Approximate analytical solutions using hyperbolic functions for the generalized Blasius problem. (English) Zbl 1256.65078

Summary: We propose simple forms of approximate analytical solutions for the generalized Blasius problem based on the given boundary conditions and some known properties of the solution. The efficiency of the proposed solutions is shown for various cases. As a result, one can see that the solutions are uniformly accurate over the whole region.


65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
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[1] 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
[2] G. Adomian, “Solution of the Thomas-Fermi equation,” Applied Mathematics Letters, vol. 11, no. 3, pp. 131-133, 1998. · Zbl 0947.34501
[3] 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
[4] 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
[5] 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
[6] 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
[7] J.-H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257-262, 1999. · Zbl 0956.70017
[8] J.-H. He, “Homotopy perturbation method for solving boundary value problems,” Physics Letters A, vol. 350, no. 1-2, pp. 87-88, 2006. · Zbl 1195.65207
[9] J. Lin, “A new approximate iteration solution of Blasius equation,” Communications in Nonlinear Science & Numerical Simulation, vol. 4, no. 2, pp. 91-94, 1999. · Zbl 0928.34012
[10] J. 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
[11] 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
[12] 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
[13] B. K. Datta, “Analytic solution for the Blasius equation,” Indian Journal of Pure and Applied Mathematics, vol. 34, no. 2, pp. 237-240, 2003. · Zbl 1054.34011
[14] 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
[15] 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
[16] R. Cortell, “Numerical solutions of the classical Blasius flat-plate problem,” Applied Mathematics and Computation, vol. 170, no. 1, pp. 706-710, 2005. · Zbl 1077.76023
[17] R. Fazio, “Numerical transformation methods: Blasius problem and its variants,” Applied Mathematics and Computation, vol. 215, no. 4, pp. 1513-1521, 2009. · Zbl 1422.76051
[18] L. Howarth, “On the solution of the laminar boundary equations,” Proceedings of the Royal Society London A, vol. 164, no. 919, pp. 547-579, 1938. · JFM 64.1452.01
[19] 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
[20] B. I. Yun, “Intuitive approach to the approximate analytical solution for the Blasius problem,” Applied Mathematics and Computation, vol. 215, no. 10, pp. 3489-3494, 2010. · Zbl 1183.65098
[21] S. Finch, “Prandtl-Blasius flow,” 2008, http://www.people.fas.harvard.edu/ sfinch/csolve/bla.pdf.
[22] H. Schlichting, Boundary Layer Theory, McGraw-Hill, New York, NY, USA, 7th edition, 1979. · Zbl 0434.76027
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