A reliable numerical method for solving a certain class of singular initial value problems using reproducing kernel algorithm

TL;DR

This paper introduces a reproducing kernel Hilbert space method for accurately and efficiently solving Lane-Emden type singular differential equations, demonstrating excellent convergence and agreement with analytical solutions.

Contribution

The study develops a novel RKHS-based numerical approach specifically tailored for singular initial value problems, enhancing accuracy and convergence over existing methods.

Findings

High accuracy in solving Lane-Emden equations

Excellent convergence properties demonstrated

Method shows strong potential for nonlinear singular problems

Abstract

The aim of this study is to present a good modernistic strategy for solving some well-known classes of Lane-Emden type singular differential equations. The proposed approach is based on the reproducing kernel Hilbert space (RKHS) and introducing the reproducing kernel properties in which the initial conditions of the problem are satisfied. The analytical solution that obtained involves in the form of a convergent series with easily computable terms in its reproducing kernel space. The approximation solution is expressed by n-term summation of reproducing kernel functions and it is converge to the analytical solution. Our investigations indicate that there is excellent agreement between the numerical results and the RKHS method, which is applied to some examples to illustrate the accuracy, efficiency, and applicability of the method. The present work shows the potential of the RKHS technique in solving such nonlinear singular initial value problems.