New approximate analytical solutions of the Falkner-Skan equation. (English) Zbl 1251.65149

Summary: We propose an iterative method for solving the Falkner-Skan equation. The method provides approximate analytical solutions which consist of coefficients of the previous iterate solution. By some examples, we show that the presented method with a small number of iterations is competitive with the existing method such as Adomian decomposition method. Furthermore, to improve the accuracy of the proposed method, we suggest an efficient correction method. In practice, for some examples one can observe that the correction method results in highly improved approximate solutions.


65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
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[1] S. Finch, “Prandtl-Blasius flow,” 2010, http://algo.inria.fr/csolve/bla.pdf.
[2] M. B. Zaturska and W. H. H. Banks, “A new solution branch of the Falkner-Skan equation,” Acta Mechanica, vol. 152, no. 1-4, pp. 197-201, 2001. · Zbl 0992.76027
[3] S. P. Hastings, “Reversed flow solutions of the Falkner-Skan equation,” SIAM Journal on Applied Mathematics, vol. 22, no. 2, pp. 329-334, 1972. · Zbl 0243.34026
[4] E. F. F. Botta, F. J. Hut, and A. E. P. Veldman, “The role of periodic solutions in the Falkner-Skan problem for \lambda >0,” Journal of Engineering Mathematics, vol. 20, no. 1, pp. 81-93, 1986. · Zbl 0594.34036
[5] P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York, NY, USA, 1964. · Zbl 0125.32102
[6] G. Adomian, “A review of the decomposition method in applied mathematics,” Journal of Mathematical Analysis and Applications, vol. 135, no. 2, pp. 501-544, 1988. · Zbl 0671.34053
[7] G. Adomian, “Solution of the Thomas-Fermi equation,” Applied Mathematics Letters, vol. 11, no. 3, pp. 131-133, 1998. · Zbl 0947.34501
[8] E. Alizadeh, M. Farhadi, K. Sedighi, H. R. Ebrahimi-Kebria, and A. Ghafourian, “Solution of the Falkner-Skan equation for wedge by Adomian Decomposition Method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 3, pp. 724-733, 2009. · Zbl 1221.76136
[9] J. Biazar, M. G. Porshokuhi, and B. Ghanbari, “Extracting a general iterative method from an Adomian decomposition method and comparing it to the variational iteration method,” Computers & Mathematics with Applications, vol. 59, no. 2, pp. 622-628, 2010. · Zbl 1189.65245
[10] X. G. Luo, Q. B. Wu, and B. Q. Zhang, “Revisit on partial solutions in the Adomian decomposition method: solving heat and wave equations,” Journal of Mathematical Analysis and Applications, vol. 321, no. 1, pp. 353-363, 2006. · Zbl 1103.65106
[11] J. H. He, “Approximate analytical solution of Blasius’ equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 3, no. 4, pp. 260-263, 1998. · Zbl 0918.34016
[12] J. H. He, “A review on some new recently developed nonlinear analytical techniques,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 1, no. 1, pp. 51-70, 2000. · Zbl 0966.65056
[13] J. H. He, “A simple perturbation approach to Blasius equation,” Applied Mathematics and Computation, vol. 140, no. 2-3, pp. 217-222, 2003. · Zbl 1028.65085
[14] J. Y. Parlange, R. D. Braddock, and G. Sander, “Analytical approximations to the solution of the Blasius equation,” Acta Mechanica, vol. 38, no. 1-2, pp. 119-125, 1981. · Zbl 0463.76042
[15] A. M. Wazwaz, “The variational iteration method for solving two forms of Blasius equation on a half-infinite domain,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 485-491, 2007. · Zbl 1114.76055
[16] F. M. Allan and M. I. Syam, “On the analytic solutions of the nonhomogeneous Blasius problem,” Journal of Computational and Applied Mathematics, vol. 182, no. 2, pp. 362-371, 2005. · Zbl 1071.65108
[17] S. J. Liao, “A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat plate,” Journal of Fluid Mechanics, vol. 385, pp. 101-128, 1999. · Zbl 0931.76017
[18] S. J. Liao, “An explicit, totally analytic approximate solution for Blasius’ viscous flow problems,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 759-778, 1999. · Zbl 1342.74180
[19] B. I. Yun, “An iteration method generating analytical solutions for Blasius problem,” Journal of Applied Mathematics, vol. 2011, Article ID 925649, 8 pages, 2011. · Zbl 1330.65117
[20] D. R. Hartree, “On an equation occurring in Falkner and Skan’s approximate treatment of the equations of the boundary layer,” Proceedings of the Cambridge Philosophical Society, vol. 33, no. 2, pp. 223-239, 1937. · Zbl 0017.08004
[21] I. Hashim, “Comments on: “A new algorithm for solving classical Blasius equation by L. Wang”,” Applied Mathematics and Computation, vol. 176, no. 2, pp. 700-703, 2006. · Zbl 1331.65103
[22] J. P. Boyd, “The Blasius function in the complex plane,” Experimental Mathematics, vol. 8, no. 4, pp. 381-394, 1999. · Zbl 0980.34053
[23] J. P. Boyd, “The Blasius function: computations before computers, the value of tricks, undergraduate projects, and open research problems,” SIAM Review, vol. 50, no. 4, pp. 791-804, 2008. · Zbl 1152.76024
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