A fast, spectrally accurate solver for the Falkner--Skan equation

Authors

DOI:

https://doi.org/10.21914/anziamj.v58i0.11746

Keywords:

nonlinear, numerical methods, differential equation, boundary value problem, Falkner-Skan, Blasius,

Abstract

We present a new numerical technique, the Gegenbauer homotopy analysis method, which allows for the construction of iterative solutions to nonlinear differential equations. This technique is a numerical extension of the semi-analytic homotopy analysis method that exhibits spectral convergence while performing sparse matrix operations in Gegenbauer space. This technique is used to present solutions to the Falkner--Skan equation, a well known problem in boundary layer fluid dynamics. These solutions are compared to previously published works, and the convergence properties exhibited by this new technique are considered. References
  • N. S. Asaithambi. A numerical method for the solution of the Falkner–Skan equation. Applied Mathematics and Computation, 81:259–264, 1997. doi:10.1016/S0096-3003(95)00325-8
  • H. Blasius. Grenzschichten in flussigkeiten mit kleiner reibung. Zeitschrift fur Angewandte Mathematik und Physik, 56:1–37, 1908.
  • T. Cebeci and H. B. Keller. Shooting and parallel shooting methods for solving the Falkner–Skan boundary-layer equation. Journal of Computational Physics, 7:289–300, 1971. doi:10.1016/0021-9991(71)90090-8
  • V. M. Falkner and S. W. Skan. Some approximate solutions of the boundary layer equations. Philosophical Magazine, 12:865–896, 1930.
  • R. Fazio. Blasius problem and Falkner–Skan model: Topfer's algorithm and its extension. Computers and Fluid, 75:202–209, 2013. doi:10.1016/j.compfluid.2012.12.012
  • S. J. Liao. The proposed homotopy analysis technique for the solution of nonlinear problems. PhD thesis, Shanghai Jiao Tong University, 1992.
  • S. S. Motsa, G. T. Marewo, P. Sibanda and S. Shateyi. An improved spectral homotopy analysis method for solving boundary layer problems. Boundary Value Problems, 2011(1):3, 2011. doi:10.1186/1687-2770-2011-3
  • S. Olver and A. Townsend. A fast and well-conditioned spectral method. SIAM Review, 55(3):462–489, 2013. doi:10.1137/120865458
  • S. M. Rassoulinejad-Mousavi and S. Abbasbandy. Analysis of forced convection in a circular tube filled with a Darcy–Brinkman–Forcheimer porous medium using spectral homotopy analysis method. Journal of Fluids Engineering, 133(10):101207–101207–9, 2011. doi:10.1115/1.4004998
  • H. Saberi Nik, S. Effati, S. S. Motsa, and M. Shirazian. Spectral homotopy analysis method and its convergence for solving a class of nonlinear optimal control problems. Numerical Algorithms, 65(1):171–194, 2014. doi:10.1007/s11075-013-9700-4

Published

2017-10-10

Issue

Section

Proceedings Computational Techniques and Applications Conference