A Rapid-Converging Analytical Iterative Scheme for Solving Singular Initial Value Problems of Lane–Emden Type

Abstract

In this paper, the approximate analytical solutions of singular initial value problems of Lane–Emden type, which model various physical applications in astrophysics and mathematical physics, are presented using a rapid-converging analytical scheme. The principle behind this method of obtaining the solution to such problems is to reduce the singular problem to an integral problem. There are two phases to the proposed method. To design an iterative scheme for such problems, first, define an integral operator for the problem, then apply the Normal-S approach to the obtained integral formulation. The results show that the proposed method reduces computational work, is computationally efficient, converges quickly, and provides an accurate approximate solution. The iterative method’s convergence is also analyzed. We consider a variety of numerical examples occurring in physical models, including real-world problems, to illustrate the method’s efficacy. The method’s superiority is demonstrated by numerical simulations, and the obtained results are compared to existing schemes. Furthermore, it is an effective method for dealing with a variety of problems involving strong nonlinearity. The comparisons show that the proposed algorithm is applicable and accurate.