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

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.

Keywords

• Lane–Emden equation
• isothermal gas spheres
• collocation method
• Jacobi rational–Gauss quadrature
• Jacobi rational polynomials