Advances in Differential Equations and Control Processes

The Advances in Differential Equations and Control Processes is an esteemed international journal indexed in the Emerging Sources Citation Index (ESCI). It publishes original research articles related to recent developments in both theory and applications of ordinary and partial differential equations, integral equations, and control theory. The journal highlights the interdisciplinary nature of these topics, with applications in physical, biological, environmental, and health sciences, mechanics, and engineering. It also considers survey articles that identify future avenues of advancement in the field.

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CHOICE OF A BASIS TO SOLVE THE LANE-EMDEN EQUATION

Authors

  • Obaid J. Algahtani

Keywords:

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

DOI:

https://doi.org/10.17654/0974324322031

Abstract

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.

Received: July 7, 2022
Accepted: August 29, 2022

References

C. I. Gheorghiu, Spectral Methods for Differential Equations, Cluj-Napoca, Romania, 2007.

J. P. Boyd, Chebyshev and Fourier Spectral Methods, DOVER Publications, 2000.

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.

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.

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.

E. Deeba and S. A. Khuri, A decomposition method for solving the nonlinear Klein-Gordon equation, J. Comput. Phys. 124 (1996), 442-448.

J. P. Boyd, The Blasius function in the complex plane, Experiment. Math. 8 (2012), 381-394.

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.

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.

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.

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.

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.

S. Chandrasekhar, Introduction to the Study of Stellar Structure, Dover, New York, 1969.

A. M. Wazwaz, A new algorithm for solving differential equations of Lane-Emden type, Appl. Math. Comput. 118 (2001), 287-310.

S. A. Yousefi, Legendre wavelets method for solving differential equations of Lane-Emden type, Appl. Math. Comput. 181 (2006), 1417-1422.

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.

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.

R. E. Bellman, Stability Theory of Differential Equations, Dover, New York, 2008.

W. G. Kelley and A. C. Peterson, The Theory of Differential Equations: Classical and Qualitative, Springer-Verlag, New York, 2010.

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Published

2022-09-12

Issue

Vol. 29 (2022)

Section

Articles

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How to Cite

CHOICE OF A BASIS TO SOLVE THE LANE-EMDEN EQUATION. (2022). Advances in Differential Equations and Control Processes, 29, 13-26. https://doi.org/10.17654/0974324322031

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