Solving Differential Equations by New Wavelet Transform Method Based on the Quasi-Wavelets and Differential Invariants

Abstract

In harmonic analysis, wavelets are useful and important tools for analyzing problems and equations. As far as we know, the wavelet applications for solving differential equations are limited to solving either ODE or PDE by numerical means. In this paper, the new mother wavelets with two independent variables are designed in accordance with differential invariants. A new method based on the wavelets is proposed and, new mother wavelets are introduced, while the corresponding wavelet transforms are calculated and applied to differential equations. A lot of methods such as the wavelet-Galerkin method, the wavelet method of moment lead to approximate or numerical solutions. Our method can be used for ODEs and PDEs from every order and accordingly the analytic solutions are obtained.