Haar wavelets collocation on a class of Emden-Fowler equation via Newton's quasilinearization and Newton-Raphson techniques

TL;DR

This paper introduces a numerical method combining Haar wavelet collocation with quasilinearization and Newton-Raphson techniques to solve generalized Emden-Fowler boundary value problems, including the Thomas-Fermi case, demonstrating robustness and accuracy.

Contribution

The paper presents a novel approach integrating Haar wavelet collocation with quasilinearization and Newton-Raphson methods for solving generalized Emden-Fowler equations.

Findings

Method effectively solves boundary value problems with high accuracy.

Small perturbations in initial guesses do not significantly affect solutions.

Results are consistent with existing literature and demonstrate robustness.

Abstract

In this paper we have considered generalized Emden-Fowler equation, \begin{equation*} y''(t)+\sigma t^\gamma y^\beta(t)=0, ~~~~~~~~t \in ]0,1[ \end{equation*} subject to the following boundary conditions \begin{equation*} y(0)=1,~y(1)=0;~~\&~~y(0)=1,~y'(1)=y(1), \end{equation*} where $\gamma,\beta$ and $\sigma$ are real numbers, $\gamma<-2$, $\beta>1$. We propsoed to solve the above BVPs with the aid of Haar wavelet coupled with quasilinearization approach as well as Newton-Raphson approach. We have also considered the special case of Emden-Fowler equation ($\sigma=-1$,$\gamma=\frac{-1}{2}$ and $\beta=\frac{3}{2}$) which is popularly, known as Thomas-Fermi equation. We have analysed different cases of generalised Emden-Fowler equation and compared our results with existing results in literature. We observe that small perturbations in initial guesses does not affect the the final solution significantly.