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

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.