Computational Techniques for Accurate Solutions of Astrophysical Problems Using Transform-Based Collocation

Abstract

This study applies three advanced techniques based on transforms to find approximate solutions to the Lane-Emden type equation, which is often encountered in mathematical physics and astrophysics. The proposed methods utilize new trial functions derived from expressing the second-order derivative of the variable function y(x) using Bernoulli polynomials, and applying Laplace, Sumudu, and differential transforms. To assess the effectiveness of the proposed methods, the study establishes an error analysis and stability analysis, and provides numerical examples demonstrating their accuracy and efficiency. In addition, a comparison of the absolute errors is made among the three methods, namely, Laplace Transform Bernoulli Collocation Method (LTBCM), Sumudu Transform Bernoulli Collocation Method (STBCM), and Differential Transform Bernoulli Collocation Method (DTBCM), and with those obtained from prior literature. The results show that all three methods perform very well in terms of efficiency and accuracy, and can be considered as suitable techniques for solving the Lane-Emden type equation.