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.
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