Research On The Non-classical Regularization Methods For Some Ill-posed Problems

In the present thesis, we study three kinds of ill-posed problems in mathematical physics, including identification of unknown source in the heat equation, the Cauchy problems for the elliptic equation, numerical differentiation and etc., by some new non-classical regularization methods.Identification of unknown source term in the heat equation is one of the focal points in the research area of inverse problem for its wide physical application. In this thesis we work over some non-classical regularization methods for this problem. We analyze the ill-posedness for the problem. Under an a priori condition we answer the question concerning the best possible accuracy. Then we study the problem in the unbounded and bounded domain by the Fourier method, wavelet-Galerkin method, wavelet dual least square method and quasi-reversibility method, respectively.The Cauchy problems of elliptic equation are severely ill-posed. We solve several Cauchy problems of elliptic equation, including the Cauchy problem of the Laplace equation, the Cauchy problem of the Helmholtz equation and the Cauchy problem of the modified Helmholtz equation (Yukawa) by multidimensional Meyer wavelet method.We also study numerical differentiation further by the wavelet-Galerkin method, and obtain some preferable results in both theory and numerical aspects.We discuss the stability of all the above regularization methods for solving these ill-posed problems in mathematical physics, and prove the convergence estimates for the exact solutions and their regularized approximation. Moreover, we give lots of examples and make numerical tests. The numerical results show the efficiency and accuracy of the proposed methods.