If the set of basis functions is chosen by overlooking physics of a problem, then the results can be misleading. It is shown that for the Lane-Emden equation, a set of functions with semi-infinite domain sometimes fails to produce results of desired accuracy. A qualitative analysis of the problem shows that the solution is bounded when m is an odd integer but is unbounded when m is even. Solution of the Lane-Emden equation with rational Legendre functions, as basis, is poorer in accuracy when m = 2 as compared with the one when m = 3 with the same basis. Since the physically important region is contained in a finite interval, a set of scaled Legendre polynomials, as basis, produces results which are much more accurate on the interval of interest.

spectral methods, collocation, orthogonal polynomials, interpolation, rational Legendre functions.

Received: July 7, 2022; Accepted: August 29, 2022; Published: September 12, 2022

How to cite this article: Obaid J. Algahtani, Choice of a basis to solve the Lane-Emden equation, Advances in Differential Equations and Control Processes 29 (2022), 13-26. http://dx.doi.org/10.17654/0974324322031

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] C. I. Gheorghiu, Spectral Methods for Differential Equations, Cluj-Napoca, Romania, 2007.
[2] J. P. Boyd, Chebyshev and Fourier Spectral Methods, DOVER Publications, 2000.
[3] R. B. Meyers, T. D. Taylor and J. W. Murdock, Pseudo-spectral simulation of a two dimensional vortex flow in a stratified, incompressible fluid, J. Comput. Fluid 43 (1981), 180-188.
[4] C. Basdevant, M. Deville, P. Haldenwang, J. M. Lacroix, J. Ouazzani, R. Peyret, P. Orlandi and A. T. Patera, Spectral and finite difference solutions of the Burgers equation, Computers and Fluids 14 (1986), 23-41.
[5] J. Shen and R. Temam, Nonlinear Galerkin method using Chebyshev and Legendre polynomials. I. The one-dimensional case, SIAM J. Numer. Anal. 32 (1995), 215-234.
[6] E. Deeba and S. A. Khuri, A decomposition method for solving the nonlinear Klein-Gordon equation, J. Comput. Phys. 124 (1996), 442-448.
[7] J. P. Boyd, The Blasius function in the complex plane, Experiment. Math. 8 (2012), 381-394.
[8] K. Parand, M. Dehgan and A. Taghavi, Modified generalized Laguerre function tau method for solving laminar viscous flow: the Blasius equation, Internat. J. Numer. Methods Heat Fluid Flow 20 (2010), 728-743.
[9] K. Parand, M. Dehgan and F. Baharifard, Solving a laminar boundary layer equation with the rational Gegenbauer functions, Appl. Math. Model. 37 (2013), 851-863.
[10] K. Parand, M. Nikaya and J. A. Rad, Solving nonlinear Lane-Emden type equations using Bessel orthogonal functions collocation method, Celestial Mechanics and Dynamical 116 (2013), 97-107.
[11] K. Parand, M. Dehgan and A. Pirkhedri, The sinc-collocation method for solving the Thomas-Fermi equation, J. Comput. Appl. Math. 237 (2013), 244-252.
[12] J. A. Rad, S. Kazem, M. Shaban, K. Parand and A. Yildrim, Numerical solution of fractional differential equations with a tau method based on Legendre and Bernstein polynomials, Math. Methods Appl. Sci. 37 (2014), 329-342.
[13] S. Chandrasekhar, Introduction to the Study of Stellar Structure, Dover, New York, 1969.
[14] A. M. Wazwaz, A new algorithm for solving differential equations of Lane-Emden type, Appl. Math. Comput. 118 (2001), 287-310.
[15] S. A. Yousefi, Legendre wavelets method for solving differential equations of Lane-Emden type, Appl. Math. Comput. 181 (2006), 1417-1422.
[16] K. Parand, M. Shahini and M. Dehghan, Rational Legendre pseudospectral approach for solving nonlinear equations of Lane-Emden type, J. Comput. Phys. 228 (2009), 8830-8840.
[17] K. Parand, M. Dehghan, A. R. Rezai and S. M. Ghaderi, An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method, Comput. Phys. Comm. 181 (2010), 1096-1108.
[18] R. E. Bellman, Stability Theory of Differential Equations, Dover, New York, 2008.
[19] W. G. Kelley and A. C. Peterson, The Theory of Differential Equations: Classical and Qualitative, Springer-Verlag, New York, 2010.