Two efficient computational algorithms to solve the singularly perturbed Lane-Emden problem

TL;DR

This paper introduces and compares two spectral collocation methods based on Rational and Exponential Bessel functions for efficiently solving nonlinear Lane-Emden equations with singularities, demonstrating their effectiveness.

Contribution

The paper presents two novel spectral collocation algorithms using RBs and EBs for solving singular Lane-Emden problems, with a focus on efficiency and applicability.

Findings

The methods accurately solve Lane-Emden equations with singularities.

RBs and EBs methods outperform some existing approaches.

The approaches are applicable to models in physics and fluid mechanics.

Abstract

In this paper, we decide to compare two new approaches based on Rational and Exponential Bessel functions (RBs and EBs) to solve several well-known class of Lane-Emden type models. The problems, which define in some models of non-Newtonian fluid mechanics and mathematical physics, are nonlinear ordinary differential equations of second-order over the semiinfinite interval and have singularity at x = 0. We have converted the nonlinear Lane-Emden equation to a sequence of linear equations by utilizing the quasilinearization method (QLM) and then, these linear equations have been solved by RBs and EBs collocation-spectral methods. Afterward, the obtained results are compared with the solution of other methods for demonstrating the efficiency and applicability of the proposed methods.