A rational optimal block hybrid method for enhanced accuracy in solving Lane–Emden equations

Abstract

This paper introduces a block hybrid method designed for the effective resolution of Lane–Emden equations, which are characterized as second-order boundary value problems incorporating a singularity at the origin. Utilizing a strategic selection of grid points through the rational approximation of optimal points, this method aims at minimizing local truncation errors, thereby enhancing the precision of solutions. Extensive numerical experimentation reveals that this approach, hereinafter referred to as the Rational Optimal Block Hybrid Method (ROBHM), offers improved accuracy and convergence rates over traditional methods. The analysis underscores the critical role of the rational approximation parameter (denoted as $d$) in optimizing both accuracy and computational efficiency. By maintaining a balance between computational demands and the quality of solutions, the Rational Optimal Block Hybrid Method opens new avenues for tackling complex differential equations, thus contributing to the advancement of numerical analysis of boundary value problems marked by singularities.