Jacobi rational–Gauss collocation method for Lane–Emden equations of astrophysical significance

Abstract

Abstract. In this paper, a new spectral collocation method is applied to solve Lane-Emden equations on a semi-infinite domain. The method allows us to overcome difficulty in both the nonlinearity and the singularity inherent in such problems. This Jacobi rational-Gauss method, based on Jacobi rational functions and Gauss quadrature integration, is implemented for the nonlinear Lane-Emden equation. Once we have developed the method, numerical results are provided to demonstrate the method. Physically interesting examples include Lane-Emden equations of both first and second kind. In the examples given, by selecting relatively few Jacobi rational-Gauss collocation points, we are able to get very accurate approximations, and we are thus able to demonstrate the utility of our approach over other analytical or numerical methods. In this way, the numerical examples provided demonstrate the accuracy, efficiency, and versatility of the method.

A Domain Decomposition Based Spectral Collocation Method for Lane-Emden Equations

“…However, the spectral methods devel- oped until now (cf. [5][6][7]14]) are not very suitable for long-term computation. In this article, we design a domain decomposition based spectral collocation method for solving the Lane-Emden equation.…”

“…The typical and effective approach is the spectral method. Concretely speaking, Parand et al proposed in [14] a Hermite function collocation method, Doha et al proposed respectively in [5,6] a Jacobi rational Gauss collocation/pseudospectral method, and Gürbü and Sezer proposed in [7] a Laguerre collocation method, to solve the Lane-Emden type equations. The other recent methods include the homotopy perturbation method (cf.…”

Exponential Jacobi spectral method for hyperbolic partial differential equations

“…These matrices were jointly implemented with the collocation approach to evaluate the solutions of the HPDEs. Collocation method [20][21][22][23][24] is an effective technique for numerically approximating different kinds of equations.…”

An approximate solution of the MHD flows of UCM fluids over porous stretching sheets by rational Legendre collocation method

“…Another effective method is based on rational orthogonal functions, such as the rational Legendre functions and rational Chebyshev functions which are mutually orthogonal in semi-infinite intervals (Guo et al , 2000, 2002; Shen and Wang, 2009; Boyd et al , 2003). Also we refer the interested reader to Doha et al (2014a, b, c, d), Bhrawy et al (2015), Bhrawy and Zaky (2015) and Abbasbandy et al (2014) for more research works in the solution of differential equations in semi-infinite domains. A brief review on some of the recent advances in the spectral methods for unbounded domains is presented in Shen and Wang (2009).…”