High-order wavelet-based numerical algorithms for nonlinear singular Lane–Emden–Fowler equations: Applications to physical models in astrophysics

Abstract

This research paper aims to present reliable numerical techniques to deal with singular differential equations, specifically the Lane–Emden equation. The challenges that arise from the singularity at point $t=0$ often leads to the difficulties in common schemes like the Euler and Runge–Kutta methods. To handle this, we implement the Haar wavelet collocation method (HWCM) and its advanced version, the higher-order Haar wavelet collocation method (HOHWCM). These methods are capable of effectively managing singularities and producing approximate solutions for Lane–Emden equations under a range of standard initial conditions, two point Robin conditions, two point mixed conditions, and two point integral conditions. The study incorporates both linear and nonlinear Lane–Emden equations. In nonlinear case, first the linearization technique based on Taylor series expansion is applied to the nonlinear Lane–Emden equation and then the linearized Lane–Emden equation has been solved iteratively with the help of Haar functions. The developed methods are simple to use and are computationally efficient. Results of numerical simulations reveal strong consistency between the numerical and exact results. Incorporating HOHWCM further increases solution accuracy without significantly escalating computational effort, making the method a valuable choice for tackling nonlinear boundary problems. An analysis covering convergence is conducted to support the fast approaching the numerical results towards the exact solutions (here HWCM convergence is of second order while HOHWCM is of ${(2+2\mu )}^{th}$ order). These results are verified by implementing on different benchmark cases of Lane–Emden equations.