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.
