Enhancing efficiency in solving coupled Lane-Emden-Fowler equations with a novel Tricomi-Carlitz wavelet method

This paper introduces a novel wavelet-based method utilizing the Tricomi-Carlitz orthogonal polynomials for solving the challenging coupled Lane-Emden-Fowler equations, which are prevalent in astrophysics and various physical sciences. These equations are notoriously difficult to solve numerically due to their singularity and nonlinearity, particularly at t=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = 0$$\end{document}. The proposed approach transforms the coupled differential equations into a nonlinear system of equations, which is subsequently solved using the Newton-Raphson method. Computational results from standard examples demonstrate that the Tricomi-Carlitz wavelet method offers superior accuracy compared to existing techniques, requiring fewer basis functions and eliminating the need for adjustable parameters. The method provides precise solutions across the entire range of relevant physical parameters, significantly improving both computational efficiency and simplicity. This work establishes the Tricomi-Carlitz wavelet method as an effective and powerful tool for addressing nonlinear differential equations in scientific and engineering applications.