On The Numerical Solution Of Partial Differential Equations In Wavelet Bases

Fourier analysis has become a mathematic tool favored by many scientists in almost every research fields, while the wavelet analysis is the development and perfection of the Fourier analysis. Since the development of the wavelet analysis is the basis to solve some practical problems, and then, it develops into a radioactive multi-disciplined theory, now it has become a hot field in the research internationally.Beginning with some basic theories and important conclusions, the paper uses the compactly supported wavelet representation of differential operator and discusses applications of wavelet in the numerical solutions of partial differential equations. By selecting Daubechies wavelets and using MATLAB language, the paper conducts the process of finding numerical solution to several partial differential equations. The outcome of the experiment shows it is effective.The paper consists four chapters: The chapterl is an introduction which summarizes the emergence, development of wavelet analysis and the application in finding the numerical solution to partial differential equations .The chapter 2 presents the basic theory of wavelet analysis which includes the definition and properties of wavelet transformation , the definition of one-dimensional and two-dimensional multiresolution analysis , the theory of wavelet basis and we can only use cutoff function when applying the infinite wavelet. However, a new type of wavelet —can avoid cutoff, so the errors can be eliminated. One example of this wavelet is Daubechies wavelets ,it is finite wavelets that is it takes nozero value only in finite intervals. With a detailed introduction to Daubechies wavelets, the chapter lays theoretical basis for using it in the following ones.The chapter 3 summarizes the relevant conclusions of wavelet representation of differential operator .A very important step 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. To reduce calculation quantity in the process of presenting differential operator by means of wavelet is an important question. To this problem, The paper presents a better way of wavelet representation of operator-nonstandard form of an operator .The nonstandard form of an operator T defined as an set of operators...