015840
119903+1 119910101584010158401015840
119903+1)
= Φ (119910119903 1199101015840
119903 11991010158401015840
119903 ) + ℎL1 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1 119910101584010158401015840
0119903+1)
+ ℎ1198770 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1)
(41)
subject to
119910119903+1 (0) = 1199101015840
119903+1 (0) = 0 1199101015840
119903+1 (infin) = 1 (42)
where
1198770 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1)
= 1205731199100119903+111991010158401015840
0119903+1 minus 120573111991010158402
0119903+1 + 1205731 + 212057311199101015840
1199031199101015840
0119903+1
minus 120573 (11991011990311991010158401015840
0119903+1 + 11991010158401015840
119903 1199100119903+1)
(43)
and 1199100119903+1 is the solution of
L1 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1 119910101584010158401015840
0119903+1) = Φ (119910119903 1199101015840
119903 11991010158401015840
119903 ) (44)
subject to
1199100 119903+1 (0) = 1199101015840
0119903+1 (0) = 0 1199101015840
0119903+1 (infin) = 1 (45)
33 Scheme-2 The complexity of the defining equationsincreases with the order of the scheme For Scheme-2 we have
L1 (119910119903+1 1199101015840
119903+1 11991010158401015840
119903+1 119910101584010158401015840
119903+1)
= Φ (119910119903 1199101015840
119903 11991010158401015840
119903 ) + ℎL1 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1 119910101584010158401015840
0119903+1)
+ (1 + ℎ)L1 (1199101119903+1 1199101015840
1119903+1 11991010158401015840
1119903+1 119910101584010158401015840
1119903+1)
+ ℎ1198770 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1)
+ ℎ1198771 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1 1199101119903+1 1199101015840
1119903+1 11991010158401015840
1119903+1)
(46)
Journal of Applied Mathematics 5
subject to
119910119903+1 (0) = 1199101015840
119903+1 (0) = 0 1199101015840
119903+1 (infin) = 1 (47)
where
1198771 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1 1199101119903+1 1199101015840
1119903+1 11991010158401015840
1119903+1)
= 120573 (1199100119903+111991010158401015840
1119903+1 + 1199101119903+111991010158401015840
0119903+1) minus 212057311199101015840
0119903+11199101119903+1
+ 212057311199101015840
1199031199101015840
1119903+1 minus 120573 (11991011990311991010158401015840
1119903+1 + 11991010158401015840
119903 1199101119903+1)
(48)
and 1199101119903+1 is the solution of
L1 (1199101119903+1 1199101015840
1119903+1 11991010158401015840
1119903+1 119910101584010158401015840
1119903+1)
= ℎL1 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1 119910101584010158401015840
0119903+1)
+ ℎ1198770 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1)
(49)
with
1199101119903+1 (0) = 1199101015840
1119903+1 (0) = 0 1199101015840
1119903+1 (infin) = 0 (50)
Equations (39) (41) and (46) describing the three solutionschemes can be solved numerically using standard meth-ods such as finite difference finite elements and splinecollocation methods In this study we use the Chebyshevspectral collocation method to solve the iteration schemes(see [33ndash36]) To allow for numerical implementation ofthe pseudospectral method the physical region [0infin) istruncated to [0 119871] where 119871 is chosen to be sufficiently largeThe truncated region is further transformed to the space[minus1 1] using the transformation
120585 =
2
119871infin
120578 minus 1 (51)
As with any other numerical approximation method somesort of discretization is introduced in the interval [minus1 1] Wechoose the Gauss-Lobatto collocation points to define thenodes in [minus1 1] as
120585119895 = cos(120587119895
119873
) 119895 = 0 1 119873 (52)
where (119873 + 1) is the number of collocation points Theessence of the Chebyshev spectral collocation method isthe idea of introducing a differentiation matrix 119863 Thedifferentiation matrix maps a vector of the function valuesY = [119910(1205850) 119910(120585119873)]
119879 at the collocation points to a vectorY1015840 defined as
Starting from a suitable initial guess 1199100(120578) the iterationscheme (60) can be used to iteratively give approximatesolutions of the governing equation (32) for Scheme-0 Theapplication of the pseudospectral method for Scheme-1 andScheme-2 can be done in a similar manner The initialapproximation used in all the algorithms is
The number of collocation points used in all the resultspresented here is119873 = 200 with 119871infin = 20
4 Results and Discussion
In this section we present solutions of the Falkner-Skan equa-tion (32) using the QLM-SHAM hybrid iteration schemesNumerical simulations were conducted for the followingspecial classes of the F-S equations
To assess the accuracy and performance of our schemesthe numerical results were compared to the recently reportedresults of Ganapol [29] To date these results are the most
accurate results for the Blasius and Falkner-Skan class ofequations Ganapol [29] reported highly accurate resultsbetween 10 and 30 decimal places using a robust algorithmbased onMaclaurin series with convergence acceleration andanalytical continuation techniques
The comparison between the present findings and theresults in the literature is made for the skin friction which isproportional to 11989110158401015840(0) Table 1 shows a comparison betweenthe computed skin friction values of the Blasius equationusing the three QLM-SHAM iteration schemes The resultsare comparedwith the results reported inGanapol [29] whichare accurate to 29 decimal places We observe that all theiteration schemes rapidly converge to the results of [29] toall 29 decimal places Full convergence is achieved after 6iterations when using Scheme-0 4 iterations when usingScheme-1 and after 3 iterations when using Scheme-2 Itis worth noting that the results of [29] were achieved after104 decimal places Prior to Ganapol [29] the most accurateBlasius skin friction results had been published to 17 decimalplaces by Boyd [28] as 11989110158401015840(0) = 033205733621519630 Thisresult was obtained after 5 iterations using Scheme-0 and3 iterations for both Schemes-1 and -2 The value reportedafter 52 iterations in [29] is 11989110158401015840(0) = 03320573362151965It is clear that the proposed iteration schemes convergesignificantly faster than the method of [29] That the results
8 Journal of Applied Mathematics
of Boyd [28] and Ganapol [29] were obtained only after a fewiterations validates both the higher order convergence and theaccuracy of the present solution methods
Table 2 shows the computed values of 11989110158401015840(0) using
Schemes-0 and -1 for the Pohlhausen flow (120573 = 0 1205731 = 1)For this particular flow the exact value of 11989110158401015840(0) is knownto be 2radic3 The iteration Scheme-0 matches the exact resultafter only 5 iterations and Scheme-1 converges after only 3iterationsThemethod used in Ganapol [29] converged to theexact result after 104 iterations This again demonstrates thesuperior convergence of the present method
Table 3 gives the numerical simulations of the skin fric-tion results for theHomann flowWe observe that the 29-digitresults reported in [29] are achieved in 5 iterations 4 itera-tions and 3 iterations for Scheme-0 -1 and -2 respectivelyThis result indicates that adding an additional level in theQLM-SHAM schemewould further significantly increase theconvergence of the iteration scheme We further note fromTables 1ndash3 that all three schemes converge significantly muchfaster than the spectral homotopy analysismethod on its own
5 Conclusion
In this study we presented three hybrid QLM-SHAM itera-tion schemes for the solution of Falkner-Skan type boundarylayer equations We have shown through numerical exper-imentation that the proposed numerical schemes signifi-cantly enhance the convergence rate of the quasilinearizationmethod By comparison with the most accurate solutionsof the Falkner-Skan equations currently available in theliterature we have shown that the schemes are highly accurateand efficient in terms of the number of iterations requiredto determine the solution to the required level of accuracyThe schemes presented provide robust tools for the efficientsolution of nonlinear equations by offering superior accuracyto many existing methods In addition the approach usedin deriving these schemes provides a suitable framework forextension to higher level schemes by adding more terms ofthe SHAM component of the method
References
[1] R E Bellman and R E Kalaba Quasilinearization and Non-linear Boundary-Value Problems Elsevier New York NY USA1965
[2] K Maleknejad and E Najafi ldquoNumerical solution of nonlinearVolterra integral equations using the idea of quasilinearizationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 1 pp 93ndash100 2011
[3] R Krivec and V B Mandelzweig ldquoNumerical investigation ofquasilinearization method in quantum mechanicsrdquo ComputerPhysics Communications vol 138 pp 69ndash79 2001
[4] V BMandelzweig ldquoQuasilinearizationmethod and its verifica-tion on exactly solvablemodels in quantummechanicsrdquo Journalof Mathematical Physics vol 40 no 12 pp 6266ndash6291 1999
[5] V B Mandelzweig and F Tabakin ldquoQuasilinearizationapproach to nonlinear problems in physics with application tononlinear ODEsrdquo Computer Physics Communications vol 141no 2 pp 268ndash281 2001
[6] V B Mandelzweig ldquoQuasilinearization method nonperturba-tive approach to physical problemsrdquo Physics of Atomic Nucleivol 68 no 7 pp 1227ndash1258 2005
[7] V Lakshmikantham S Leela and S Sivasundaram ldquoExtensionsof the method of quasilinearizationrdquo Journal of OptimizationTheory and Applications vol 87 no 2 pp 379ndash401 1995
[8] V Lakshmikantham ldquoFurther improvement of generalizedquasilinearization methodrdquo Nonlinear Analysis Theory Meth-ods amp Applications vol 27 no 2 pp 223ndash227 1996
[9] K Parand M Ghasemi S Rezazadeh A Peiravi A Ghor-banpour and A Tavakoli Golpaygani ldquoQuasilinearizationapproach for solving Volterrarsquos population modelrdquo Applied andComputational Mathematics vol 9 no 1 pp 95ndash103 2010
[10] J Jiang and A S Vatsala ldquoThe quasilinearization method in thesystem of reaction diffusion equationsrdquo Applied Mathematicsand Computation vol 97 no 2-3 pp 223ndash235 1998
[11] B Ahmad ldquoA quasilinearization method for a class of integro-differential equations with mixed nonlinearitiesrdquo NonlinearAnalysis Real World Applications vol 7 no 5 pp 997ndash10042006
[12] S G Pandit ldquoQuadratically converging iterative schemes fornonlinear Volterra integral equations and an applicationrdquo Jour-nal of Applied Mathematics and Stochastic Analysis vol 10 no2 pp 169ndash178 1997
[13] J I Ramos ldquoPiecewise-quasilinearization techniques for singu-larly perturbed Volterra integro-differential equationsrdquo AppliedMathematics and Computation vol 188 no 2 pp 1221ndash12332007
[14] S Tuffuor and J W Labadie ldquoA nonlinear time variant rainfall-runoff model for augmenting monthly datardquo Water ResourcesResearch vol 10 pp 1161ndash1166 1974
[15] 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
[16] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers amp Fluids vol 39 no 7 pp 1219ndash1225 2010
[17] S J Liao Beyond Perturbation Introduction to HomotopyAnalysis Method Chapman amp HallCRC Press 2003
[18] S Li and S-J Liao ldquoAn analytic approach to solve multiplesolutions of a strongly nonlinear problemrdquoAppliedMathematicsand Computation vol 169 no 2 pp 854ndash865 2005
[19] P Sibanda S Motsa and Z Makukula ldquoA spectral-homotopyanalysis method for heat transfer flow of a third grade fluidbetween parallel platesrdquo International Journal of NumericalMethods for Heat amp Fluid Flow vol 22 no 1 pp 4ndash23 2012
[20] S S Motsa and P Sibanda ldquoOn the solution of MHD flowover a nonlinear stretching sheet by an efficient semi-analyticaltechniquerdquo International Journal for Numerical Methods inFluids vol 68 no 12 pp 1524ndash1537 2012
[21] S Abbasbandy ldquoImproving Newton-Raphson method for non-linear equations bymodifiedAdomiandecompositionmethodrdquoApplied Mathematics and Computation vol 145 no 2-3 pp887ndash893 2003
[22] C Chun ldquoIterative methods improving Newtonrsquos method bythe decomposition methodrdquo Computers amp Mathematics withApplications vol 50 no 10ndash12 pp 1559ndash1568 2005
[23] G Adomian ldquoA review of the decomposition method andsome recent results for nonlinear equationsrdquoMathematical andComputer Modelling vol 13 no 7 pp 17ndash43 1990
Journal of Applied Mathematics 9
[24] G Adomian and R Rach ldquoNoise terms in decompositionsolution seriesrdquo Computers amp Mathematics with Applicationsvol 24 no 11 pp 61ndash64 1992
[25] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer Academic Publishers Group BostonMass USA 1994
[26] C Chun ldquoConstruction of Newton-like iteration methods forsolving nonlinear equationsrdquoNumerische Mathematik vol 104no 3 pp 297ndash315 2006
[27] M A Noor ldquoIterative methods for nonlinear equations usinghomotopy perturbation techniquerdquo Applied Mathematics ampInformation Sciences vol 4 no 2 pp 227ndash235 2010
[28] J P Boyd ldquoThe Blasius function computations before com-puters the value of tricks undergraduate projects and openresearch problemsrdquo SIAM Review vol 50 no 4 pp 791ndash8042008
[29] B D Ganapol ldquoHighly accurate solutions of the BlasiusandFalkner-Skan boundary layer equations via convergence accel-erationrdquo httparxivorgabs10063888
[30] A S Bataineh M S M Noorani and I Hashim ldquoSeriessolution of the multispecies Lotka-Volterra equations by meansof the homotopy analysis methodrdquo Differential Equations andNonlinear Mechanics vol 2008 Article ID 816787 14 pages2008
[31] A S Bataineh M S M Noorani and I Hashim ldquoHomotopyanalysis method for singular IVPs of Emden-Fowler typerdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 4 pp 1121ndash1131 2009
[32] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[33] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Berlin Germany1988
[34] W S Don and A Solomonoff ldquoAccuracy and speed in com-puting the Chebyshev collocation derivativerdquo SIAM Journal onScientific Computing vol 16 no 6 pp 1253ndash1268 1995
[35] L N Trefethen Spectral Methods in MATLAB SIAM 2000[36] J A CWeideman and S C Reddy ldquoAMATLAB differentiation
matrix suiterdquoACMTransactions onMathematical Software vol26 no 4 pp 465ndash519 2000
1119903+1 minus 120573 (11991011990311991010158401015840
1119903+1 + 11991010158401015840
119903 1199101119903+1)
(48)
and 1199101119903+1 is the solution of
L1 (1199101119903+1 1199101015840
1119903+1 11991010158401015840
1119903+1 119910101584010158401015840
1119903+1)
= ℎL1 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1 119910101584010158401015840
0119903+1)
+ ℎ1198770 (1199100119903+1 1199101015840
0119903+1 11991010158401015840
0119903+1)
(49)
with
1199101119903+1 (0) = 1199101015840
1119903+1 (0) = 0 1199101015840
1119903+1 (infin) = 0 (50)
Equations (39) (41) and (46) describing the three solutionschemes can be solved numerically using standard meth-ods such as finite difference finite elements and splinecollocation methods In this study we use the Chebyshevspectral collocation method to solve the iteration schemes(see [33ndash36]) To allow for numerical implementation ofthe pseudospectral method the physical region [0infin) istruncated to [0 119871] where 119871 is chosen to be sufficiently largeThe truncated region is further transformed to the space[minus1 1] using the transformation
120585 =
2
119871infin
120578 minus 1 (51)
As with any other numerical approximation method somesort of discretization is introduced in the interval [minus1 1] Wechoose the Gauss-Lobatto collocation points to define thenodes in [minus1 1] as
120585119895 = cos(120587119895
119873
) 119895 = 0 1 119873 (52)
where (119873 + 1) is the number of collocation points Theessence of the Chebyshev spectral collocation method isthe idea of introducing a differentiation matrix 119863 Thedifferentiation matrix maps a vector of the function valuesY = [119910(1205850) 119910(120585119873)]
119879 at the collocation points to a vectorY1015840 defined as
Starting from a suitable initial guess 1199100(120578) the iterationscheme (60) can be used to iteratively give approximatesolutions of the governing equation (32) for Scheme-0 Theapplication of the pseudospectral method for Scheme-1 andScheme-2 can be done in a similar manner The initialapproximation used in all the algorithms is
The number of collocation points used in all the resultspresented here is119873 = 200 with 119871infin = 20
4 Results and Discussion
In this section we present solutions of the Falkner-Skan equa-tion (32) using the QLM-SHAM hybrid iteration schemesNumerical simulations were conducted for the followingspecial classes of the F-S equations
To assess the accuracy and performance of our schemesthe numerical results were compared to the recently reportedresults of Ganapol [29] To date these results are the most
accurate results for the Blasius and Falkner-Skan class ofequations Ganapol [29] reported highly accurate resultsbetween 10 and 30 decimal places using a robust algorithmbased onMaclaurin series with convergence acceleration andanalytical continuation techniques
The comparison between the present findings and theresults in the literature is made for the skin friction which isproportional to 11989110158401015840(0) Table 1 shows a comparison betweenthe computed skin friction values of the Blasius equationusing the three QLM-SHAM iteration schemes The resultsare comparedwith the results reported inGanapol [29] whichare accurate to 29 decimal places We observe that all theiteration schemes rapidly converge to the results of [29] toall 29 decimal places Full convergence is achieved after 6iterations when using Scheme-0 4 iterations when usingScheme-1 and after 3 iterations when using Scheme-2 Itis worth noting that the results of [29] were achieved after104 decimal places Prior to Ganapol [29] the most accurateBlasius skin friction results had been published to 17 decimalplaces by Boyd [28] as 11989110158401015840(0) = 033205733621519630 Thisresult was obtained after 5 iterations using Scheme-0 and3 iterations for both Schemes-1 and -2 The value reportedafter 52 iterations in [29] is 11989110158401015840(0) = 03320573362151965It is clear that the proposed iteration schemes convergesignificantly faster than the method of [29] That the results
8 Journal of Applied Mathematics
of Boyd [28] and Ganapol [29] were obtained only after a fewiterations validates both the higher order convergence and theaccuracy of the present solution methods
Table 2 shows the computed values of 11989110158401015840(0) using
Schemes-0 and -1 for the Pohlhausen flow (120573 = 0 1205731 = 1)For this particular flow the exact value of 11989110158401015840(0) is knownto be 2radic3 The iteration Scheme-0 matches the exact resultafter only 5 iterations and Scheme-1 converges after only 3iterationsThemethod used in Ganapol [29] converged to theexact result after 104 iterations This again demonstrates thesuperior convergence of the present method
Table 3 gives the numerical simulations of the skin fric-tion results for theHomann flowWe observe that the 29-digitresults reported in [29] are achieved in 5 iterations 4 itera-tions and 3 iterations for Scheme-0 -1 and -2 respectivelyThis result indicates that adding an additional level in theQLM-SHAM schemewould further significantly increase theconvergence of the iteration scheme We further note fromTables 1ndash3 that all three schemes converge significantly muchfaster than the spectral homotopy analysismethod on its own
5 Conclusion
In this study we presented three hybrid QLM-SHAM itera-tion schemes for the solution of Falkner-Skan type boundarylayer equations We have shown through numerical exper-imentation that the proposed numerical schemes signifi-cantly enhance the convergence rate of the quasilinearizationmethod By comparison with the most accurate solutionsof the Falkner-Skan equations currently available in theliterature we have shown that the schemes are highly accurateand efficient in terms of the number of iterations requiredto determine the solution to the required level of accuracyThe schemes presented provide robust tools for the efficientsolution of nonlinear equations by offering superior accuracyto many existing methods In addition the approach usedin deriving these schemes provides a suitable framework forextension to higher level schemes by adding more terms ofthe SHAM component of the method
References
[1] R E Bellman and R E Kalaba Quasilinearization and Non-linear Boundary-Value Problems Elsevier New York NY USA1965
[2] K Maleknejad and E Najafi ldquoNumerical solution of nonlinearVolterra integral equations using the idea of quasilinearizationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 1 pp 93ndash100 2011
[3] R Krivec and V B Mandelzweig ldquoNumerical investigation ofquasilinearization method in quantum mechanicsrdquo ComputerPhysics Communications vol 138 pp 69ndash79 2001
[4] V BMandelzweig ldquoQuasilinearizationmethod and its verifica-tion on exactly solvablemodels in quantummechanicsrdquo Journalof Mathematical Physics vol 40 no 12 pp 6266ndash6291 1999
[5] V B Mandelzweig and F Tabakin ldquoQuasilinearizationapproach to nonlinear problems in physics with application tononlinear ODEsrdquo Computer Physics Communications vol 141no 2 pp 268ndash281 2001
[6] V B Mandelzweig ldquoQuasilinearization method nonperturba-tive approach to physical problemsrdquo Physics of Atomic Nucleivol 68 no 7 pp 1227ndash1258 2005
[7] V Lakshmikantham S Leela and S Sivasundaram ldquoExtensionsof the method of quasilinearizationrdquo Journal of OptimizationTheory and Applications vol 87 no 2 pp 379ndash401 1995
[8] V Lakshmikantham ldquoFurther improvement of generalizedquasilinearization methodrdquo Nonlinear Analysis Theory Meth-ods amp Applications vol 27 no 2 pp 223ndash227 1996
[9] K Parand M Ghasemi S Rezazadeh A Peiravi A Ghor-banpour and A Tavakoli Golpaygani ldquoQuasilinearizationapproach for solving Volterrarsquos population modelrdquo Applied andComputational Mathematics vol 9 no 1 pp 95ndash103 2010
[10] J Jiang and A S Vatsala ldquoThe quasilinearization method in thesystem of reaction diffusion equationsrdquo Applied Mathematicsand Computation vol 97 no 2-3 pp 223ndash235 1998
[11] B Ahmad ldquoA quasilinearization method for a class of integro-differential equations with mixed nonlinearitiesrdquo NonlinearAnalysis Real World Applications vol 7 no 5 pp 997ndash10042006
[12] S G Pandit ldquoQuadratically converging iterative schemes fornonlinear Volterra integral equations and an applicationrdquo Jour-nal of Applied Mathematics and Stochastic Analysis vol 10 no2 pp 169ndash178 1997
[13] J I Ramos ldquoPiecewise-quasilinearization techniques for singu-larly perturbed Volterra integro-differential equationsrdquo AppliedMathematics and Computation vol 188 no 2 pp 1221ndash12332007
[14] S Tuffuor and J W Labadie ldquoA nonlinear time variant rainfall-runoff model for augmenting monthly datardquo Water ResourcesResearch vol 10 pp 1161ndash1166 1974
[15] 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
[16] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers amp Fluids vol 39 no 7 pp 1219ndash1225 2010
[17] S J Liao Beyond Perturbation Introduction to HomotopyAnalysis Method Chapman amp HallCRC Press 2003
[18] S Li and S-J Liao ldquoAn analytic approach to solve multiplesolutions of a strongly nonlinear problemrdquoAppliedMathematicsand Computation vol 169 no 2 pp 854ndash865 2005
[19] P Sibanda S Motsa and Z Makukula ldquoA spectral-homotopyanalysis method for heat transfer flow of a third grade fluidbetween parallel platesrdquo International Journal of NumericalMethods for Heat amp Fluid Flow vol 22 no 1 pp 4ndash23 2012
[20] S S Motsa and P Sibanda ldquoOn the solution of MHD flowover a nonlinear stretching sheet by an efficient semi-analyticaltechniquerdquo International Journal for Numerical Methods inFluids vol 68 no 12 pp 1524ndash1537 2012
[21] S Abbasbandy ldquoImproving Newton-Raphson method for non-linear equations bymodifiedAdomiandecompositionmethodrdquoApplied Mathematics and Computation vol 145 no 2-3 pp887ndash893 2003
[22] C Chun ldquoIterative methods improving Newtonrsquos method bythe decomposition methodrdquo Computers amp Mathematics withApplications vol 50 no 10ndash12 pp 1559ndash1568 2005
[23] G Adomian ldquoA review of the decomposition method andsome recent results for nonlinear equationsrdquoMathematical andComputer Modelling vol 13 no 7 pp 17ndash43 1990
Journal of Applied Mathematics 9
[24] G Adomian and R Rach ldquoNoise terms in decompositionsolution seriesrdquo Computers amp Mathematics with Applicationsvol 24 no 11 pp 61ndash64 1992
[25] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer Academic Publishers Group BostonMass USA 1994
[26] C Chun ldquoConstruction of Newton-like iteration methods forsolving nonlinear equationsrdquoNumerische Mathematik vol 104no 3 pp 297ndash315 2006
[27] M A Noor ldquoIterative methods for nonlinear equations usinghomotopy perturbation techniquerdquo Applied Mathematics ampInformation Sciences vol 4 no 2 pp 227ndash235 2010
[28] J P Boyd ldquoThe Blasius function computations before com-puters the value of tricks undergraduate projects and openresearch problemsrdquo SIAM Review vol 50 no 4 pp 791ndash8042008
[29] B D Ganapol ldquoHighly accurate solutions of the BlasiusandFalkner-Skan boundary layer equations via convergence accel-erationrdquo httparxivorgabs10063888
[30] A S Bataineh M S M Noorani and I Hashim ldquoSeriessolution of the multispecies Lotka-Volterra equations by meansof the homotopy analysis methodrdquo Differential Equations andNonlinear Mechanics vol 2008 Article ID 816787 14 pages2008
[31] A S Bataineh M S M Noorani and I Hashim ldquoHomotopyanalysis method for singular IVPs of Emden-Fowler typerdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 4 pp 1121ndash1131 2009
[32] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[33] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Berlin Germany1988
[34] W S Don and A Solomonoff ldquoAccuracy and speed in com-puting the Chebyshev collocation derivativerdquo SIAM Journal onScientific Computing vol 16 no 6 pp 1253ndash1268 1995
[35] L N Trefethen Spectral Methods in MATLAB SIAM 2000[36] J A CWeideman and S C Reddy ldquoAMATLAB differentiation
matrix suiterdquoACMTransactions onMathematical Software vol26 no 4 pp 465ndash519 2000
Starting from a suitable initial guess 1199100(120578) the iterationscheme (60) can be used to iteratively give approximatesolutions of the governing equation (32) for Scheme-0 Theapplication of the pseudospectral method for Scheme-1 andScheme-2 can be done in a similar manner The initialapproximation used in all the algorithms is
The number of collocation points used in all the resultspresented here is119873 = 200 with 119871infin = 20
4 Results and Discussion
In this section we present solutions of the Falkner-Skan equa-tion (32) using the QLM-SHAM hybrid iteration schemesNumerical simulations were conducted for the followingspecial classes of the F-S equations
To assess the accuracy and performance of our schemesthe numerical results were compared to the recently reportedresults of Ganapol [29] To date these results are the most
accurate results for the Blasius and Falkner-Skan class ofequations Ganapol [29] reported highly accurate resultsbetween 10 and 30 decimal places using a robust algorithmbased onMaclaurin series with convergence acceleration andanalytical continuation techniques
The comparison between the present findings and theresults in the literature is made for the skin friction which isproportional to 11989110158401015840(0) Table 1 shows a comparison betweenthe computed skin friction values of the Blasius equationusing the three QLM-SHAM iteration schemes The resultsare comparedwith the results reported inGanapol [29] whichare accurate to 29 decimal places We observe that all theiteration schemes rapidly converge to the results of [29] toall 29 decimal places Full convergence is achieved after 6iterations when using Scheme-0 4 iterations when usingScheme-1 and after 3 iterations when using Scheme-2 Itis worth noting that the results of [29] were achieved after104 decimal places Prior to Ganapol [29] the most accurateBlasius skin friction results had been published to 17 decimalplaces by Boyd [28] as 11989110158401015840(0) = 033205733621519630 Thisresult was obtained after 5 iterations using Scheme-0 and3 iterations for both Schemes-1 and -2 The value reportedafter 52 iterations in [29] is 11989110158401015840(0) = 03320573362151965It is clear that the proposed iteration schemes convergesignificantly faster than the method of [29] That the results
8 Journal of Applied Mathematics
of Boyd [28] and Ganapol [29] were obtained only after a fewiterations validates both the higher order convergence and theaccuracy of the present solution methods
Table 2 shows the computed values of 11989110158401015840(0) using
Schemes-0 and -1 for the Pohlhausen flow (120573 = 0 1205731 = 1)For this particular flow the exact value of 11989110158401015840(0) is knownto be 2radic3 The iteration Scheme-0 matches the exact resultafter only 5 iterations and Scheme-1 converges after only 3iterationsThemethod used in Ganapol [29] converged to theexact result after 104 iterations This again demonstrates thesuperior convergence of the present method
Table 3 gives the numerical simulations of the skin fric-tion results for theHomann flowWe observe that the 29-digitresults reported in [29] are achieved in 5 iterations 4 itera-tions and 3 iterations for Scheme-0 -1 and -2 respectivelyThis result indicates that adding an additional level in theQLM-SHAM schemewould further significantly increase theconvergence of the iteration scheme We further note fromTables 1ndash3 that all three schemes converge significantly muchfaster than the spectral homotopy analysismethod on its own
5 Conclusion
In this study we presented three hybrid QLM-SHAM itera-tion schemes for the solution of Falkner-Skan type boundarylayer equations We have shown through numerical exper-imentation that the proposed numerical schemes signifi-cantly enhance the convergence rate of the quasilinearizationmethod By comparison with the most accurate solutionsof the Falkner-Skan equations currently available in theliterature we have shown that the schemes are highly accurateand efficient in terms of the number of iterations requiredto determine the solution to the required level of accuracyThe schemes presented provide robust tools for the efficientsolution of nonlinear equations by offering superior accuracyto many existing methods In addition the approach usedin deriving these schemes provides a suitable framework forextension to higher level schemes by adding more terms ofthe SHAM component of the method
References
[1] R E Bellman and R E Kalaba Quasilinearization and Non-linear Boundary-Value Problems Elsevier New York NY USA1965
[2] K Maleknejad and E Najafi ldquoNumerical solution of nonlinearVolterra integral equations using the idea of quasilinearizationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 1 pp 93ndash100 2011
[3] R Krivec and V B Mandelzweig ldquoNumerical investigation ofquasilinearization method in quantum mechanicsrdquo ComputerPhysics Communications vol 138 pp 69ndash79 2001
[4] V BMandelzweig ldquoQuasilinearizationmethod and its verifica-tion on exactly solvablemodels in quantummechanicsrdquo Journalof Mathematical Physics vol 40 no 12 pp 6266ndash6291 1999
[5] V B Mandelzweig and F Tabakin ldquoQuasilinearizationapproach to nonlinear problems in physics with application tononlinear ODEsrdquo Computer Physics Communications vol 141no 2 pp 268ndash281 2001
[6] V B Mandelzweig ldquoQuasilinearization method nonperturba-tive approach to physical problemsrdquo Physics of Atomic Nucleivol 68 no 7 pp 1227ndash1258 2005
[7] V Lakshmikantham S Leela and S Sivasundaram ldquoExtensionsof the method of quasilinearizationrdquo Journal of OptimizationTheory and Applications vol 87 no 2 pp 379ndash401 1995
[8] V Lakshmikantham ldquoFurther improvement of generalizedquasilinearization methodrdquo Nonlinear Analysis Theory Meth-ods amp Applications vol 27 no 2 pp 223ndash227 1996
[9] K Parand M Ghasemi S Rezazadeh A Peiravi A Ghor-banpour and A Tavakoli Golpaygani ldquoQuasilinearizationapproach for solving Volterrarsquos population modelrdquo Applied andComputational Mathematics vol 9 no 1 pp 95ndash103 2010
[10] J Jiang and A S Vatsala ldquoThe quasilinearization method in thesystem of reaction diffusion equationsrdquo Applied Mathematicsand Computation vol 97 no 2-3 pp 223ndash235 1998
[11] B Ahmad ldquoA quasilinearization method for a class of integro-differential equations with mixed nonlinearitiesrdquo NonlinearAnalysis Real World Applications vol 7 no 5 pp 997ndash10042006
[12] S G Pandit ldquoQuadratically converging iterative schemes fornonlinear Volterra integral equations and an applicationrdquo Jour-nal of Applied Mathematics and Stochastic Analysis vol 10 no2 pp 169ndash178 1997
[13] J I Ramos ldquoPiecewise-quasilinearization techniques for singu-larly perturbed Volterra integro-differential equationsrdquo AppliedMathematics and Computation vol 188 no 2 pp 1221ndash12332007
[14] S Tuffuor and J W Labadie ldquoA nonlinear time variant rainfall-runoff model for augmenting monthly datardquo Water ResourcesResearch vol 10 pp 1161ndash1166 1974
[15] 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
[16] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers amp Fluids vol 39 no 7 pp 1219ndash1225 2010
[17] S J Liao Beyond Perturbation Introduction to HomotopyAnalysis Method Chapman amp HallCRC Press 2003
[18] S Li and S-J Liao ldquoAn analytic approach to solve multiplesolutions of a strongly nonlinear problemrdquoAppliedMathematicsand Computation vol 169 no 2 pp 854ndash865 2005
[19] P Sibanda S Motsa and Z Makukula ldquoA spectral-homotopyanalysis method for heat transfer flow of a third grade fluidbetween parallel platesrdquo International Journal of NumericalMethods for Heat amp Fluid Flow vol 22 no 1 pp 4ndash23 2012
[20] S S Motsa and P Sibanda ldquoOn the solution of MHD flowover a nonlinear stretching sheet by an efficient semi-analyticaltechniquerdquo International Journal for Numerical Methods inFluids vol 68 no 12 pp 1524ndash1537 2012
[21] S Abbasbandy ldquoImproving Newton-Raphson method for non-linear equations bymodifiedAdomiandecompositionmethodrdquoApplied Mathematics and Computation vol 145 no 2-3 pp887ndash893 2003
[22] C Chun ldquoIterative methods improving Newtonrsquos method bythe decomposition methodrdquo Computers amp Mathematics withApplications vol 50 no 10ndash12 pp 1559ndash1568 2005
[23] G Adomian ldquoA review of the decomposition method andsome recent results for nonlinear equationsrdquoMathematical andComputer Modelling vol 13 no 7 pp 17ndash43 1990
Journal of Applied Mathematics 9
[24] G Adomian and R Rach ldquoNoise terms in decompositionsolution seriesrdquo Computers amp Mathematics with Applicationsvol 24 no 11 pp 61ndash64 1992
[25] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer Academic Publishers Group BostonMass USA 1994
[26] C Chun ldquoConstruction of Newton-like iteration methods forsolving nonlinear equationsrdquoNumerische Mathematik vol 104no 3 pp 297ndash315 2006
[27] M A Noor ldquoIterative methods for nonlinear equations usinghomotopy perturbation techniquerdquo Applied Mathematics ampInformation Sciences vol 4 no 2 pp 227ndash235 2010
[28] J P Boyd ldquoThe Blasius function computations before com-puters the value of tricks undergraduate projects and openresearch problemsrdquo SIAM Review vol 50 no 4 pp 791ndash8042008
[29] B D Ganapol ldquoHighly accurate solutions of the BlasiusandFalkner-Skan boundary layer equations via convergence accel-erationrdquo httparxivorgabs10063888
[30] A S Bataineh M S M Noorani and I Hashim ldquoSeriessolution of the multispecies Lotka-Volterra equations by meansof the homotopy analysis methodrdquo Differential Equations andNonlinear Mechanics vol 2008 Article ID 816787 14 pages2008
[31] A S Bataineh M S M Noorani and I Hashim ldquoHomotopyanalysis method for singular IVPs of Emden-Fowler typerdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 4 pp 1121ndash1131 2009
[32] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[33] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Berlin Germany1988
[34] W S Don and A Solomonoff ldquoAccuracy and speed in com-puting the Chebyshev collocation derivativerdquo SIAM Journal onScientific Computing vol 16 no 6 pp 1253ndash1268 1995
[35] L N Trefethen Spectral Methods in MATLAB SIAM 2000[36] J A CWeideman and S C Reddy ldquoAMATLAB differentiation
matrix suiterdquoACMTransactions onMathematical Software vol26 no 4 pp 465ndash519 2000
Starting from a suitable initial guess 1199100(120578) the iterationscheme (60) can be used to iteratively give approximatesolutions of the governing equation (32) for Scheme-0 Theapplication of the pseudospectral method for Scheme-1 andScheme-2 can be done in a similar manner The initialapproximation used in all the algorithms is
The number of collocation points used in all the resultspresented here is119873 = 200 with 119871infin = 20
4 Results and Discussion
In this section we present solutions of the Falkner-Skan equa-tion (32) using the QLM-SHAM hybrid iteration schemesNumerical simulations were conducted for the followingspecial classes of the F-S equations
To assess the accuracy and performance of our schemesthe numerical results were compared to the recently reportedresults of Ganapol [29] To date these results are the most
accurate results for the Blasius and Falkner-Skan class ofequations Ganapol [29] reported highly accurate resultsbetween 10 and 30 decimal places using a robust algorithmbased onMaclaurin series with convergence acceleration andanalytical continuation techniques
The comparison between the present findings and theresults in the literature is made for the skin friction which isproportional to 11989110158401015840(0) Table 1 shows a comparison betweenthe computed skin friction values of the Blasius equationusing the three QLM-SHAM iteration schemes The resultsare comparedwith the results reported inGanapol [29] whichare accurate to 29 decimal places We observe that all theiteration schemes rapidly converge to the results of [29] toall 29 decimal places Full convergence is achieved after 6iterations when using Scheme-0 4 iterations when usingScheme-1 and after 3 iterations when using Scheme-2 Itis worth noting that the results of [29] were achieved after104 decimal places Prior to Ganapol [29] the most accurateBlasius skin friction results had been published to 17 decimalplaces by Boyd [28] as 11989110158401015840(0) = 033205733621519630 Thisresult was obtained after 5 iterations using Scheme-0 and3 iterations for both Schemes-1 and -2 The value reportedafter 52 iterations in [29] is 11989110158401015840(0) = 03320573362151965It is clear that the proposed iteration schemes convergesignificantly faster than the method of [29] That the results
8 Journal of Applied Mathematics
of Boyd [28] and Ganapol [29] were obtained only after a fewiterations validates both the higher order convergence and theaccuracy of the present solution methods
Table 2 shows the computed values of 11989110158401015840(0) using
Schemes-0 and -1 for the Pohlhausen flow (120573 = 0 1205731 = 1)For this particular flow the exact value of 11989110158401015840(0) is knownto be 2radic3 The iteration Scheme-0 matches the exact resultafter only 5 iterations and Scheme-1 converges after only 3iterationsThemethod used in Ganapol [29] converged to theexact result after 104 iterations This again demonstrates thesuperior convergence of the present method
Table 3 gives the numerical simulations of the skin fric-tion results for theHomann flowWe observe that the 29-digitresults reported in [29] are achieved in 5 iterations 4 itera-tions and 3 iterations for Scheme-0 -1 and -2 respectivelyThis result indicates that adding an additional level in theQLM-SHAM schemewould further significantly increase theconvergence of the iteration scheme We further note fromTables 1ndash3 that all three schemes converge significantly muchfaster than the spectral homotopy analysismethod on its own
5 Conclusion
In this study we presented three hybrid QLM-SHAM itera-tion schemes for the solution of Falkner-Skan type boundarylayer equations We have shown through numerical exper-imentation that the proposed numerical schemes signifi-cantly enhance the convergence rate of the quasilinearizationmethod By comparison with the most accurate solutionsof the Falkner-Skan equations currently available in theliterature we have shown that the schemes are highly accurateand efficient in terms of the number of iterations requiredto determine the solution to the required level of accuracyThe schemes presented provide robust tools for the efficientsolution of nonlinear equations by offering superior accuracyto many existing methods In addition the approach usedin deriving these schemes provides a suitable framework forextension to higher level schemes by adding more terms ofthe SHAM component of the method
References
[1] R E Bellman and R E Kalaba Quasilinearization and Non-linear Boundary-Value Problems Elsevier New York NY USA1965
[2] K Maleknejad and E Najafi ldquoNumerical solution of nonlinearVolterra integral equations using the idea of quasilinearizationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 1 pp 93ndash100 2011
[3] R Krivec and V B Mandelzweig ldquoNumerical investigation ofquasilinearization method in quantum mechanicsrdquo ComputerPhysics Communications vol 138 pp 69ndash79 2001
[4] V BMandelzweig ldquoQuasilinearizationmethod and its verifica-tion on exactly solvablemodels in quantummechanicsrdquo Journalof Mathematical Physics vol 40 no 12 pp 6266ndash6291 1999
[5] V B Mandelzweig and F Tabakin ldquoQuasilinearizationapproach to nonlinear problems in physics with application tononlinear ODEsrdquo Computer Physics Communications vol 141no 2 pp 268ndash281 2001
[6] V B Mandelzweig ldquoQuasilinearization method nonperturba-tive approach to physical problemsrdquo Physics of Atomic Nucleivol 68 no 7 pp 1227ndash1258 2005
[7] V Lakshmikantham S Leela and S Sivasundaram ldquoExtensionsof the method of quasilinearizationrdquo Journal of OptimizationTheory and Applications vol 87 no 2 pp 379ndash401 1995
[8] V Lakshmikantham ldquoFurther improvement of generalizedquasilinearization methodrdquo Nonlinear Analysis Theory Meth-ods amp Applications vol 27 no 2 pp 223ndash227 1996
[9] K Parand M Ghasemi S Rezazadeh A Peiravi A Ghor-banpour and A Tavakoli Golpaygani ldquoQuasilinearizationapproach for solving Volterrarsquos population modelrdquo Applied andComputational Mathematics vol 9 no 1 pp 95ndash103 2010
[10] J Jiang and A S Vatsala ldquoThe quasilinearization method in thesystem of reaction diffusion equationsrdquo Applied Mathematicsand Computation vol 97 no 2-3 pp 223ndash235 1998
[11] B Ahmad ldquoA quasilinearization method for a class of integro-differential equations with mixed nonlinearitiesrdquo NonlinearAnalysis Real World Applications vol 7 no 5 pp 997ndash10042006
[12] S G Pandit ldquoQuadratically converging iterative schemes fornonlinear Volterra integral equations and an applicationrdquo Jour-nal of Applied Mathematics and Stochastic Analysis vol 10 no2 pp 169ndash178 1997
[13] J I Ramos ldquoPiecewise-quasilinearization techniques for singu-larly perturbed Volterra integro-differential equationsrdquo AppliedMathematics and Computation vol 188 no 2 pp 1221ndash12332007
[14] S Tuffuor and J W Labadie ldquoA nonlinear time variant rainfall-runoff model for augmenting monthly datardquo Water ResourcesResearch vol 10 pp 1161ndash1166 1974
[15] 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
[16] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers amp Fluids vol 39 no 7 pp 1219ndash1225 2010
[17] S J Liao Beyond Perturbation Introduction to HomotopyAnalysis Method Chapman amp HallCRC Press 2003
[18] S Li and S-J Liao ldquoAn analytic approach to solve multiplesolutions of a strongly nonlinear problemrdquoAppliedMathematicsand Computation vol 169 no 2 pp 854ndash865 2005
[19] P Sibanda S Motsa and Z Makukula ldquoA spectral-homotopyanalysis method for heat transfer flow of a third grade fluidbetween parallel platesrdquo International Journal of NumericalMethods for Heat amp Fluid Flow vol 22 no 1 pp 4ndash23 2012
[20] S S Motsa and P Sibanda ldquoOn the solution of MHD flowover a nonlinear stretching sheet by an efficient semi-analyticaltechniquerdquo International Journal for Numerical Methods inFluids vol 68 no 12 pp 1524ndash1537 2012
[21] S Abbasbandy ldquoImproving Newton-Raphson method for non-linear equations bymodifiedAdomiandecompositionmethodrdquoApplied Mathematics and Computation vol 145 no 2-3 pp887ndash893 2003
[22] C Chun ldquoIterative methods improving Newtonrsquos method bythe decomposition methodrdquo Computers amp Mathematics withApplications vol 50 no 10ndash12 pp 1559ndash1568 2005
[23] G Adomian ldquoA review of the decomposition method andsome recent results for nonlinear equationsrdquoMathematical andComputer Modelling vol 13 no 7 pp 17ndash43 1990
Journal of Applied Mathematics 9
[24] G Adomian and R Rach ldquoNoise terms in decompositionsolution seriesrdquo Computers amp Mathematics with Applicationsvol 24 no 11 pp 61ndash64 1992
[25] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer Academic Publishers Group BostonMass USA 1994
[26] C Chun ldquoConstruction of Newton-like iteration methods forsolving nonlinear equationsrdquoNumerische Mathematik vol 104no 3 pp 297ndash315 2006
[27] M A Noor ldquoIterative methods for nonlinear equations usinghomotopy perturbation techniquerdquo Applied Mathematics ampInformation Sciences vol 4 no 2 pp 227ndash235 2010
[28] J P Boyd ldquoThe Blasius function computations before com-puters the value of tricks undergraduate projects and openresearch problemsrdquo SIAM Review vol 50 no 4 pp 791ndash8042008
[29] B D Ganapol ldquoHighly accurate solutions of the BlasiusandFalkner-Skan boundary layer equations via convergence accel-erationrdquo httparxivorgabs10063888
[30] A S Bataineh M S M Noorani and I Hashim ldquoSeriessolution of the multispecies Lotka-Volterra equations by meansof the homotopy analysis methodrdquo Differential Equations andNonlinear Mechanics vol 2008 Article ID 816787 14 pages2008
[31] A S Bataineh M S M Noorani and I Hashim ldquoHomotopyanalysis method for singular IVPs of Emden-Fowler typerdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 4 pp 1121ndash1131 2009
[32] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[33] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Berlin Germany1988
[34] W S Don and A Solomonoff ldquoAccuracy and speed in com-puting the Chebyshev collocation derivativerdquo SIAM Journal onScientific Computing vol 16 no 6 pp 1253ndash1268 1995
[35] L N Trefethen Spectral Methods in MATLAB SIAM 2000[36] J A CWeideman and S C Reddy ldquoAMATLAB differentiation
matrix suiterdquoACMTransactions onMathematical Software vol26 no 4 pp 465ndash519 2000
of Boyd [28] and Ganapol [29] were obtained only after a fewiterations validates both the higher order convergence and theaccuracy of the present solution methods
Table 2 shows the computed values of 11989110158401015840(0) using
Schemes-0 and -1 for the Pohlhausen flow (120573 = 0 1205731 = 1)For this particular flow the exact value of 11989110158401015840(0) is knownto be 2radic3 The iteration Scheme-0 matches the exact resultafter only 5 iterations and Scheme-1 converges after only 3iterationsThemethod used in Ganapol [29] converged to theexact result after 104 iterations This again demonstrates thesuperior convergence of the present method
Table 3 gives the numerical simulations of the skin fric-tion results for theHomann flowWe observe that the 29-digitresults reported in [29] are achieved in 5 iterations 4 itera-tions and 3 iterations for Scheme-0 -1 and -2 respectivelyThis result indicates that adding an additional level in theQLM-SHAM schemewould further significantly increase theconvergence of the iteration scheme We further note fromTables 1ndash3 that all three schemes converge significantly muchfaster than the spectral homotopy analysismethod on its own
5 Conclusion
In this study we presented three hybrid QLM-SHAM itera-tion schemes for the solution of Falkner-Skan type boundarylayer equations We have shown through numerical exper-imentation that the proposed numerical schemes signifi-cantly enhance the convergence rate of the quasilinearizationmethod By comparison with the most accurate solutionsof the Falkner-Skan equations currently available in theliterature we have shown that the schemes are highly accurateand efficient in terms of the number of iterations requiredto determine the solution to the required level of accuracyThe schemes presented provide robust tools for the efficientsolution of nonlinear equations by offering superior accuracyto many existing methods In addition the approach usedin deriving these schemes provides a suitable framework forextension to higher level schemes by adding more terms ofthe SHAM component of the method
References
[1] R E Bellman and R E Kalaba Quasilinearization and Non-linear Boundary-Value Problems Elsevier New York NY USA1965
[2] K Maleknejad and E Najafi ldquoNumerical solution of nonlinearVolterra integral equations using the idea of quasilinearizationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 1 pp 93ndash100 2011
[3] R Krivec and V B Mandelzweig ldquoNumerical investigation ofquasilinearization method in quantum mechanicsrdquo ComputerPhysics Communications vol 138 pp 69ndash79 2001
[4] V BMandelzweig ldquoQuasilinearizationmethod and its verifica-tion on exactly solvablemodels in quantummechanicsrdquo Journalof Mathematical Physics vol 40 no 12 pp 6266ndash6291 1999
[5] V B Mandelzweig and F Tabakin ldquoQuasilinearizationapproach to nonlinear problems in physics with application tononlinear ODEsrdquo Computer Physics Communications vol 141no 2 pp 268ndash281 2001
[6] V B Mandelzweig ldquoQuasilinearization method nonperturba-tive approach to physical problemsrdquo Physics of Atomic Nucleivol 68 no 7 pp 1227ndash1258 2005
[7] V Lakshmikantham S Leela and S Sivasundaram ldquoExtensionsof the method of quasilinearizationrdquo Journal of OptimizationTheory and Applications vol 87 no 2 pp 379ndash401 1995
[8] V Lakshmikantham ldquoFurther improvement of generalizedquasilinearization methodrdquo Nonlinear Analysis Theory Meth-ods amp Applications vol 27 no 2 pp 223ndash227 1996
[9] K Parand M Ghasemi S Rezazadeh A Peiravi A Ghor-banpour and A Tavakoli Golpaygani ldquoQuasilinearizationapproach for solving Volterrarsquos population modelrdquo Applied andComputational Mathematics vol 9 no 1 pp 95ndash103 2010
[10] J Jiang and A S Vatsala ldquoThe quasilinearization method in thesystem of reaction diffusion equationsrdquo Applied Mathematicsand Computation vol 97 no 2-3 pp 223ndash235 1998
[11] B Ahmad ldquoA quasilinearization method for a class of integro-differential equations with mixed nonlinearitiesrdquo NonlinearAnalysis Real World Applications vol 7 no 5 pp 997ndash10042006
[12] S G Pandit ldquoQuadratically converging iterative schemes fornonlinear Volterra integral equations and an applicationrdquo Jour-nal of Applied Mathematics and Stochastic Analysis vol 10 no2 pp 169ndash178 1997
[13] J I Ramos ldquoPiecewise-quasilinearization techniques for singu-larly perturbed Volterra integro-differential equationsrdquo AppliedMathematics and Computation vol 188 no 2 pp 1221ndash12332007
[14] S Tuffuor and J W Labadie ldquoA nonlinear time variant rainfall-runoff model for augmenting monthly datardquo Water ResourcesResearch vol 10 pp 1161ndash1166 1974
[15] 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
[16] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers amp Fluids vol 39 no 7 pp 1219ndash1225 2010
[17] S J Liao Beyond Perturbation Introduction to HomotopyAnalysis Method Chapman amp HallCRC Press 2003
[18] S Li and S-J Liao ldquoAn analytic approach to solve multiplesolutions of a strongly nonlinear problemrdquoAppliedMathematicsand Computation vol 169 no 2 pp 854ndash865 2005
[19] P Sibanda S Motsa and Z Makukula ldquoA spectral-homotopyanalysis method for heat transfer flow of a third grade fluidbetween parallel platesrdquo International Journal of NumericalMethods for Heat amp Fluid Flow vol 22 no 1 pp 4ndash23 2012
[20] S S Motsa and P Sibanda ldquoOn the solution of MHD flowover a nonlinear stretching sheet by an efficient semi-analyticaltechniquerdquo International Journal for Numerical Methods inFluids vol 68 no 12 pp 1524ndash1537 2012
[21] S Abbasbandy ldquoImproving Newton-Raphson method for non-linear equations bymodifiedAdomiandecompositionmethodrdquoApplied Mathematics and Computation vol 145 no 2-3 pp887ndash893 2003
[22] C Chun ldquoIterative methods improving Newtonrsquos method bythe decomposition methodrdquo Computers amp Mathematics withApplications vol 50 no 10ndash12 pp 1559ndash1568 2005
[23] G Adomian ldquoA review of the decomposition method andsome recent results for nonlinear equationsrdquoMathematical andComputer Modelling vol 13 no 7 pp 17ndash43 1990
Journal of Applied Mathematics 9
[24] G Adomian and R Rach ldquoNoise terms in decompositionsolution seriesrdquo Computers amp Mathematics with Applicationsvol 24 no 11 pp 61ndash64 1992
[25] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer Academic Publishers Group BostonMass USA 1994
[26] C Chun ldquoConstruction of Newton-like iteration methods forsolving nonlinear equationsrdquoNumerische Mathematik vol 104no 3 pp 297ndash315 2006
[27] M A Noor ldquoIterative methods for nonlinear equations usinghomotopy perturbation techniquerdquo Applied Mathematics ampInformation Sciences vol 4 no 2 pp 227ndash235 2010
[28] J P Boyd ldquoThe Blasius function computations before com-puters the value of tricks undergraduate projects and openresearch problemsrdquo SIAM Review vol 50 no 4 pp 791ndash8042008
[29] B D Ganapol ldquoHighly accurate solutions of the BlasiusandFalkner-Skan boundary layer equations via convergence accel-erationrdquo httparxivorgabs10063888
[30] A S Bataineh M S M Noorani and I Hashim ldquoSeriessolution of the multispecies Lotka-Volterra equations by meansof the homotopy analysis methodrdquo Differential Equations andNonlinear Mechanics vol 2008 Article ID 816787 14 pages2008
[31] A S Bataineh M S M Noorani and I Hashim ldquoHomotopyanalysis method for singular IVPs of Emden-Fowler typerdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 4 pp 1121ndash1131 2009
[32] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[33] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Berlin Germany1988
[34] W S Don and A Solomonoff ldquoAccuracy and speed in com-puting the Chebyshev collocation derivativerdquo SIAM Journal onScientific Computing vol 16 no 6 pp 1253ndash1268 1995
[35] L N Trefethen Spectral Methods in MATLAB SIAM 2000[36] J A CWeideman and S C Reddy ldquoAMATLAB differentiation
matrix suiterdquoACMTransactions onMathematical Software vol26 no 4 pp 465ndash519 2000
[24] G Adomian and R Rach ldquoNoise terms in decompositionsolution seriesrdquo Computers amp Mathematics with Applicationsvol 24 no 11 pp 61ndash64 1992
[25] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer Academic Publishers Group BostonMass USA 1994
[26] C Chun ldquoConstruction of Newton-like iteration methods forsolving nonlinear equationsrdquoNumerische Mathematik vol 104no 3 pp 297ndash315 2006
[27] M A Noor ldquoIterative methods for nonlinear equations usinghomotopy perturbation techniquerdquo Applied Mathematics ampInformation Sciences vol 4 no 2 pp 227ndash235 2010
[28] J P Boyd ldquoThe Blasius function computations before com-puters the value of tricks undergraduate projects and openresearch problemsrdquo SIAM Review vol 50 no 4 pp 791ndash8042008
[29] B D Ganapol ldquoHighly accurate solutions of the BlasiusandFalkner-Skan boundary layer equations via convergence accel-erationrdquo httparxivorgabs10063888
[30] A S Bataineh M S M Noorani and I Hashim ldquoSeriessolution of the multispecies Lotka-Volterra equations by meansof the homotopy analysis methodrdquo Differential Equations andNonlinear Mechanics vol 2008 Article ID 816787 14 pages2008
[31] A S Bataineh M S M Noorani and I Hashim ldquoHomotopyanalysis method for singular IVPs of Emden-Fowler typerdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 4 pp 1121ndash1131 2009
[32] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[33] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Berlin Germany1988
[34] W S Don and A Solomonoff ldquoAccuracy and speed in com-puting the Chebyshev collocation derivativerdquo SIAM Journal onScientific Computing vol 16 no 6 pp 1253ndash1268 1995
[35] L N Trefethen Spectral Methods in MATLAB SIAM 2000[36] J A CWeideman and S C Reddy ldquoAMATLAB differentiation
matrix suiterdquoACMTransactions onMathematical Software vol26 no 4 pp 465ndash519 2000
