Application Of Wavelet In The Numerical Solution Of Partial Differential Equations

Wavelet Analysis is one of the most popular fields in science research recently in the world which has offered a powerful tool and brought the original ideas to some of the correlative subjects. Wavelet Analysis is a breakthrough progress after Fourier analysis and has caused the extensive concern in science and technology. Wavelet analysis theory is a arisen science which was applied extensively to every domain. Being a time-frequency analysis tool in 1980s', wavelet transform has succeeded to be applied to the signal and imageprocessing domain.This paper first presents the development of the wavelet analysis and some classic subjects such as Multiresolution analysis and Mallat arithmetic,which includes wavelet's definition and all sorts of wavelet transformation.Besides,it notices some theories on two-dimensional wavelet transformation. Then it also introduces supported wavelet, Daubechies wavelet, the chapter lays theoretical basis for using it in the following ones.Secondly, this chapter summarizes the relevant conclusion of wavelet representation of differential operator. And in the solution of partial differential equations by means of wavelet listed in paper is to project the differential operator to wavelet basis, so it can find the numerical solution. The paper presents a better way of wavelet representation of operator-nonstandard form of an operator and also introduces the matrix listed of the frequent operator in the differential equation in the wavelet coefficient, then considers the solution of sparse equation in the wavelet coefficient. Besides, it studies the heat-conduction equation solved by Daubechies wavelet, and provides an error analysis to relevant algorithm and get the outcome of the numerical experical experiments, which shows application of wavelet basis in the solving process of partial differential equtions is very effective.In the last chapter, on the base of Galerkin method, the interval wavelet-Galerkin method are presented and their properties are discussed in detail. The numerical results on two point boundary problems test and verify the algorithms.