Wavelet Methods Of Boundary Value Problems Of Biharmonic Equations On Rectangular Domain

In this paper,Wavelet methods for boundary value problems of one- and two-dimensional biharmonic equations is presented.An Haar operational matrix of integration, which can make numerical procedure simpler because of generating a sparse matrix,is applied for solving one-dimensional biharmonic problem.Based on Haar wavelet method for one dimensional biharmonic equation,two-dimensional biharmonic equation,that the unknown's variables is separated,is solved.Chapter 1 report the origin of wavelet theory,historical background of formalization and emergence,how the researches about wavelet is,what happen in the future about wavelet in its research field.How wavelet is applied to numerical calculation of ODEs and of PDEs.Meanwhile,the histories,developments,changes of partial differential equations are presented.In addition,the contents and methodology of this paper is given.Before finished the chapter,How this paper are organized is showed.The basic theories of wavelet are introduced in chapter 2.It consists of conceptions of wavelet,continuous wavelet transform,discrete wavelet transform,wavelet framework,multi-resolution analysis and related important theorems and properties on wavelet etc.This chapter is theoretical basis of our paper.In chapter 3,The wavelet methods for solving boundary value problems of one-and two-dimensional biharmonic equations is proposed.Haar wavelet method is used to solve one-dimensional biharmonic problem and to the two-dimensional that the variables,the unknown is dependent on,is separated.Chapter 4 gives several numerical examples,which show that the suggested methods is effcctive and applicable,furthermore,it is numerically steady.A brief conclusion and an acknowledgement is stated finally.