Haar wavelet collocation approach for Lane-Emden equations arising in mathematical physics and astrophysics

In this paper, we propose the Haar wavelet collocation method for the numerical solution of the Lane-Emden equation with boundary conditions. To overcome the singular behavior at the origin, we first transform the Lane-Emden equation into an integral equation. The Haar wavelet collocation method is utilized to reduce the integral equation into a system of algebraic equations in the unknown expansion coefficients, then Newton's iterative process is employed for numerical solutions. We also discuss the convergence and error analysis of the technique. Six singular problems are presented to demonstrate the accuracy and applicability of the method, including; thermal explosion, oxygen-diffusion in a spherical cell, heat conduction through a solid with heat generation and shallow membrane caps problems. The numerical results are compared with existing numerical and exact solutions. The use of the Haar wavelet is found to be accurate, fast, flexible and convenient.