An improved spectral homotopy analysis method for solving boundary layer problems
 Sandile Sydney Motsa^{1},
 Gerald T Marewo^{1},
 Precious Sibanda^{2} and
 Stanford Shateyi^{3}Email author
DOI: 10.1186/1687277020113
© Motsa et al; licensee Springer. 2011
Received: 10 November 2010
Accepted: 22 June 2011
Published: 22 June 2011
Abstract
This article presents an improved spectralhomotopy analysis method (ISHAM) for solving nonlinear differential equations. The implementation of this new technique is shown by solving the FalknerSkan and magnetohydrodynamic boundary layer problems. The results obtained are compared to numerical solutions in the literature and MATLAB's bvp4c solver. The results show that the ISHAM converges faster and gives accurate results.
Keywords
FalknerSkan flow MHD flow improved spectralhomotopy analysis methodIntroduction
Boundary layer flow problems have wide applications in fluid mechanics. In this article, we propose an improved spectralhomotopy analysis method (ISHAM) for solving general boundary layer problems. Three boundary layer problems are considered and solved in this study using the novel technique. The first problem considered is the classical twopoint nonlinear boundary value Blasius problem which models viscous fluid flow over a semiinfinite flat plate. Although solutions for this problem had been obtained as far back as 1908 by Blasius [1], the problem is still of great interest to many researchers as can be seen from the several recent studies [2–5].
The second problem considered in this article is the thirdorder nonlinear FalknerSkan equation. The FalknerSkan boundary layer equation has been studied by several researchers from as early as 1931 [6]. More recent studies of the solutions of the The FalknerSkan equation include those of Harries et al. [7], Pade [8] and Pantokratoras [9]. The third problem considered is magnetohydrodynamic (MHD) boundary layer flow. Such boundary layer problems arise in the study of the flow of electrically conducting fluids such as liquid metal. Owing to its many applications such as power generators, flow meters, and the cooling of reactors, MHD flow has been studied by many researchers, for example [10, 11].
Owing to the nonlinearity of equations that describe most engineering and science phenomena, many authors traditionally resort to numerical methods such as finite difference methods [12], RungeKutta methods [13], finite element methods [14] and spectral methods [4] to solve the governing equations. However, in recent years, several analytical or semianalytical methods have been proposed and used to find solutions to most nonlinear equations. These methods include the Adomian decomposition method [15–17], differential transform method [18], variational iteration method [19], homotopy analysis method (HAM) [20–23], and the spectralhomotopy analysis (SHAM) (see Motsa et al. [24, 25]) which sought to remove some of the perceived limitations of the HAM. More recently, successive linearization method [26–28], has been used successfully to solve nonlinear equations that govern the flow of fluids in bounded domains.
In this article, boundary layer equations are solved using the ISHAM. The ISHAM is a modified version of the SHAM [24, 25]. One strength of the SHAM is that it removes restrictions of the HAM such as the requirement for the solution to conform to the socalled rule of solution expression and the rule of coefficient ergodicity. Also, the SHAM inherits the strengths of the HAM, for example, it does not depend on the existence of a small parameter in the equation to be solved, it avoids discretization, and the solution obtained is in terms of an auxiliary parameter ħ which can conveniently be chosen to determine the convergence rate of the solution.
Mathematical formulation
where c_{ i } , b_{ j } (i = 1, ..., 4 j = 1, 2, 3) are constants.
Equation 2.1 can be solved easily using methods such as the HAM and the SHAM. In each of these methods, an initial approximation f_{0}(η) is sought, which satisfies the boundary conditions. The speed of convergence of the method depends on whether f_{0}(η) is a good approximation of f (η) or not. The approach proposed here seeks to find an optimal initial approximation f_{0} that would lead to faster convergence of the method to the true solution. We thus first seek to improve the initial approximation that is used later in the SHAM to solve the governing nonlinear equation.
where q ∈ 0[1] is the embedding parameter, and F_{ i } (η; q) is an unknown function.
where Ã_{i1}is the modified matrix A_{i1}after incorporating the boundary conditions (2.30). Thus, starting from the initial approximation, which is obtained from (2.28), higherorder approximations f_{i,m}(ξ) for m ≥ 1, can be obtained through the recursive formula (2.33).
The [i, m] approximate solution for f (η) is then obtained by substituting f_{ i } (obtained from 2.34) in equation 2.3.
Results and discussion
Order [i, m] ISHAM approximate results for f" (0) of the Blasius boundary layer flow (Example 1) using L = 30, ħ = 1 and N = 80
m  1  2  3  4  10  15 

i  
1  0.33849743  0.33398878  0.33272105  0.33230382  0.33205863  0.33205736 
2  0.33205889  0.33205734  0.33205734  0.33205734  0.33205734  0.33205734 
3  0.33205734  0.33205734  0.33205734  0.33205734  0.33205734  0.33205734 
4  0.33205734  0.33205734  0.33205734  0.33205734  0.33205734  0.33205734 
5  0.33205734  0.33205734  0.33205734  0.33205734  0.33205734  0.33205734 
Comparison between the [m, m] ISHAM results and the bvp4c numerical results for the velocity pro le f' (η) at selected values of η for the Blasius boundary layer flow (Example 1) using L = 30, ħ = 1 and N = 200
η  [1,1]  [2,2]  [3,3]  [4,4]  Numerical 

0.0  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 
0.4  0.1353503  0.1327642  0.1327642  0.1327642  0.1327642 
0.8  0.2699826  0.2647092  0.2647092  0.2647092  0.2647091 
1.6  0.5279353  0.5167568  0.5167568  0.5167568  0.5167568 
2.0  0.6436159  0.6297657  0.6297657  0.6297657  0.6297657 
3.0  0.8609681  0.8460445  0.8460445  0.8460445  0.8460444 
4.0  0.9635769  0.9555182  0.9555182  0.9555182  0.9555182 
5.0  0.9937558  0.9915420  0.9915420  0.9915420  0.9915419 
6.0  0.9992643  0.9989729  0.9989729  0.9989729  0.9989729 
8.0  0.9999880  0.9999963  0.9999963  0.9999963  0.9999963 
10.0  0.9999991  1.0000000  1.0000000  1.0000000  1.0000000 
Order [m, m] ISHAM approximate results for f" (0) of the FalknerSkan boundary layer flow (Example 2) using L = 30, ħ = 1 and N = 80
β  [1,1]  [2,2]  [3,3]  [4,4]  Numerical 

0.4  0.85435667  0.85442123  0.85442123  0.85442123  0.85442123 
0.8  1.11956168  1.12026766  1.12026766  1.12026766  1.12026766 
1.2  1.33311019  1.33572147  1.33572147  1.33572147  1.33572147 
1.6  1.51553054  1.52151400  1.52151400  1.52151400  1.52151400 
2.0  1.67637221  1.68721817  1.68721817  1.68721817  1.68721817 
Order [m, m] ISHAM approximate results for the velocity profile f' (η) of the MHD boundary layer flow (Example 3) when M = 10 using L = 10, ħ = 1 and N = 200
η  f' (η)  Exact  Absolute error  

[1,1]  [2,2]  [3,3]  [1,1]  [2,2]  [3,3]  
0.0  1.00000000  1.00000000  1.00000000  1.00000000  0.00000000  0.00000000  0.00000000 
0.5  0.19106051  0.19046007  0.19046007  0.19046013  0.00060038  0.00000006  0.00000006 
1.0  0.03731355  0.03627506  0.03627506  0.03627506  0.00103849  0.00000000  0.00000000 
1.5  0.00795438  0.00690893  0.00690893  0.00690895  0.00104543  0.00000002  0.00000002 
2.0  0.00212716  0.00131588  0.00131588  0.00131588  0.00081128  0.00000000  0.00000000 
2.5  0.00080280  0.00025062  0.00025062  0.00025062  0.00055218  0.00000000  0.00000000 
3.0  0.00040021  0.00004773  0.00004773  0.00004773  0.00035248  0.00000000  0.00000000 
3.5  0.00022752  0.00000909  0.00000909  0.00000909  0.00021843  0.00000000  0.00000000 
4.0  0.00013536  0.00000173  0.00000173  0.00000173  0.00013363  0.00000000  0.00000000 
5.0  0.00004944  0.00000006  0.00000006  0.00000006  0.00004938  0.00000000  0.00000000 
6.0  0.00001818  0.00000000  0.00000000  0.00000000  0.00001818  0.00000000  0.00000000 
Order [m, m] ISHAM approximate results for f" (η) of the MHD boundary layer flow (Example 3) for different values of M using L = 10, ħ = 1 and N = 200
M  f" (0)  Exact  Absolute error  

[1,1]  [2,2]  [1,1]  [2,2]  
5  2.44812872  2.44948974  2.44948974  0.00136102  0.00000000 
10  3.31554301  3.31662479  3.31662479  0.00108178  0.00000000 
20  4.58188947  4.58257570  4.58257569  0.00068622  0.00000001 
50  7.14113929  7.14142843  7.14142843  0.00028914  0.00000000 
100  10.04974330  10.04987562  10.04987562  0.00013232  0.00000000 
200  14.17739008  14.17744688  14.17744688  0.00005680  0.00000000 
500  22.38301286  22.38302928  22.38302929  0.00001643  0.00000001 
1000  31.63857773  31.63858404  31.63858404  0.00000631  0.00000000 
Conclusion
In this article, we have proposed an ISHAM for solving general nonlinear differential equations. This novel technique was compared against both numerical approximations and the MATLAB bvp4c routine for solving FalknerSkan and MHD boundary layer problems. The results demonstrate the relatively more rapid convergence of the ISHAM, and they show that the ISHAM is highly accurate.
Abbreviations
 HAM:

homotopy analysis method
 ISHAM:

improved spectralhomotopy analysis method
 MHD:

magnetohydrodynamic
 SHAM:

spectralhomotopy analysis.
Declarations
Acknowledgements
The authors wish to acknowledge financial support from the University of Swaziland, University of KwaZuluNatal, University of Venda, and the National Research Foundation (NRF).
Authors’ Affiliations
References
 Blasius H: Grenzschichten in Flussigkeiten mit kleiner Reibung. Z Math Phys 1908, 56: 137.Google Scholar
 Ahmad F, AlBarakati WH: An analytic solution of the Blasius problem. Commun Nonlinear Sci Numer Simul 2009, 14: 10201024.MathSciNetGoogle Scholar
 AlizadehPahlavan A, BorjianBoroujeni S: On the analytic solution of viscous fluid flow past a flat plate. Phys Lett A 2008, 372: 36783682. 10.1016/j.physleta.2008.02.050View ArticleGoogle Scholar
 Parand K, Shahini M, Dehghan M: Solution of a laminar boundary layer flow via a numerical method. Commun Nonlinear Sci Numer Simulat 2010, 15: 360367. 10.1016/j.cnsns.2009.04.007View ArticleGoogle Scholar
 Yun BI: Intuitive approach to the approximate analytical solution for the Blasius problem. Appl Math Comput 2009, 208: 156164. 10.1016/j.amc.2008.11.028View ArticleMathSciNetGoogle Scholar
 Falkner VM, Skan SW: Some approximate solutions of the boundary layer equations. Philos Mag 1931, 12: 865896.View ArticleGoogle Scholar
 Harris SD, Ingham DB, Pop I: Unsteady heat transfer in impulsive FalknerSkan flows: constant wall temperature case. Eur J Mech B 2002, 21: 447468. 10.1016/S09977546(02)011937View ArticleMathSciNetGoogle Scholar
 Padé O: On the solution of FalknerSkan equations. J Math Anal Appl 2003, 285: 264274. 10.1016/S0022247X(03)004025View ArticleMathSciNetGoogle Scholar
 Pantokratoras A: The FalknerSkan flow with constant wall temperature and variable viscosity. Int J Thermal Sci 2006, 45: 378389. 10.1016/j.ijthermalsci.2005.06.004View ArticleGoogle Scholar
 Rashidi MM: The modified differential transform method for solving MHD boundarylayer equations. Comput Phys Commun 2009, 180: 22102217. 10.1016/j.cpc.2009.06.029View ArticleMathSciNetGoogle Scholar
 Parand K, Rezai AR, Ghaderi SM: An approximate solution of the MHD FalknerSkan flow by Hermite functions pseudospectral method. Commun Nonlinear Sci Numer Simulat 2010.Google Scholar
 Asaithambi A: A secondorder finitedifference method for the FalknerSkan equation. Appl Math Comput 2004, 156: 779786. 10.1016/j.amc.2003.06.020View ArticleMathSciNetGoogle Scholar
 Cortell R: Numerical solutions of the classical Blasius flatplate problem. Appl Math Comput 2005, 170: 706710. 10.1016/j.amc.2004.12.037View ArticleMathSciNetGoogle Scholar
 Asaithambi A: Numerical solution of the FalknerSkan equation using piecewise linear functions. Appl Math Comput 2004, 159: 267273. 10.1016/j.amc.2003.10.047View ArticleMathSciNetGoogle Scholar
 Elgazery NS: Numerical solution for the FalknerSkan equation. Chaos Soliton Fract 2008, 35: 738746. 10.1016/j.chaos.2006.05.040View ArticleGoogle Scholar
 Wang L: A new algorithm for solving classical Blasius equation. Appl Math Comput 2004, 157: 19. 10.1016/j.amc.2003.06.011View ArticleMathSciNetGoogle Scholar
 Abbasbandy S: A numerical solution of Blasius equation by Adomian's decomposition method and comparison with homotopy perturbation method. Chaos Soliton Fract 2007, 31: 257260. 10.1016/j.chaos.2005.10.071View ArticleGoogle Scholar
 Kuo B: Heat analysis for the FalknerSkan wedge flow by the differential transformation method. Int J Heat Mass Transfer 2005, 48: 50365046. 10.1016/j.ijheatmasstransfer.2003.10.046View ArticleGoogle Scholar
 Wazwaz A: The variational iteration method for solving two forms of Blasius equation on a halfinfinite domain. Appl Math Comput 2007, 188: 485491. 10.1016/j.amc.2006.10.009View ArticleMathSciNetGoogle Scholar
 Liao SJ: Beyond Perturbation: Introduction to Homotopy Analysis Method. Chapman & Hall/CRC Press; 2003.View ArticleGoogle Scholar
 Yao B, Chen J: A new analytical solution branch for the Blasius equation with a shrinking sheet. Appl Math Comput 2009, 215: 11461153. 10.1016/j.amc.2009.06.057View ArticleMathSciNetGoogle Scholar
 Yao B, Chen J: Series solution to the FalknerSkan equation with stretching boundary. Appl Math Comput 2009, 215: 11461153. 10.1016/j.amc.2009.06.057View ArticleMathSciNetGoogle Scholar
 Yao B: Approximate analytical solution to the FalknerSkan wedge flow with the permeable wall of uniform suction. Commun Nonlinear Sci Numer Simulat 2009, 14: 33203326. 10.1016/j.cnsns.2009.01.014View ArticleGoogle Scholar
 Motsa SS, Sibanda P, Shateyi S: A new spectralhomotopy analysis method for solving a nonlinear second order BVP. Commun Nonlinear Sci Numer Simulat 2010, 15: 22932302. 10.1016/j.cnsns.2009.09.019View ArticleMathSciNetGoogle Scholar
 Motsa SS, Sibanda P, Awad FG, Shateyi S: A new spectralhomotopy analysis method for the MHD JefferyHamel problem. Comput Fluids 2010, 39: 12191225. 10.1016/j.compfluid.2010.03.004View ArticleMathSciNetGoogle Scholar
 Makukula Z, Motsa SS, Sibanda P: On a new solution for the viscoelastic squeezing flow between two parallel plates. J Adv Res Appl Math 2010, 2(4):3138. 10.5373/jaram.455.060310View ArticleMathSciNetGoogle Scholar
 Makukula ZG, Sibanda P, Motsa SS: A novel numerical technique for twodimensional laminar flow between two moving porous walls. Math Problems Eng 2010, 15: Article ID 528956.MathSciNetGoogle Scholar
 Shateyi S, Motsa SS: Variable viscosity on magnetohydrodynamic fluid flow and heat transfer over an unsteady stretching surface with Hall effect. Boundary Value Problems 2010, 2010: 120.View ArticleMathSciNetGoogle Scholar
 Canuto C, Hussaini MY, Quarteroni A, Zang TA: Spectral Methods in Fluid Dynamics. SpringerVerlag, Berlin; 1988.View ArticleGoogle Scholar
 Don WS, Solomonoff A: Accuracy and speed in computing the Chebyshev Collocation Derivative. SIAM J Sci Comput 1995, 16(6):12531268. 10.1137/0916073View ArticleMathSciNetGoogle Scholar
 Trefethen LN: Spectral Methods in MATLAB. SIAM 2000.Google Scholar
 Howarth L: On the solution of the laminar boundary layer equations. Proc R Soc Lond A 1938, 164: 547579. 10.1098/rspa.1938.0037View ArticleGoogle Scholar
 Asaithambi A: Solution of the FalkneSkan equation by recursive evaluation of Taylor coefficients. J Comput Appl Math 2005, 176: 20314. 10.1016/j.cam.2004.07.013View ArticleMathSciNetGoogle Scholar
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