Research ArticleA Simple Modification of Homotopy Perturbation Method forthe Solution of Blasius Equation in Semi-Infinite Domains
M Aghakhani1 M Suhatril2 M Mohammadhassani2 M Daie3 and A Toghroli2
1Young Researchers and Elite Club Islamic Azad University Ilkhchi Branch Ilkhchi Iran2Department of Civil Engineering University of Malaya 50603 Kuala Lumpur Malaysia3Department of Civil Engineering University of Tabriz Tabriz Iran
Correspondence should be addressed to M Mohammadhassani drmohammadmohammadhassanigmailcom
Received 8 July 2015 Revised 28 August 2015 Accepted 9 September 2015
Academic Editor Gerhard-WilhelmWeber
Copyright copy 2015 M Aghakhani et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
A simple modification of the homotopy perturbationmethod is proposed for the solution of the Blasius equation with two differentboundary conditions Pade approximate is used to deal with the boundary condition at infinity The results obtained from theanalytical method are compared to Howarthrsquos numerical solution and fifth order Runge-Kutta Fehlberg method indicating a verygood agreement The proposed method is a simple and reliable modification of homotopy perturbation method which does notrequire the existence of a small parameter linearization of the equation or computation of Adomianrsquos polynomials
1 Introduction
Nonlinear differential equations are ubiquitous in scienceand technology However finding analytical solutions for thisclass of equations always has been a challenging task In therecent years several approximate methods were proposed forthe analytical solution of nonlinear differential equations thatdo not depend on the existence of a small or large parameterin the equation Among them Homotopy Analysis Method(HAM) [1] Adomian Decomposition Method (ADM) [2]Variational Iteration Method (VIM) [3] Differential Trans-formation Method (DTM) [4] and Homotopy Perturba-tion Method (HPM) [5] can be mentioned The homotopyperturbation method first proposed by He [5] combinesease of implementation of perturbation methods with theflexibility of the homotopy analysis method In additionin the recent years some novel methods for approximatesolution of nonlinear differential equations emerged suchas Optimal Homotopy Asymptotic Method (OHAM) [6]Generalized Homotopy Method (GHM) [7] and (119866
10158401015840119866)-expansion method [8]
In the past many scientists attempted to suggest animprovement to the homotopy perturbation method Theirstudies weremainly focused on enlarging convergence radiusand accelerating convergence of the solution and new
suggestions for homotopy construction as well as alternationof linear and nonlinear part of homotopy based on theapplied problem Jafari and Aminataei [9] introduced anew treatment for homotopy perturbation method whichimproves results from HPM They discussed convergency ofthe proposed method In order to demonstrate efficiencyaccuracy and superiority of the suggested method the newmodification was applied to some experiments in their paperYusufoglu [10] purposed an alternation of HPM for exactsolution of system of linear equations He introduced anaccelerating parameter to solve the system linear equationThrough embedding the accelerating parameters conver-gence improved and required iterations reduced to oneiteration An effective method for convergence improvementof HPM for the solution of fractional differential equationswas suggested by Hosseinnia et al [11] They proposed amethod to select the linear part in the HPM to keep theinherent stability of fractional equations To illustrate theimprovement Riccati fractional differential equations weresolved Results indicated the accuracy and effectiveness ofthe method compared to traditional HPM A modificationof HPM to solve fractional multidimensional diffusion equa-tions is introduced by Kumar et al [12] In their studySumudu transform was utilized for transformation of partialdifferential equations into a new form Dong et al [13]
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 671527 7 pageshttpdxdoiorg1011552015671527
2 Mathematical Problems in Engineering
obtained the solution of strongly nonlinear mixed Volterra-Fredholm integral equation by using an improvement to thehomotopy perturbation method Although the traditionalHPM is divergent in these kinds of problems the proposedmethod is convergent and leads to the exact solution
In the present paper a simple and reliable modificationof HPM is employed for the solution of two different formsof nonlinear Blasius equation in a semi-infinite domain Forthat we consider two forms of the Blasius equation arising influid flow inside the velocity boundary layer as follows
The first form of the Blasius equation is as follows
1198893
1198891205783119891 (120578) +
1
2119891 (120578)
1198892
1198891205782119891 (120578) = 0
119891 (0) = 0 1198911015840
(0) = 0 1198911015840(120578) = 1 as 120578 997888rarr infin
(1)
And the second form is as follows
1198893
1198891205783119891 (120578) +
1
2119891 (120578)
1198892
1198891205782119891 (120578) = 0
119891 (0) = 0 1198911015840
(0) = 1 1198911015840(120578) = 0 as 120578 997888rarr infin
(2)
It can be seen that two equations are the same exceptfor boundary conditions The first form of the equation isthe well-known classical Blasius first derived by Blasius [14]and dates back about a century which describes the velocityprofile of two-dimensional viscous laminar flow over a finiteflat plate This form of the Blasius equation is the simplestform and the origin of all boundary layer equations in fluidmechanics The second form of the equation proposed morerecently arises in the steady free convection about a verticalflat plate embedded in a saturated porous medium [15]Laminar boundary layers at the interface of cocurrent parallelstreams [16] or the flow near the leading edge of a verylong steadily operating conveyor belt [17] Despite the factthat the Blasius equation is a simple third order nonlineardifferential equation with smooth and monotonic solutionthe hope that solution techniques employed to handle thisequation can be extended to the difficult equations motivatedseveral researchers to conduct abundant studies around it bymeans of different analytical and numerical approaches [18ndash21] It is worth mentioning that despite that this problem isa century old it is still being investigated by several authorsand some recent paper was published about it [22ndash26]
In the recent years the homotopy perturbationmethod issuccessfully applied to a wide variety of problems in scienceand technology Analytical solution of the Blasius equationplays a very important role in the design and optimizationof fluid devices In the present paper a simple modificationof this analytical method was employed for the solution ofthe Blasius equation with two different forms of boundaryconditions For that first the equations are transferredto corresponding initial value problems with appropriateboundary conditions Then the initial value problems aresolved by the homotopy perturbation method Finally thediagonal Pade transformation is used to handle the boundarycondition at infinity and enlarge the convergence radius ofthe resulting series For the convenience of the reader we
first introduce the homotopy perturbation method in thefollowing section
2 Homotopy Perturbation Method
The combination of the perturbation method and the homo-topy method is called the homotopy perturbation method(HPM) This method lacks the limitations of the traditionalperturbation methods and can take the full advantage of thetraditional perturbation techniques To illustrate the basicidea of this method [5] consider the following nonlineardifferential equation
where 119860 is a general differential operator 119861 is a boundaryoperator 119891(119903) is a known analytical function and Γ isthe boundary of the domain 120597119906120597119899 denotes differentiationalong the normal drawn outwards from Ω The operator 119860generally speaking can be divided into two parts of 119871 and119873where 119871 is the linear part while119873 is a nonlinear one Hence(3) can be rewritten as follows
Solution of (3) can be written as a power series in119901 as follows
] = ]0+ 119901]1+ 1199012]2+ sdot sdot sdot (8)
Setting 119901 = 1 results in the approximate solution in the formof
119881 = lim119901rarr1
] = ]0+ ]1+ ]2+ sdot sdot sdot (9)
The convergence of (9) has been proven by He [27]
3 Application of HPM-Padeacute to the First Formof the Blasius Equation
In this section we consider the following form of the Blasiusequation
1198893
1198891205783119891 (120578) +
1
2119891 (120578)
1198892
1198891205782119891 (120578) = 0
119891 (0) = 0 1198911015840
(0) = 0 1198911015840(120578) = 1 as 120578 997888rarr infin
(10)
Mathematical Problems in Engineering 3
Due to the boundary condition at infinity direct appli-cation homotopy perturbation method is not appropriatefor this equation and will not yield the desired resultsConsequently a modification in the equation or boundarycondition is necessary Equation (10) is reformulated asfollows
1198893
1198891205783119891 (120578) +
1
2119891 (120578)
1198892
1198891205782119891 (120578) = 0
119891 (0) = 0 1198911015840
(0) = 0 11989110158401015840
(0) = 120590
(11)
where 120590 is an unknown to be determined later This modifi-cation of boundary conditions converts (10) from a boundaryvalue problem to the corresponding initial value problemwhich can be easily handled by HPM This procedure issimilar to the shooting method for numerical solution ofboundary value problems in which the given boundary valueproblem is converted into an initial value problem
Following the outline given in Section 2 a homotopy isconstructed for (11)
119867(119891 119901) = (1 minus 119901) (119891101584010158401015840) + 119901 (119891
101584010158401015840+
1
211989111989110158401015840) = 0 (12)
It is assumed that (12) has a solution in the following form
Substituting 119891 from (13) into (12) and equating terms withidentical powers of 119901 we have
1199010 119891101584010158400
= 0
1198910(0) = 0 119891
1015840
0(0) = 0 119891
10158401015840
0(0) = 120590
1199011 119891101584010158401+
1
2119891011989110158401015840
0= 0
1198911(0) = 0 119891
1015840
1(0) = 0 119891
10158401015840
1(0) = 0
1199012 119891101584010158402+
1
2(119891011989110158401015840
1+ 119891111989110158401015840
0) = 0
1198912(0) = 0 119891
1015840
2(0) = 0 119891
10158401015840
2(0) = 0
1199013 119891101584010158403+
1
2(119891011989110158401015840
2+ 119891111989110158401015840
1+ 119891211989110158401015840
0) = 0
1198913(0) = 0 119891
1015840
3(0) = 0 119891
10158401015840
3(0) = 0
(14)
and Equations (14) can be rapidly solved resulting in the
following
1198910=
1
21205901205782
1198911= minus
1
24012059021205785
1198912=
1145833333
1680000000000012059031205788
1198913= minus
232514881
198000000000000120590412057811
(15)
and
Table 1 Root of the Pade approximates for 120590 = 11989110158401015840(0) the first formof the Blasius equation
Order 11989110158401015840(0) Error
[2 2] 033548 342119864 minus 3
[4 4] 033267 613119864 minus 4
[6 6] 033213 727119864 minus 5
[8 8] 0332052 534119864 minus 6
[10 10] 03320549 244119864 minus 6
Numerical [28] 033205734
Therefore according to (9) the solution of (12) reads
119891 (120578) =1
21205901205782minus
1
24012059021205785+
1145833333
1680000000000012059031205788
minus232514881
198000000000000120590412057811
+181999653
9100000000000000120590512057814
+ sdot sdot sdot
(16)
The unknown initial curvature in (16) 120590 can be deter-mined by imposing the boundary condition of (10) at infinitythat is 1198911015840(infin) = 1 For that Pade approximants of (16) wereformed Pade approximants have the advantage of convertinga function 119891(120578) into a rational function in order to obtainmore information about 119891(120578) A [119872119873] Pade approximateto function 119891(120578) is the quotient of two polynomials 119875(120578)
119872
and 119876(120578)119873
of degrees 119872 and 119873 respectively It is wellknown that if a function is free of singularities on the realaxis the Pade approximant will usually converge on theentire real axis [30] Following the procedure suggested byBoyd [31] in order to determine the unknown parameter120590 in (16) the boundary condition at infinity was imposedto the diagonal Pade approximant Then the roots of Padeapproximant are used for the calculation of the unknownvalue The result of this calculation is presented in Table 1It can be seen from this table that the value obtained for120590 = 119891
10158401015840(0) agrees very well with the numerical results InTables 2ndash4 119891 1198911015840 and 11989110158401015840 obtained from 12th order HPM-Pade approximation are compared with Howarthrsquos numericalsolution [28] Furthermore as it can be seen from Tables2ndash4 purposed modification of HPM-Pade method is moreaccurate than the variational iterationmethod byHe [29] andis valid for a wider range of the solution domain
4 Application of HPM-Padeacute to the SecondForm of the Blasius Equation
The second form of the Blasius equation is as follows
1198893
1198891205783119891 (120578) +
1
2119891 (120578)
1198892
1198891205782119891 (120578) = 0
119891 (0) = 0 1198911015840
(0) = 1 1198911015840(120578) = 0 as 120578 997888rarr infin
(17)
4 Mathematical Problems in Engineering
Table 2 Comparison between 119891(120578) obtained from HPM-Pade with VIM and numerical method first form of the Blasius equation
Similar to the previous section in order to solve (17) thisequation was modified as follows
1198893
1198891205783119891 (120578) +
1
2119891 (120578)
1198892
1198891205782119891 (120578) = 0
119891 (0) = 0 1198911015840
(0) = 1 11989110158401015840
(0) = 120572
(18)
In this equation 120572 is unknown initial curvature of (17)and will be determined later by the application of the bound-ary condition of (17) at infinity According to the instructionsin Section 2 by embedding 119901 isin [0 1] a homotopy wasconstructed for (17) as follows
119867(119891 119901) = (1 minus 119901) (119891101584010158401015840) + 119901(119891
101584010158401015840+
1
211989111989110158401015840) = 0 (19)
assuming that this equation has a solution in the followingform
Table 5 Root of the Pade approximates for 120572 = 11989110158401015840(0) the secondform of the Blasius equation
Order 11989110158401015840(0) Error
[4 4] minus052270 78119864 minus 2
[6 6] minus048944 45119864 minus 2
[8 8] minus046407 20119864 minus 2
[12 12] minus044654 27119864 minus 3
[16 16] minus044565 19119864 minus 4
[20 20] minus044372 19119864 minus 5
5th order Runge-Kutta Fehlberg(present) minus044374
Consequently the solution of (18) can be written asfollows
119891 (120578) = 120578 +1
21205721205782minus
1
481205721205784+
1
24012057221205785+
1
9601205721205786
+11
2016012057221205787+ sdot sdot sdot
(23)
In order to determine the unknown initial curvature 120572 in(23) boundary conditions of (17) at infinity 1198911015840(120578) = 0 mustbe applied For that Pade approximants of (23) which enlargeconvergence radius of the solution were used Then 120572 wasdetermined from 119891
1015840(120578) = 0 to the Pade approximants Initialcurvature of (23) obtained from this method is comparedto fifth order Runge-Kutta Fehlberg numerical method inTable 5 It is worthmentioning that examining the behavior of(17) reveals that its initial curvature must be negative There-fore the negative root of Pade approximants is selected InTables 6ndash8 the result of 8th order HPM-Pade approximationis presented against that of exact (numerical) method It canbe seen that there is a good agreement between the results ofthe proposed method and numerical solution
5 Conclusion
In the present paper a simple modification of the homotopyperturbation method is proposed for the solution of theBlasius equation in semi-infinite domains The equation insemi-infinite domain is transferred into equivalent initialvalue problems which results in appearance of an unknown
Table 6 Comparison between 119891(120578) obtained fromHPM-Pade withthe numerical method second form of the Blasius equation
coefficient In order to determine the unknown coefficientthe boundary condition of the problem at infinity is imposedto Pade approximant of the solution The results are in verygood agreement with numerical and previous data availablein the literature
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
6 Mathematical Problems in Engineering
Table 8 Comparison between 11989110158401015840(120578) obtained from HPM-Padewith the numerical method second form of the Blasius equation
The study presented herein was made possible by the Univer-sity of Malaya Research Grant UMRG RP004D-11AET Theauthors would like to acknowledge the support
References
[1] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004
[2] G Adomian Solving Frontier Problems of Physics The Decom-position Methoc [ie Method] Kluwer Academic PublishersDordrecht The Netherlands 2013
[3] J-H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[4] M-J Jang C-L Chen and Y-C Liy ldquoOn solving the initial-value problems using the differential transformation methodrdquoAppliedMathematics andComputation vol 115 no 2-3 pp 145ndash160 2000
[5] J-H He ldquoHomotopy perturbation method a new nonlinearanalytical techniquerdquo Applied Mathematics and Computationvol 135 no 1 pp 73ndash79 2003
[6] N Herisanu V Marinca and Gh Madescu ldquoAn analyticalapproach to non-linear dynamical model of a permanentmagnet synchronous generatorrdquoWind Energy vol 18 pp 1657ndash1670 2015
[7] H Vazquez-Leal ldquoGeneralized homotopy method for solvingnonlinear differential equationsrdquo Computational and AppliedMathematics vol 33 no 1 pp 275ndash288 2014
[8] M N Alam M A Akbar and S T Mohyud-Din ldquoA novel(G1015840G)-expansionmethod and its application to the Boussinesqequationrdquo Chinese Physics B vol 23 no 2 Article ID 0202032014
[9] M A Jafari and A Aminataei ldquoImproved homotopy perturba-tionmethodrdquo InternationalMathematical Forum vol 5 no 29ndash32 pp 1567ndash1579 2010
[10] E Yusufoglu ldquoAn improvement to homotopy perturbationmethod for solving system of linear equationsrdquo Computers ampMathematics with Applications vol 58 no 11-12 pp 2231ndash22352009
[11] S H Hosseinnia A Ranjbar and S Momani ldquoUsing anenhanced homotopy perturbation method in fractional differ-ential equations via deforming the linear partrdquo Computers andMathematics with Applications vol 56 no 12 pp 3138ndash31492008
[12] D Kumar J Singh and S Kumar ldquoNumerical computationof fractional multi-dimensional diffusion equations by usinga modified homotopy perturbation methodrdquo Journal of theAssociation of Arab Universities for Basic and Applied Sciencesvol 17 pp 20ndash26 2015
[13] C Dong Z Chen and W Jiang ldquoA modified homotopyperturbation method for solving the nonlinear mixed Volterra-Fredholm integral equationrdquo Journal of Computational andApplied Mathematics vol 239 pp 359ndash366 2013
[14] H Blasius ldquoGrenzschichten in Flussigkeiten mit kleiner Rei-bungrdquo Zeitschrift fur Angewandte Mathematik und Physik vol56 pp 1ndash37 1908
[15] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application toheat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[16] O E Potter ldquoLaminar boundary layers at the interface of co-current parallel streamsrdquo The Quarterly Journal of Mechanicsand Applied Mathematics vol 10 pp 302ndash311 1957
[17] J A Ackroyd ldquoOn the laminar compressible boundary layerwith stationary origin on a moving flat wallrdquo MathematicalProceedings of the Cambridge Philosophical Society vol 63 no3 pp 871ndash888 1967
[18] J-H He ldquoA simple perturbation approach to Blasius equationrdquoApplied Mathematics and Computation vol 140 no 2-3 pp217ndash222 2003
[19] S Abbasbandy ldquoA numerical solution of Blasius equationby Adomianrsquos decomposition method and comparison withhomotopy perturbation methodrdquo Chaos Solitons amp Fractalsvol 31 no 1 pp 257ndash260 2007
[20] L-T Yu and C-K Chen ldquoThe solution of the blasius equationby the differential transformation methodrdquo Mathematical andComputer Modelling vol 28 no 1 pp 101ndash111 1998
[21] J H He ldquoApproximate analytical solution of Blasiusrsquo equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 4 no 1 pp 75ndash78 1999
[22] B I Yun ldquoConstructing uniform approximate analytical solu-tions for the Blasius problemrdquo Abstract and Applied Analysisvol 2014 Article ID 495734 6 pages 2014
[23] I Ahmad andM Bilal ldquoNumerical solution of blasius equationthrough neural networks algorithmrdquoAmerican Journal of Com-putational Mathematics vol 4 no 3 pp 223ndash232 2014
[24] V Marinca and N Herisanu ldquoThe optimal homotopy asymp-toticmethod for solving Blasius equationrdquoAppliedMathematicsand Computation vol 231 pp 134ndash139 2014
[25] A Ebaid and N Al-Armani ldquoA new approach for a class of theblasius problem via a transformation and adomianrsquos methodrdquoAbstract and Applied Analysis vol 2013 Article ID 753049 8pages 2013
[26] O Costin T E Kim and S Tanveer ldquoA quasi-solution approachto nonlinear problemsmdashthe case of the Blasius similaritysolutionrdquo Fluid Dynamics Research vol 46 no 3 2014
[27] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
Mathematical Problems in Engineering 7
[28] L Howarth ldquoOn the solution of the laminar boundary layerequationsrdquo Proceedings of the Royal Society A vol 164 no 919pp 547ndash579 1938
[29] J H He ldquoApproximate analytical solution of Blasiusrsquo equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 3 no 4 pp 260ndash263 1998
[30] G A Baker and P R Graves-Morris Essentials of Pade Approx-imants Academic Press New York NY USA 1975
[31] J P Boyd ldquoPade approximant algorithm for solving nonlinearordinary differential equation boundary value problems on anunbounded domainrdquo Computers in Physics vol 11 no 3 article299 1997
obtained the solution of strongly nonlinear mixed Volterra-Fredholm integral equation by using an improvement to thehomotopy perturbation method Although the traditionalHPM is divergent in these kinds of problems the proposedmethod is convergent and leads to the exact solution
In the present paper a simple and reliable modificationof HPM is employed for the solution of two different formsof nonlinear Blasius equation in a semi-infinite domain Forthat we consider two forms of the Blasius equation arising influid flow inside the velocity boundary layer as follows
The first form of the Blasius equation is as follows
1198893
1198891205783119891 (120578) +
1
2119891 (120578)
1198892
1198891205782119891 (120578) = 0
119891 (0) = 0 1198911015840
(0) = 0 1198911015840(120578) = 1 as 120578 997888rarr infin
(1)
And the second form is as follows
1198893
1198891205783119891 (120578) +
1
2119891 (120578)
1198892
1198891205782119891 (120578) = 0
119891 (0) = 0 1198911015840
(0) = 1 1198911015840(120578) = 0 as 120578 997888rarr infin
(2)
It can be seen that two equations are the same exceptfor boundary conditions The first form of the equation isthe well-known classical Blasius first derived by Blasius [14]and dates back about a century which describes the velocityprofile of two-dimensional viscous laminar flow over a finiteflat plate This form of the Blasius equation is the simplestform and the origin of all boundary layer equations in fluidmechanics The second form of the equation proposed morerecently arises in the steady free convection about a verticalflat plate embedded in a saturated porous medium [15]Laminar boundary layers at the interface of cocurrent parallelstreams [16] or the flow near the leading edge of a verylong steadily operating conveyor belt [17] Despite the factthat the Blasius equation is a simple third order nonlineardifferential equation with smooth and monotonic solutionthe hope that solution techniques employed to handle thisequation can be extended to the difficult equations motivatedseveral researchers to conduct abundant studies around it bymeans of different analytical and numerical approaches [18ndash21] It is worth mentioning that despite that this problem isa century old it is still being investigated by several authorsand some recent paper was published about it [22ndash26]
In the recent years the homotopy perturbationmethod issuccessfully applied to a wide variety of problems in scienceand technology Analytical solution of the Blasius equationplays a very important role in the design and optimizationof fluid devices In the present paper a simple modificationof this analytical method was employed for the solution ofthe Blasius equation with two different forms of boundaryconditions For that first the equations are transferredto corresponding initial value problems with appropriateboundary conditions Then the initial value problems aresolved by the homotopy perturbation method Finally thediagonal Pade transformation is used to handle the boundarycondition at infinity and enlarge the convergence radius ofthe resulting series For the convenience of the reader we
first introduce the homotopy perturbation method in thefollowing section
2 Homotopy Perturbation Method
The combination of the perturbation method and the homo-topy method is called the homotopy perturbation method(HPM) This method lacks the limitations of the traditionalperturbation methods and can take the full advantage of thetraditional perturbation techniques To illustrate the basicidea of this method [5] consider the following nonlineardifferential equation
where 119860 is a general differential operator 119861 is a boundaryoperator 119891(119903) is a known analytical function and Γ isthe boundary of the domain 120597119906120597119899 denotes differentiationalong the normal drawn outwards from Ω The operator 119860generally speaking can be divided into two parts of 119871 and119873where 119871 is the linear part while119873 is a nonlinear one Hence(3) can be rewritten as follows
Solution of (3) can be written as a power series in119901 as follows
] = ]0+ 119901]1+ 1199012]2+ sdot sdot sdot (8)
Setting 119901 = 1 results in the approximate solution in the formof
119881 = lim119901rarr1
] = ]0+ ]1+ ]2+ sdot sdot sdot (9)
The convergence of (9) has been proven by He [27]
3 Application of HPM-Padeacute to the First Formof the Blasius Equation
In this section we consider the following form of the Blasiusequation
1198893
1198891205783119891 (120578) +
1
2119891 (120578)
1198892
1198891205782119891 (120578) = 0
119891 (0) = 0 1198911015840
(0) = 0 1198911015840(120578) = 1 as 120578 997888rarr infin
(10)
Mathematical Problems in Engineering 3
Due to the boundary condition at infinity direct appli-cation homotopy perturbation method is not appropriatefor this equation and will not yield the desired resultsConsequently a modification in the equation or boundarycondition is necessary Equation (10) is reformulated asfollows
1198893
1198891205783119891 (120578) +
1
2119891 (120578)
1198892
1198891205782119891 (120578) = 0
119891 (0) = 0 1198911015840
(0) = 0 11989110158401015840
(0) = 120590
(11)
where 120590 is an unknown to be determined later This modifi-cation of boundary conditions converts (10) from a boundaryvalue problem to the corresponding initial value problemwhich can be easily handled by HPM This procedure issimilar to the shooting method for numerical solution ofboundary value problems in which the given boundary valueproblem is converted into an initial value problem
Following the outline given in Section 2 a homotopy isconstructed for (11)
119867(119891 119901) = (1 minus 119901) (119891101584010158401015840) + 119901 (119891
101584010158401015840+
1
211989111989110158401015840) = 0 (12)
It is assumed that (12) has a solution in the following form
Substituting 119891 from (13) into (12) and equating terms withidentical powers of 119901 we have
1199010 119891101584010158400
= 0
1198910(0) = 0 119891
1015840
0(0) = 0 119891
10158401015840
0(0) = 120590
1199011 119891101584010158401+
1
2119891011989110158401015840
0= 0
1198911(0) = 0 119891
1015840
1(0) = 0 119891
10158401015840
1(0) = 0
1199012 119891101584010158402+
1
2(119891011989110158401015840
1+ 119891111989110158401015840
0) = 0
1198912(0) = 0 119891
1015840
2(0) = 0 119891
10158401015840
2(0) = 0
1199013 119891101584010158403+
1
2(119891011989110158401015840
2+ 119891111989110158401015840
1+ 119891211989110158401015840
0) = 0
1198913(0) = 0 119891
1015840
3(0) = 0 119891
10158401015840
3(0) = 0
(14)
and Equations (14) can be rapidly solved resulting in the
following
1198910=
1
21205901205782
1198911= minus
1
24012059021205785
1198912=
1145833333
1680000000000012059031205788
1198913= minus
232514881
198000000000000120590412057811
(15)
and
Table 1 Root of the Pade approximates for 120590 = 11989110158401015840(0) the first formof the Blasius equation
Order 11989110158401015840(0) Error
[2 2] 033548 342119864 minus 3
[4 4] 033267 613119864 minus 4
[6 6] 033213 727119864 minus 5
[8 8] 0332052 534119864 minus 6
[10 10] 03320549 244119864 minus 6
Numerical [28] 033205734
Therefore according to (9) the solution of (12) reads
119891 (120578) =1
21205901205782minus
1
24012059021205785+
1145833333
1680000000000012059031205788
minus232514881
198000000000000120590412057811
+181999653
9100000000000000120590512057814
+ sdot sdot sdot
(16)
The unknown initial curvature in (16) 120590 can be deter-mined by imposing the boundary condition of (10) at infinitythat is 1198911015840(infin) = 1 For that Pade approximants of (16) wereformed Pade approximants have the advantage of convertinga function 119891(120578) into a rational function in order to obtainmore information about 119891(120578) A [119872119873] Pade approximateto function 119891(120578) is the quotient of two polynomials 119875(120578)
119872
and 119876(120578)119873
of degrees 119872 and 119873 respectively It is wellknown that if a function is free of singularities on the realaxis the Pade approximant will usually converge on theentire real axis [30] Following the procedure suggested byBoyd [31] in order to determine the unknown parameter120590 in (16) the boundary condition at infinity was imposedto the diagonal Pade approximant Then the roots of Padeapproximant are used for the calculation of the unknownvalue The result of this calculation is presented in Table 1It can be seen from this table that the value obtained for120590 = 119891
10158401015840(0) agrees very well with the numerical results InTables 2ndash4 119891 1198911015840 and 11989110158401015840 obtained from 12th order HPM-Pade approximation are compared with Howarthrsquos numericalsolution [28] Furthermore as it can be seen from Tables2ndash4 purposed modification of HPM-Pade method is moreaccurate than the variational iterationmethod byHe [29] andis valid for a wider range of the solution domain
4 Application of HPM-Padeacute to the SecondForm of the Blasius Equation
The second form of the Blasius equation is as follows
1198893
1198891205783119891 (120578) +
1
2119891 (120578)
1198892
1198891205782119891 (120578) = 0
119891 (0) = 0 1198911015840
(0) = 1 1198911015840(120578) = 0 as 120578 997888rarr infin
(17)
4 Mathematical Problems in Engineering
Table 2 Comparison between 119891(120578) obtained from HPM-Pade with VIM and numerical method first form of the Blasius equation
Similar to the previous section in order to solve (17) thisequation was modified as follows
1198893
1198891205783119891 (120578) +
1
2119891 (120578)
1198892
1198891205782119891 (120578) = 0
119891 (0) = 0 1198911015840
(0) = 1 11989110158401015840
(0) = 120572
(18)
In this equation 120572 is unknown initial curvature of (17)and will be determined later by the application of the bound-ary condition of (17) at infinity According to the instructionsin Section 2 by embedding 119901 isin [0 1] a homotopy wasconstructed for (17) as follows
119867(119891 119901) = (1 minus 119901) (119891101584010158401015840) + 119901(119891
101584010158401015840+
1
211989111989110158401015840) = 0 (19)
assuming that this equation has a solution in the followingform
Table 5 Root of the Pade approximates for 120572 = 11989110158401015840(0) the secondform of the Blasius equation
Order 11989110158401015840(0) Error
[4 4] minus052270 78119864 minus 2
[6 6] minus048944 45119864 minus 2
[8 8] minus046407 20119864 minus 2
[12 12] minus044654 27119864 minus 3
[16 16] minus044565 19119864 minus 4
[20 20] minus044372 19119864 minus 5
5th order Runge-Kutta Fehlberg(present) minus044374
Consequently the solution of (18) can be written asfollows
119891 (120578) = 120578 +1
21205721205782minus
1
481205721205784+
1
24012057221205785+
1
9601205721205786
+11
2016012057221205787+ sdot sdot sdot
(23)
In order to determine the unknown initial curvature 120572 in(23) boundary conditions of (17) at infinity 1198911015840(120578) = 0 mustbe applied For that Pade approximants of (23) which enlargeconvergence radius of the solution were used Then 120572 wasdetermined from 119891
1015840(120578) = 0 to the Pade approximants Initialcurvature of (23) obtained from this method is comparedto fifth order Runge-Kutta Fehlberg numerical method inTable 5 It is worthmentioning that examining the behavior of(17) reveals that its initial curvature must be negative There-fore the negative root of Pade approximants is selected InTables 6ndash8 the result of 8th order HPM-Pade approximationis presented against that of exact (numerical) method It canbe seen that there is a good agreement between the results ofthe proposed method and numerical solution
5 Conclusion
In the present paper a simple modification of the homotopyperturbation method is proposed for the solution of theBlasius equation in semi-infinite domains The equation insemi-infinite domain is transferred into equivalent initialvalue problems which results in appearance of an unknown
Table 6 Comparison between 119891(120578) obtained fromHPM-Pade withthe numerical method second form of the Blasius equation
coefficient In order to determine the unknown coefficientthe boundary condition of the problem at infinity is imposedto Pade approximant of the solution The results are in verygood agreement with numerical and previous data availablein the literature
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
6 Mathematical Problems in Engineering
Table 8 Comparison between 11989110158401015840(120578) obtained from HPM-Padewith the numerical method second form of the Blasius equation
The study presented herein was made possible by the Univer-sity of Malaya Research Grant UMRG RP004D-11AET Theauthors would like to acknowledge the support
References
[1] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004
[2] G Adomian Solving Frontier Problems of Physics The Decom-position Methoc [ie Method] Kluwer Academic PublishersDordrecht The Netherlands 2013
[3] J-H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[4] M-J Jang C-L Chen and Y-C Liy ldquoOn solving the initial-value problems using the differential transformation methodrdquoAppliedMathematics andComputation vol 115 no 2-3 pp 145ndash160 2000
[5] J-H He ldquoHomotopy perturbation method a new nonlinearanalytical techniquerdquo Applied Mathematics and Computationvol 135 no 1 pp 73ndash79 2003
[6] N Herisanu V Marinca and Gh Madescu ldquoAn analyticalapproach to non-linear dynamical model of a permanentmagnet synchronous generatorrdquoWind Energy vol 18 pp 1657ndash1670 2015
[7] H Vazquez-Leal ldquoGeneralized homotopy method for solvingnonlinear differential equationsrdquo Computational and AppliedMathematics vol 33 no 1 pp 275ndash288 2014
[8] M N Alam M A Akbar and S T Mohyud-Din ldquoA novel(G1015840G)-expansionmethod and its application to the Boussinesqequationrdquo Chinese Physics B vol 23 no 2 Article ID 0202032014
[9] M A Jafari and A Aminataei ldquoImproved homotopy perturba-tionmethodrdquo InternationalMathematical Forum vol 5 no 29ndash32 pp 1567ndash1579 2010
[10] E Yusufoglu ldquoAn improvement to homotopy perturbationmethod for solving system of linear equationsrdquo Computers ampMathematics with Applications vol 58 no 11-12 pp 2231ndash22352009
[11] S H Hosseinnia A Ranjbar and S Momani ldquoUsing anenhanced homotopy perturbation method in fractional differ-ential equations via deforming the linear partrdquo Computers andMathematics with Applications vol 56 no 12 pp 3138ndash31492008
[12] D Kumar J Singh and S Kumar ldquoNumerical computationof fractional multi-dimensional diffusion equations by usinga modified homotopy perturbation methodrdquo Journal of theAssociation of Arab Universities for Basic and Applied Sciencesvol 17 pp 20ndash26 2015
[13] C Dong Z Chen and W Jiang ldquoA modified homotopyperturbation method for solving the nonlinear mixed Volterra-Fredholm integral equationrdquo Journal of Computational andApplied Mathematics vol 239 pp 359ndash366 2013
[14] H Blasius ldquoGrenzschichten in Flussigkeiten mit kleiner Rei-bungrdquo Zeitschrift fur Angewandte Mathematik und Physik vol56 pp 1ndash37 1908
[15] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application toheat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[16] O E Potter ldquoLaminar boundary layers at the interface of co-current parallel streamsrdquo The Quarterly Journal of Mechanicsand Applied Mathematics vol 10 pp 302ndash311 1957
[17] J A Ackroyd ldquoOn the laminar compressible boundary layerwith stationary origin on a moving flat wallrdquo MathematicalProceedings of the Cambridge Philosophical Society vol 63 no3 pp 871ndash888 1967
[18] J-H He ldquoA simple perturbation approach to Blasius equationrdquoApplied Mathematics and Computation vol 140 no 2-3 pp217ndash222 2003
[19] S Abbasbandy ldquoA numerical solution of Blasius equationby Adomianrsquos decomposition method and comparison withhomotopy perturbation methodrdquo Chaos Solitons amp Fractalsvol 31 no 1 pp 257ndash260 2007
[20] L-T Yu and C-K Chen ldquoThe solution of the blasius equationby the differential transformation methodrdquo Mathematical andComputer Modelling vol 28 no 1 pp 101ndash111 1998
[21] J H He ldquoApproximate analytical solution of Blasiusrsquo equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 4 no 1 pp 75ndash78 1999
[22] B I Yun ldquoConstructing uniform approximate analytical solu-tions for the Blasius problemrdquo Abstract and Applied Analysisvol 2014 Article ID 495734 6 pages 2014
[23] I Ahmad andM Bilal ldquoNumerical solution of blasius equationthrough neural networks algorithmrdquoAmerican Journal of Com-putational Mathematics vol 4 no 3 pp 223ndash232 2014
[24] V Marinca and N Herisanu ldquoThe optimal homotopy asymp-toticmethod for solving Blasius equationrdquoAppliedMathematicsand Computation vol 231 pp 134ndash139 2014
[25] A Ebaid and N Al-Armani ldquoA new approach for a class of theblasius problem via a transformation and adomianrsquos methodrdquoAbstract and Applied Analysis vol 2013 Article ID 753049 8pages 2013
[26] O Costin T E Kim and S Tanveer ldquoA quasi-solution approachto nonlinear problemsmdashthe case of the Blasius similaritysolutionrdquo Fluid Dynamics Research vol 46 no 3 2014
[27] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
Mathematical Problems in Engineering 7
[28] L Howarth ldquoOn the solution of the laminar boundary layerequationsrdquo Proceedings of the Royal Society A vol 164 no 919pp 547ndash579 1938
[29] J H He ldquoApproximate analytical solution of Blasiusrsquo equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 3 no 4 pp 260ndash263 1998
[30] G A Baker and P R Graves-Morris Essentials of Pade Approx-imants Academic Press New York NY USA 1975
[31] J P Boyd ldquoPade approximant algorithm for solving nonlinearordinary differential equation boundary value problems on anunbounded domainrdquo Computers in Physics vol 11 no 3 article299 1997
Due to the boundary condition at infinity direct appli-cation homotopy perturbation method is not appropriatefor this equation and will not yield the desired resultsConsequently a modification in the equation or boundarycondition is necessary Equation (10) is reformulated asfollows
1198893
1198891205783119891 (120578) +
1
2119891 (120578)
1198892
1198891205782119891 (120578) = 0
119891 (0) = 0 1198911015840
(0) = 0 11989110158401015840
(0) = 120590
(11)
where 120590 is an unknown to be determined later This modifi-cation of boundary conditions converts (10) from a boundaryvalue problem to the corresponding initial value problemwhich can be easily handled by HPM This procedure issimilar to the shooting method for numerical solution ofboundary value problems in which the given boundary valueproblem is converted into an initial value problem
Following the outline given in Section 2 a homotopy isconstructed for (11)
119867(119891 119901) = (1 minus 119901) (119891101584010158401015840) + 119901 (119891
101584010158401015840+
1
211989111989110158401015840) = 0 (12)
It is assumed that (12) has a solution in the following form
Substituting 119891 from (13) into (12) and equating terms withidentical powers of 119901 we have
1199010 119891101584010158400
= 0
1198910(0) = 0 119891
1015840
0(0) = 0 119891
10158401015840
0(0) = 120590
1199011 119891101584010158401+
1
2119891011989110158401015840
0= 0
1198911(0) = 0 119891
1015840
1(0) = 0 119891
10158401015840
1(0) = 0
1199012 119891101584010158402+
1
2(119891011989110158401015840
1+ 119891111989110158401015840
0) = 0
1198912(0) = 0 119891
1015840
2(0) = 0 119891
10158401015840
2(0) = 0
1199013 119891101584010158403+
1
2(119891011989110158401015840
2+ 119891111989110158401015840
1+ 119891211989110158401015840
0) = 0
1198913(0) = 0 119891
1015840
3(0) = 0 119891
10158401015840
3(0) = 0
(14)
and Equations (14) can be rapidly solved resulting in the
following
1198910=
1
21205901205782
1198911= minus
1
24012059021205785
1198912=
1145833333
1680000000000012059031205788
1198913= minus
232514881
198000000000000120590412057811
(15)
and
Table 1 Root of the Pade approximates for 120590 = 11989110158401015840(0) the first formof the Blasius equation
Order 11989110158401015840(0) Error
[2 2] 033548 342119864 minus 3
[4 4] 033267 613119864 minus 4
[6 6] 033213 727119864 minus 5
[8 8] 0332052 534119864 minus 6
[10 10] 03320549 244119864 minus 6
Numerical [28] 033205734
Therefore according to (9) the solution of (12) reads
119891 (120578) =1
21205901205782minus
1
24012059021205785+
1145833333
1680000000000012059031205788
minus232514881
198000000000000120590412057811
+181999653
9100000000000000120590512057814
+ sdot sdot sdot
(16)
The unknown initial curvature in (16) 120590 can be deter-mined by imposing the boundary condition of (10) at infinitythat is 1198911015840(infin) = 1 For that Pade approximants of (16) wereformed Pade approximants have the advantage of convertinga function 119891(120578) into a rational function in order to obtainmore information about 119891(120578) A [119872119873] Pade approximateto function 119891(120578) is the quotient of two polynomials 119875(120578)
119872
and 119876(120578)119873
of degrees 119872 and 119873 respectively It is wellknown that if a function is free of singularities on the realaxis the Pade approximant will usually converge on theentire real axis [30] Following the procedure suggested byBoyd [31] in order to determine the unknown parameter120590 in (16) the boundary condition at infinity was imposedto the diagonal Pade approximant Then the roots of Padeapproximant are used for the calculation of the unknownvalue The result of this calculation is presented in Table 1It can be seen from this table that the value obtained for120590 = 119891
10158401015840(0) agrees very well with the numerical results InTables 2ndash4 119891 1198911015840 and 11989110158401015840 obtained from 12th order HPM-Pade approximation are compared with Howarthrsquos numericalsolution [28] Furthermore as it can be seen from Tables2ndash4 purposed modification of HPM-Pade method is moreaccurate than the variational iterationmethod byHe [29] andis valid for a wider range of the solution domain
4 Application of HPM-Padeacute to the SecondForm of the Blasius Equation
The second form of the Blasius equation is as follows
1198893
1198891205783119891 (120578) +
1
2119891 (120578)
1198892
1198891205782119891 (120578) = 0
119891 (0) = 0 1198911015840
(0) = 1 1198911015840(120578) = 0 as 120578 997888rarr infin
(17)
4 Mathematical Problems in Engineering
Table 2 Comparison between 119891(120578) obtained from HPM-Pade with VIM and numerical method first form of the Blasius equation
Similar to the previous section in order to solve (17) thisequation was modified as follows
1198893
1198891205783119891 (120578) +
1
2119891 (120578)
1198892
1198891205782119891 (120578) = 0
119891 (0) = 0 1198911015840
(0) = 1 11989110158401015840
(0) = 120572
(18)
In this equation 120572 is unknown initial curvature of (17)and will be determined later by the application of the bound-ary condition of (17) at infinity According to the instructionsin Section 2 by embedding 119901 isin [0 1] a homotopy wasconstructed for (17) as follows
119867(119891 119901) = (1 minus 119901) (119891101584010158401015840) + 119901(119891
101584010158401015840+
1
211989111989110158401015840) = 0 (19)
assuming that this equation has a solution in the followingform
Table 5 Root of the Pade approximates for 120572 = 11989110158401015840(0) the secondform of the Blasius equation
Order 11989110158401015840(0) Error
[4 4] minus052270 78119864 minus 2
[6 6] minus048944 45119864 minus 2
[8 8] minus046407 20119864 minus 2
[12 12] minus044654 27119864 minus 3
[16 16] minus044565 19119864 minus 4
[20 20] minus044372 19119864 minus 5
5th order Runge-Kutta Fehlberg(present) minus044374
Consequently the solution of (18) can be written asfollows
119891 (120578) = 120578 +1
21205721205782minus
1
481205721205784+
1
24012057221205785+
1
9601205721205786
+11
2016012057221205787+ sdot sdot sdot
(23)
In order to determine the unknown initial curvature 120572 in(23) boundary conditions of (17) at infinity 1198911015840(120578) = 0 mustbe applied For that Pade approximants of (23) which enlargeconvergence radius of the solution were used Then 120572 wasdetermined from 119891
1015840(120578) = 0 to the Pade approximants Initialcurvature of (23) obtained from this method is comparedto fifth order Runge-Kutta Fehlberg numerical method inTable 5 It is worthmentioning that examining the behavior of(17) reveals that its initial curvature must be negative There-fore the negative root of Pade approximants is selected InTables 6ndash8 the result of 8th order HPM-Pade approximationis presented against that of exact (numerical) method It canbe seen that there is a good agreement between the results ofthe proposed method and numerical solution
5 Conclusion
In the present paper a simple modification of the homotopyperturbation method is proposed for the solution of theBlasius equation in semi-infinite domains The equation insemi-infinite domain is transferred into equivalent initialvalue problems which results in appearance of an unknown
Table 6 Comparison between 119891(120578) obtained fromHPM-Pade withthe numerical method second form of the Blasius equation
coefficient In order to determine the unknown coefficientthe boundary condition of the problem at infinity is imposedto Pade approximant of the solution The results are in verygood agreement with numerical and previous data availablein the literature
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
6 Mathematical Problems in Engineering
Table 8 Comparison between 11989110158401015840(120578) obtained from HPM-Padewith the numerical method second form of the Blasius equation
The study presented herein was made possible by the Univer-sity of Malaya Research Grant UMRG RP004D-11AET Theauthors would like to acknowledge the support
References
[1] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004
[2] G Adomian Solving Frontier Problems of Physics The Decom-position Methoc [ie Method] Kluwer Academic PublishersDordrecht The Netherlands 2013
[3] J-H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[4] M-J Jang C-L Chen and Y-C Liy ldquoOn solving the initial-value problems using the differential transformation methodrdquoAppliedMathematics andComputation vol 115 no 2-3 pp 145ndash160 2000
[5] J-H He ldquoHomotopy perturbation method a new nonlinearanalytical techniquerdquo Applied Mathematics and Computationvol 135 no 1 pp 73ndash79 2003
[6] N Herisanu V Marinca and Gh Madescu ldquoAn analyticalapproach to non-linear dynamical model of a permanentmagnet synchronous generatorrdquoWind Energy vol 18 pp 1657ndash1670 2015
[7] H Vazquez-Leal ldquoGeneralized homotopy method for solvingnonlinear differential equationsrdquo Computational and AppliedMathematics vol 33 no 1 pp 275ndash288 2014
[8] M N Alam M A Akbar and S T Mohyud-Din ldquoA novel(G1015840G)-expansionmethod and its application to the Boussinesqequationrdquo Chinese Physics B vol 23 no 2 Article ID 0202032014
[9] M A Jafari and A Aminataei ldquoImproved homotopy perturba-tionmethodrdquo InternationalMathematical Forum vol 5 no 29ndash32 pp 1567ndash1579 2010
[10] E Yusufoglu ldquoAn improvement to homotopy perturbationmethod for solving system of linear equationsrdquo Computers ampMathematics with Applications vol 58 no 11-12 pp 2231ndash22352009
[11] S H Hosseinnia A Ranjbar and S Momani ldquoUsing anenhanced homotopy perturbation method in fractional differ-ential equations via deforming the linear partrdquo Computers andMathematics with Applications vol 56 no 12 pp 3138ndash31492008
[12] D Kumar J Singh and S Kumar ldquoNumerical computationof fractional multi-dimensional diffusion equations by usinga modified homotopy perturbation methodrdquo Journal of theAssociation of Arab Universities for Basic and Applied Sciencesvol 17 pp 20ndash26 2015
[13] C Dong Z Chen and W Jiang ldquoA modified homotopyperturbation method for solving the nonlinear mixed Volterra-Fredholm integral equationrdquo Journal of Computational andApplied Mathematics vol 239 pp 359ndash366 2013
[14] H Blasius ldquoGrenzschichten in Flussigkeiten mit kleiner Rei-bungrdquo Zeitschrift fur Angewandte Mathematik und Physik vol56 pp 1ndash37 1908
[15] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application toheat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[16] O E Potter ldquoLaminar boundary layers at the interface of co-current parallel streamsrdquo The Quarterly Journal of Mechanicsand Applied Mathematics vol 10 pp 302ndash311 1957
[17] J A Ackroyd ldquoOn the laminar compressible boundary layerwith stationary origin on a moving flat wallrdquo MathematicalProceedings of the Cambridge Philosophical Society vol 63 no3 pp 871ndash888 1967
[18] J-H He ldquoA simple perturbation approach to Blasius equationrdquoApplied Mathematics and Computation vol 140 no 2-3 pp217ndash222 2003
[19] S Abbasbandy ldquoA numerical solution of Blasius equationby Adomianrsquos decomposition method and comparison withhomotopy perturbation methodrdquo Chaos Solitons amp Fractalsvol 31 no 1 pp 257ndash260 2007
[20] L-T Yu and C-K Chen ldquoThe solution of the blasius equationby the differential transformation methodrdquo Mathematical andComputer Modelling vol 28 no 1 pp 101ndash111 1998
[21] J H He ldquoApproximate analytical solution of Blasiusrsquo equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 4 no 1 pp 75ndash78 1999
[22] B I Yun ldquoConstructing uniform approximate analytical solu-tions for the Blasius problemrdquo Abstract and Applied Analysisvol 2014 Article ID 495734 6 pages 2014
[23] I Ahmad andM Bilal ldquoNumerical solution of blasius equationthrough neural networks algorithmrdquoAmerican Journal of Com-putational Mathematics vol 4 no 3 pp 223ndash232 2014
[24] V Marinca and N Herisanu ldquoThe optimal homotopy asymp-toticmethod for solving Blasius equationrdquoAppliedMathematicsand Computation vol 231 pp 134ndash139 2014
[25] A Ebaid and N Al-Armani ldquoA new approach for a class of theblasius problem via a transformation and adomianrsquos methodrdquoAbstract and Applied Analysis vol 2013 Article ID 753049 8pages 2013
[26] O Costin T E Kim and S Tanveer ldquoA quasi-solution approachto nonlinear problemsmdashthe case of the Blasius similaritysolutionrdquo Fluid Dynamics Research vol 46 no 3 2014
[27] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
Mathematical Problems in Engineering 7
[28] L Howarth ldquoOn the solution of the laminar boundary layerequationsrdquo Proceedings of the Royal Society A vol 164 no 919pp 547ndash579 1938
[29] J H He ldquoApproximate analytical solution of Blasiusrsquo equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 3 no 4 pp 260ndash263 1998
[30] G A Baker and P R Graves-Morris Essentials of Pade Approx-imants Academic Press New York NY USA 1975
[31] J P Boyd ldquoPade approximant algorithm for solving nonlinearordinary differential equation boundary value problems on anunbounded domainrdquo Computers in Physics vol 11 no 3 article299 1997
Similar to the previous section in order to solve (17) thisequation was modified as follows
1198893
1198891205783119891 (120578) +
1
2119891 (120578)
1198892
1198891205782119891 (120578) = 0
119891 (0) = 0 1198911015840
(0) = 1 11989110158401015840
(0) = 120572
(18)
In this equation 120572 is unknown initial curvature of (17)and will be determined later by the application of the bound-ary condition of (17) at infinity According to the instructionsin Section 2 by embedding 119901 isin [0 1] a homotopy wasconstructed for (17) as follows
119867(119891 119901) = (1 minus 119901) (119891101584010158401015840) + 119901(119891
101584010158401015840+
1
211989111989110158401015840) = 0 (19)
assuming that this equation has a solution in the followingform
Table 5 Root of the Pade approximates for 120572 = 11989110158401015840(0) the secondform of the Blasius equation
Order 11989110158401015840(0) Error
[4 4] minus052270 78119864 minus 2
[6 6] minus048944 45119864 minus 2
[8 8] minus046407 20119864 minus 2
[12 12] minus044654 27119864 minus 3
[16 16] minus044565 19119864 minus 4
[20 20] minus044372 19119864 minus 5
5th order Runge-Kutta Fehlberg(present) minus044374
Consequently the solution of (18) can be written asfollows
119891 (120578) = 120578 +1
21205721205782minus
1
481205721205784+
1
24012057221205785+
1
9601205721205786
+11
2016012057221205787+ sdot sdot sdot
(23)
In order to determine the unknown initial curvature 120572 in(23) boundary conditions of (17) at infinity 1198911015840(120578) = 0 mustbe applied For that Pade approximants of (23) which enlargeconvergence radius of the solution were used Then 120572 wasdetermined from 119891
1015840(120578) = 0 to the Pade approximants Initialcurvature of (23) obtained from this method is comparedto fifth order Runge-Kutta Fehlberg numerical method inTable 5 It is worthmentioning that examining the behavior of(17) reveals that its initial curvature must be negative There-fore the negative root of Pade approximants is selected InTables 6ndash8 the result of 8th order HPM-Pade approximationis presented against that of exact (numerical) method It canbe seen that there is a good agreement between the results ofthe proposed method and numerical solution
5 Conclusion
In the present paper a simple modification of the homotopyperturbation method is proposed for the solution of theBlasius equation in semi-infinite domains The equation insemi-infinite domain is transferred into equivalent initialvalue problems which results in appearance of an unknown
Table 6 Comparison between 119891(120578) obtained fromHPM-Pade withthe numerical method second form of the Blasius equation
coefficient In order to determine the unknown coefficientthe boundary condition of the problem at infinity is imposedto Pade approximant of the solution The results are in verygood agreement with numerical and previous data availablein the literature
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
6 Mathematical Problems in Engineering
Table 8 Comparison between 11989110158401015840(120578) obtained from HPM-Padewith the numerical method second form of the Blasius equation
The study presented herein was made possible by the Univer-sity of Malaya Research Grant UMRG RP004D-11AET Theauthors would like to acknowledge the support
References
[1] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004
[2] G Adomian Solving Frontier Problems of Physics The Decom-position Methoc [ie Method] Kluwer Academic PublishersDordrecht The Netherlands 2013
[3] J-H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[4] M-J Jang C-L Chen and Y-C Liy ldquoOn solving the initial-value problems using the differential transformation methodrdquoAppliedMathematics andComputation vol 115 no 2-3 pp 145ndash160 2000
[5] J-H He ldquoHomotopy perturbation method a new nonlinearanalytical techniquerdquo Applied Mathematics and Computationvol 135 no 1 pp 73ndash79 2003
[6] N Herisanu V Marinca and Gh Madescu ldquoAn analyticalapproach to non-linear dynamical model of a permanentmagnet synchronous generatorrdquoWind Energy vol 18 pp 1657ndash1670 2015
[7] H Vazquez-Leal ldquoGeneralized homotopy method for solvingnonlinear differential equationsrdquo Computational and AppliedMathematics vol 33 no 1 pp 275ndash288 2014
[8] M N Alam M A Akbar and S T Mohyud-Din ldquoA novel(G1015840G)-expansionmethod and its application to the Boussinesqequationrdquo Chinese Physics B vol 23 no 2 Article ID 0202032014
[9] M A Jafari and A Aminataei ldquoImproved homotopy perturba-tionmethodrdquo InternationalMathematical Forum vol 5 no 29ndash32 pp 1567ndash1579 2010
[10] E Yusufoglu ldquoAn improvement to homotopy perturbationmethod for solving system of linear equationsrdquo Computers ampMathematics with Applications vol 58 no 11-12 pp 2231ndash22352009
[11] S H Hosseinnia A Ranjbar and S Momani ldquoUsing anenhanced homotopy perturbation method in fractional differ-ential equations via deforming the linear partrdquo Computers andMathematics with Applications vol 56 no 12 pp 3138ndash31492008
[12] D Kumar J Singh and S Kumar ldquoNumerical computationof fractional multi-dimensional diffusion equations by usinga modified homotopy perturbation methodrdquo Journal of theAssociation of Arab Universities for Basic and Applied Sciencesvol 17 pp 20ndash26 2015
[13] C Dong Z Chen and W Jiang ldquoA modified homotopyperturbation method for solving the nonlinear mixed Volterra-Fredholm integral equationrdquo Journal of Computational andApplied Mathematics vol 239 pp 359ndash366 2013
[14] H Blasius ldquoGrenzschichten in Flussigkeiten mit kleiner Rei-bungrdquo Zeitschrift fur Angewandte Mathematik und Physik vol56 pp 1ndash37 1908
[15] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application toheat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[16] O E Potter ldquoLaminar boundary layers at the interface of co-current parallel streamsrdquo The Quarterly Journal of Mechanicsand Applied Mathematics vol 10 pp 302ndash311 1957
[17] J A Ackroyd ldquoOn the laminar compressible boundary layerwith stationary origin on a moving flat wallrdquo MathematicalProceedings of the Cambridge Philosophical Society vol 63 no3 pp 871ndash888 1967
[18] J-H He ldquoA simple perturbation approach to Blasius equationrdquoApplied Mathematics and Computation vol 140 no 2-3 pp217ndash222 2003
[19] S Abbasbandy ldquoA numerical solution of Blasius equationby Adomianrsquos decomposition method and comparison withhomotopy perturbation methodrdquo Chaos Solitons amp Fractalsvol 31 no 1 pp 257ndash260 2007
[20] L-T Yu and C-K Chen ldquoThe solution of the blasius equationby the differential transformation methodrdquo Mathematical andComputer Modelling vol 28 no 1 pp 101ndash111 1998
[21] J H He ldquoApproximate analytical solution of Blasiusrsquo equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 4 no 1 pp 75ndash78 1999
[22] B I Yun ldquoConstructing uniform approximate analytical solu-tions for the Blasius problemrdquo Abstract and Applied Analysisvol 2014 Article ID 495734 6 pages 2014
[23] I Ahmad andM Bilal ldquoNumerical solution of blasius equationthrough neural networks algorithmrdquoAmerican Journal of Com-putational Mathematics vol 4 no 3 pp 223ndash232 2014
[24] V Marinca and N Herisanu ldquoThe optimal homotopy asymp-toticmethod for solving Blasius equationrdquoAppliedMathematicsand Computation vol 231 pp 134ndash139 2014
[25] A Ebaid and N Al-Armani ldquoA new approach for a class of theblasius problem via a transformation and adomianrsquos methodrdquoAbstract and Applied Analysis vol 2013 Article ID 753049 8pages 2013
[26] O Costin T E Kim and S Tanveer ldquoA quasi-solution approachto nonlinear problemsmdashthe case of the Blasius similaritysolutionrdquo Fluid Dynamics Research vol 46 no 3 2014
[27] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
Mathematical Problems in Engineering 7
[28] L Howarth ldquoOn the solution of the laminar boundary layerequationsrdquo Proceedings of the Royal Society A vol 164 no 919pp 547ndash579 1938
[29] J H He ldquoApproximate analytical solution of Blasiusrsquo equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 3 no 4 pp 260ndash263 1998
[30] G A Baker and P R Graves-Morris Essentials of Pade Approx-imants Academic Press New York NY USA 1975
[31] J P Boyd ldquoPade approximant algorithm for solving nonlinearordinary differential equation boundary value problems on anunbounded domainrdquo Computers in Physics vol 11 no 3 article299 1997
Table 5 Root of the Pade approximates for 120572 = 11989110158401015840(0) the secondform of the Blasius equation
Order 11989110158401015840(0) Error
[4 4] minus052270 78119864 minus 2
[6 6] minus048944 45119864 minus 2
[8 8] minus046407 20119864 minus 2
[12 12] minus044654 27119864 minus 3
[16 16] minus044565 19119864 minus 4
[20 20] minus044372 19119864 minus 5
5th order Runge-Kutta Fehlberg(present) minus044374
Consequently the solution of (18) can be written asfollows
119891 (120578) = 120578 +1
21205721205782minus
1
481205721205784+
1
24012057221205785+
1
9601205721205786
+11
2016012057221205787+ sdot sdot sdot
(23)
In order to determine the unknown initial curvature 120572 in(23) boundary conditions of (17) at infinity 1198911015840(120578) = 0 mustbe applied For that Pade approximants of (23) which enlargeconvergence radius of the solution were used Then 120572 wasdetermined from 119891
1015840(120578) = 0 to the Pade approximants Initialcurvature of (23) obtained from this method is comparedto fifth order Runge-Kutta Fehlberg numerical method inTable 5 It is worthmentioning that examining the behavior of(17) reveals that its initial curvature must be negative There-fore the negative root of Pade approximants is selected InTables 6ndash8 the result of 8th order HPM-Pade approximationis presented against that of exact (numerical) method It canbe seen that there is a good agreement between the results ofthe proposed method and numerical solution
5 Conclusion
In the present paper a simple modification of the homotopyperturbation method is proposed for the solution of theBlasius equation in semi-infinite domains The equation insemi-infinite domain is transferred into equivalent initialvalue problems which results in appearance of an unknown
Table 6 Comparison between 119891(120578) obtained fromHPM-Pade withthe numerical method second form of the Blasius equation
coefficient In order to determine the unknown coefficientthe boundary condition of the problem at infinity is imposedto Pade approximant of the solution The results are in verygood agreement with numerical and previous data availablein the literature
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
6 Mathematical Problems in Engineering
Table 8 Comparison between 11989110158401015840(120578) obtained from HPM-Padewith the numerical method second form of the Blasius equation
The study presented herein was made possible by the Univer-sity of Malaya Research Grant UMRG RP004D-11AET Theauthors would like to acknowledge the support
References
[1] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004
[2] G Adomian Solving Frontier Problems of Physics The Decom-position Methoc [ie Method] Kluwer Academic PublishersDordrecht The Netherlands 2013
[3] J-H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[4] M-J Jang C-L Chen and Y-C Liy ldquoOn solving the initial-value problems using the differential transformation methodrdquoAppliedMathematics andComputation vol 115 no 2-3 pp 145ndash160 2000
[5] J-H He ldquoHomotopy perturbation method a new nonlinearanalytical techniquerdquo Applied Mathematics and Computationvol 135 no 1 pp 73ndash79 2003
[6] N Herisanu V Marinca and Gh Madescu ldquoAn analyticalapproach to non-linear dynamical model of a permanentmagnet synchronous generatorrdquoWind Energy vol 18 pp 1657ndash1670 2015
[7] H Vazquez-Leal ldquoGeneralized homotopy method for solvingnonlinear differential equationsrdquo Computational and AppliedMathematics vol 33 no 1 pp 275ndash288 2014
[8] M N Alam M A Akbar and S T Mohyud-Din ldquoA novel(G1015840G)-expansionmethod and its application to the Boussinesqequationrdquo Chinese Physics B vol 23 no 2 Article ID 0202032014
[9] M A Jafari and A Aminataei ldquoImproved homotopy perturba-tionmethodrdquo InternationalMathematical Forum vol 5 no 29ndash32 pp 1567ndash1579 2010
[10] E Yusufoglu ldquoAn improvement to homotopy perturbationmethod for solving system of linear equationsrdquo Computers ampMathematics with Applications vol 58 no 11-12 pp 2231ndash22352009
[11] S H Hosseinnia A Ranjbar and S Momani ldquoUsing anenhanced homotopy perturbation method in fractional differ-ential equations via deforming the linear partrdquo Computers andMathematics with Applications vol 56 no 12 pp 3138ndash31492008
[12] D Kumar J Singh and S Kumar ldquoNumerical computationof fractional multi-dimensional diffusion equations by usinga modified homotopy perturbation methodrdquo Journal of theAssociation of Arab Universities for Basic and Applied Sciencesvol 17 pp 20ndash26 2015
[13] C Dong Z Chen and W Jiang ldquoA modified homotopyperturbation method for solving the nonlinear mixed Volterra-Fredholm integral equationrdquo Journal of Computational andApplied Mathematics vol 239 pp 359ndash366 2013
[14] H Blasius ldquoGrenzschichten in Flussigkeiten mit kleiner Rei-bungrdquo Zeitschrift fur Angewandte Mathematik und Physik vol56 pp 1ndash37 1908
[15] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application toheat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[16] O E Potter ldquoLaminar boundary layers at the interface of co-current parallel streamsrdquo The Quarterly Journal of Mechanicsand Applied Mathematics vol 10 pp 302ndash311 1957
[17] J A Ackroyd ldquoOn the laminar compressible boundary layerwith stationary origin on a moving flat wallrdquo MathematicalProceedings of the Cambridge Philosophical Society vol 63 no3 pp 871ndash888 1967
[18] J-H He ldquoA simple perturbation approach to Blasius equationrdquoApplied Mathematics and Computation vol 140 no 2-3 pp217ndash222 2003
[19] S Abbasbandy ldquoA numerical solution of Blasius equationby Adomianrsquos decomposition method and comparison withhomotopy perturbation methodrdquo Chaos Solitons amp Fractalsvol 31 no 1 pp 257ndash260 2007
[20] L-T Yu and C-K Chen ldquoThe solution of the blasius equationby the differential transformation methodrdquo Mathematical andComputer Modelling vol 28 no 1 pp 101ndash111 1998
[21] J H He ldquoApproximate analytical solution of Blasiusrsquo equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 4 no 1 pp 75ndash78 1999
[22] B I Yun ldquoConstructing uniform approximate analytical solu-tions for the Blasius problemrdquo Abstract and Applied Analysisvol 2014 Article ID 495734 6 pages 2014
[23] I Ahmad andM Bilal ldquoNumerical solution of blasius equationthrough neural networks algorithmrdquoAmerican Journal of Com-putational Mathematics vol 4 no 3 pp 223ndash232 2014
[24] V Marinca and N Herisanu ldquoThe optimal homotopy asymp-toticmethod for solving Blasius equationrdquoAppliedMathematicsand Computation vol 231 pp 134ndash139 2014
[25] A Ebaid and N Al-Armani ldquoA new approach for a class of theblasius problem via a transformation and adomianrsquos methodrdquoAbstract and Applied Analysis vol 2013 Article ID 753049 8pages 2013
[26] O Costin T E Kim and S Tanveer ldquoA quasi-solution approachto nonlinear problemsmdashthe case of the Blasius similaritysolutionrdquo Fluid Dynamics Research vol 46 no 3 2014
[27] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
Mathematical Problems in Engineering 7
[28] L Howarth ldquoOn the solution of the laminar boundary layerequationsrdquo Proceedings of the Royal Society A vol 164 no 919pp 547ndash579 1938
[29] J H He ldquoApproximate analytical solution of Blasiusrsquo equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 3 no 4 pp 260ndash263 1998
[30] G A Baker and P R Graves-Morris Essentials of Pade Approx-imants Academic Press New York NY USA 1975
[31] J P Boyd ldquoPade approximant algorithm for solving nonlinearordinary differential equation boundary value problems on anunbounded domainrdquo Computers in Physics vol 11 no 3 article299 1997
The study presented herein was made possible by the Univer-sity of Malaya Research Grant UMRG RP004D-11AET Theauthors would like to acknowledge the support
References
[1] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004
[2] G Adomian Solving Frontier Problems of Physics The Decom-position Methoc [ie Method] Kluwer Academic PublishersDordrecht The Netherlands 2013
[3] J-H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[4] M-J Jang C-L Chen and Y-C Liy ldquoOn solving the initial-value problems using the differential transformation methodrdquoAppliedMathematics andComputation vol 115 no 2-3 pp 145ndash160 2000
[5] J-H He ldquoHomotopy perturbation method a new nonlinearanalytical techniquerdquo Applied Mathematics and Computationvol 135 no 1 pp 73ndash79 2003
[6] N Herisanu V Marinca and Gh Madescu ldquoAn analyticalapproach to non-linear dynamical model of a permanentmagnet synchronous generatorrdquoWind Energy vol 18 pp 1657ndash1670 2015
[7] H Vazquez-Leal ldquoGeneralized homotopy method for solvingnonlinear differential equationsrdquo Computational and AppliedMathematics vol 33 no 1 pp 275ndash288 2014
[8] M N Alam M A Akbar and S T Mohyud-Din ldquoA novel(G1015840G)-expansionmethod and its application to the Boussinesqequationrdquo Chinese Physics B vol 23 no 2 Article ID 0202032014
[9] M A Jafari and A Aminataei ldquoImproved homotopy perturba-tionmethodrdquo InternationalMathematical Forum vol 5 no 29ndash32 pp 1567ndash1579 2010
[10] E Yusufoglu ldquoAn improvement to homotopy perturbationmethod for solving system of linear equationsrdquo Computers ampMathematics with Applications vol 58 no 11-12 pp 2231ndash22352009
[11] S H Hosseinnia A Ranjbar and S Momani ldquoUsing anenhanced homotopy perturbation method in fractional differ-ential equations via deforming the linear partrdquo Computers andMathematics with Applications vol 56 no 12 pp 3138ndash31492008
[12] D Kumar J Singh and S Kumar ldquoNumerical computationof fractional multi-dimensional diffusion equations by usinga modified homotopy perturbation methodrdquo Journal of theAssociation of Arab Universities for Basic and Applied Sciencesvol 17 pp 20ndash26 2015
[13] C Dong Z Chen and W Jiang ldquoA modified homotopyperturbation method for solving the nonlinear mixed Volterra-Fredholm integral equationrdquo Journal of Computational andApplied Mathematics vol 239 pp 359ndash366 2013
[14] H Blasius ldquoGrenzschichten in Flussigkeiten mit kleiner Rei-bungrdquo Zeitschrift fur Angewandte Mathematik und Physik vol56 pp 1ndash37 1908
[15] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application toheat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[16] O E Potter ldquoLaminar boundary layers at the interface of co-current parallel streamsrdquo The Quarterly Journal of Mechanicsand Applied Mathematics vol 10 pp 302ndash311 1957
[17] J A Ackroyd ldquoOn the laminar compressible boundary layerwith stationary origin on a moving flat wallrdquo MathematicalProceedings of the Cambridge Philosophical Society vol 63 no3 pp 871ndash888 1967
[18] J-H He ldquoA simple perturbation approach to Blasius equationrdquoApplied Mathematics and Computation vol 140 no 2-3 pp217ndash222 2003
[19] S Abbasbandy ldquoA numerical solution of Blasius equationby Adomianrsquos decomposition method and comparison withhomotopy perturbation methodrdquo Chaos Solitons amp Fractalsvol 31 no 1 pp 257ndash260 2007
[20] L-T Yu and C-K Chen ldquoThe solution of the blasius equationby the differential transformation methodrdquo Mathematical andComputer Modelling vol 28 no 1 pp 101ndash111 1998
[21] J H He ldquoApproximate analytical solution of Blasiusrsquo equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 4 no 1 pp 75ndash78 1999
[22] B I Yun ldquoConstructing uniform approximate analytical solu-tions for the Blasius problemrdquo Abstract and Applied Analysisvol 2014 Article ID 495734 6 pages 2014
[23] I Ahmad andM Bilal ldquoNumerical solution of blasius equationthrough neural networks algorithmrdquoAmerican Journal of Com-putational Mathematics vol 4 no 3 pp 223ndash232 2014
[24] V Marinca and N Herisanu ldquoThe optimal homotopy asymp-toticmethod for solving Blasius equationrdquoAppliedMathematicsand Computation vol 231 pp 134ndash139 2014
[25] A Ebaid and N Al-Armani ldquoA new approach for a class of theblasius problem via a transformation and adomianrsquos methodrdquoAbstract and Applied Analysis vol 2013 Article ID 753049 8pages 2013
[26] O Costin T E Kim and S Tanveer ldquoA quasi-solution approachto nonlinear problemsmdashthe case of the Blasius similaritysolutionrdquo Fluid Dynamics Research vol 46 no 3 2014
[27] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
Mathematical Problems in Engineering 7
[28] L Howarth ldquoOn the solution of the laminar boundary layerequationsrdquo Proceedings of the Royal Society A vol 164 no 919pp 547ndash579 1938
[29] J H He ldquoApproximate analytical solution of Blasiusrsquo equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 3 no 4 pp 260ndash263 1998
[30] G A Baker and P R Graves-Morris Essentials of Pade Approx-imants Academic Press New York NY USA 1975
[31] J P Boyd ldquoPade approximant algorithm for solving nonlinearordinary differential equation boundary value problems on anunbounded domainrdquo Computers in Physics vol 11 no 3 article299 1997
[28] L Howarth ldquoOn the solution of the laminar boundary layerequationsrdquo Proceedings of the Royal Society A vol 164 no 919pp 547ndash579 1938
[29] J H He ldquoApproximate analytical solution of Blasiusrsquo equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 3 no 4 pp 260ndash263 1998
[30] G A Baker and P R Graves-Morris Essentials of Pade Approx-imants Academic Press New York NY USA 1975
[31] J P Boyd ldquoPade approximant algorithm for solving nonlinearordinary differential equation boundary value problems on anunbounded domainrdquo Computers in Physics vol 11 no 3 article299 1997
![1BDJGJD +PVSOBM PG .BUIFNBUJDT · simple homotopy equivalences of [16] is the notion of a simple map, i.e. a PL map of finite polyhedra with contractible point inverses. Using simple](https://static.documents.pub/doc/80x56/5f700bb582565b2c980458b6/1bdjgjd-pvsobm-pg-buifnbujdt-simple-homotopy-equivalences-of-16-is-the-notion.jpg)
