Numerical Solution of Coupled Emden-FowlerEquations Using Haar Wavelet Collocation Method

This paper presents a numerical scheme for solving coupled systems of Emden-Fowler equations based on the Haar wavelet collocation method (HWCM). This method converts the system of singular differential equations into a set of algebraic equations by approximating the higher-order derivatives using wavelet techniques. Newton's method is used to solve the resulting nonlinear system and obtain the Haar coefficients. Numerical experiments on several benchmark problems demonstrate the precision and effectiveness of the proposed approach. The results show excellent agreement with exact solutions and compare favorably with existing methods, including artificial neural network (ANN) techniques. The method is extended to coupled systems for the first time, achieving errors of order \(10^{-3}\) to \(10^{-6}\) with a theoretical convergence rate of \(O(2^{-3L/2})\). The method is implemented in Mathematica, and comprehensive error tables and graphical comparisons are provided. HWCM is shown to be a robust, deterministic alternative to data-driven methods for solving coupled singular differential systems in astrophysics and related fields. MSC: Primary 34A34, 34K37, 34K28, 65L60, 65T60.