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)+σt^γy^β(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 ${\hat{I}}^3,{\hat{I}}^2$ and $\ddot{I}ƒ$ are real numbers, ${\hat{I}}^3<-2$, ${\hat{I}}^2>1$. 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 ($\ddot{I}ƒ=-1$,${\hat{I}}^3=\frac{-1}{2}$ and ${\hat{I}}^2=\frac{3}{2}$) 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.

12 pages, 8 Figures, 8 Tables