‏‪ Comparative Study of Collocation and Spectral Methods for Solving Nonlinear Singular Initial Value Problems in Astrophysics

Abstract

    This study presents a rigorous and in-depth comparative analysis of the Chebyshev Collocation and Spectral Galerkin methods for solving nonlinear singular initial value problems, which are fundamental in modeling astrophysical phenomena, with a focus on the Lane-Emden equation as a model test case, by applying both methodologies, performance was evaluated based on precise metrics including accuracy, rate of convergence, computational efficiency, the results unequivocally demonstrate the superiority of the Spectral Galerkin method across all critical performance aspects, the method achieved true exponential convergence for smooth solutions, reaching a precision approaching machine accuracy using significantly lower approximation degrees than the collocation method. Furthermore, the Galerkin method exhibited superior robustness and a higher algebraic rate of convergence in cases where the solution is less regular, while maintaining excellent numerical stability and better-conditioned algebraic systems, we conclude that the integral nature of the orthogonal projection in the Galerkin method endows it with a structural advantage