The Higher-Order Modified Method For Two Kinds Of Ill-Posed Problems

The inverse problem of mathematical physics is one of the most valuable and the fastest growing research of mathematics in this century, and its prospects are very extensive. The difficulty of the ill-posed problem is its ill-posed characteristic, the reason of this is its solution will not be able to depend on the original measure-ment data continuously if it is exist. This will cause us not use numerical methods (such as finite element method, etc.) to solve the ill-posed problem. In this paper, by adding a higher order mixed partial derivative method, we get stable approxi-mate solutions of the backward heat conduction problem with a convection term or the Cauchy problem for the Laplace equation, they can approach the classical solutions on one hand, on the other hand the approximate solutions can depend on the original measurement data continuously. The major work of the paper is the convergence of approximate solution is much better, that is to say it can reach its convergent optimal order. This paper is divided into five chapters.In Chapter 1, we introduce the introduction.In Chapter 2, we introduce the preparation knowledge.In Chapter 3, we introduce the following backward heat conduction problem with a convection term ut(x,t)= uxx(x,t)+ux(x,t), x?R,t?[0,T) u(x,T)=?T(x),x?R. For the problem, lots of scholars have given many methods. For example, theFourier method, modified Tikhonov method, Shannon-wavelets method, iterative compensation regularization method and so on. We introduce a new method of modified equation in this paper:By selecting the positive integer k and the appropriate regularization parame-ter ?, we can reach the Holder and logarithmic type stability estimation respectively under different prior information. Numerical tests given in the end also verify our conclusions.In Chapter 4, we studied the Cauchy problem for the Laplace equation asSimilar to the previous chapter, we consider the following modified equationBy selecting the positive integer k and the appropriate regularization parame-ter ?, we can reach the Holder type stability estimation under the prior information.In Chapter 5, we introduce the conclusions and prospects.