An ‎extended Legendre wavelet ‎method‎ for solving differential equations with non analytic solutions

Abstract

A‎lthough spectral methods such as Galerkin‎, ‎Tau and pseudospectral methods do not work well for solving ordinary differential equations in which‎, ‎at least‎, ‎one of the coefficient functions or solution function is not analytic\cite{BabolianH02}‎, ‎but it is shown that the Legendre wavelet Galerkin method is suitable for solving this kind of problems provided that the singular points have the form $2^{-k}$ for some positive integer $k$[4]‎. ‎However‎, ‎for the other type of singular point the Legendre ‎wavelet ‎basis ‎are‎ not an efficient method‎. ‎To overcome this difficulty‎, ‎in this study we use the extended Legendre wavelet basis and Tau method for solving a wide range of singular boundary value problems‎. ‎T‎he ‎c‎‎onvergence ‎p‎roperties and ‎error analysis of the proposed method is ‎investigated.‎‎ ‎‎A comparison between the standard Legendre wavelets and extended Legendre wavelets methods shows the capability of the proposed method‎.