The Wavelet-Galerkin Method For Parabolic Equation

The contents of this dissertation were divided into three chapters.Chapter one introduced the preparatory knowledge of wavelet analysis briefly.First, the basic concepts of wavelet analysis were summarized systematically;second,the basic property of Daubechies' functions was introduced.Chapter two studied the Wavelet-Galerkin method tailored to solve the initial boundary value problem of one-dimention heat conduction equation.First.the wavelet basis of the subspace of H₀¹ (Ω) was given by using the anti-derivatives of Daubechies' functions;second,the development of the Wavelet-Galerkin discretization formulation of heat conduction equation was given based on the given wavelet basis; finally,by using the energy method, the convergence and error estimate of solution were proved.Chapter three studied the Wavelet-Galerkin method tailored to solve the mixed problem of one-dimention nonlinear Burgers equation.First,the wavelet basis of the sub-space of H₀¹ (0,1) was given based on the scaling functions of Daubechies' wavelet;second, the development of the Wavelet-Galerkin discretization formulation of Burgers equation was given based on the given wavelet basis;then, the definition of connection coefficients and the development of algorithms for computing them were given;finally,the numerical results were used to validate the proposed Wavelet-Galerkin method as an effective numerical algorithm to solve Burgers equation.