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.
Highlights
The fundamental goal of this paper is to develop a suitable way to approximate the singular nonlinear Lane–Emden equation on the interval x ∈ (0, ∞) using the Jacobi rational polynomials
For the sake of comparison with others methods, we consider the following two cases: (i) In the case of m = 4, we introduce Table 1, where the maximum absolute errors using the present Jacobi rational–Gauss collocation (JRC) method, those obtained by the Hermite functions collocation method (HFC, see [28]), and the values obtained by Horedt [40] are compared
By selecting relatively few Jacobi rational–Gauss collocation points, we are able to get very accurate approximations, and we are able to demonstrate the utility of our approach over other analytical or numerical methods such as other collocation methods or perturbation methods
Summary
The fundamental goal of this paper is to develop a suitable way to approximate the singular nonlinear Lane–Emden equation on the interval x ∈ (0, ∞) using the Jacobi rational polynomials. Bhrawy et al [25] proposed the shifted Jacobi collocation spectral method for solving the nonlinear Lane–Emden type equation, while the spatial approximation is based on shifted Jacobi polynomials with their parameters α and β and used the collocation nodes of shifted Jacobi–Gauss points. The main concern of this paper is to develop a spectral Jacobi rational–Gauss collocation (JRC) method to find an approximate solution uN (x) of singular Lane–Emden type initial value problems on the semi-infinite domain (0, ∞).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
