A Number Of Mathematical Physics Is The Problem With The Inverse Problem Algorithm And Its Applications

The inverse problems of mathematical physics arise from the practical problems in physics, biology, medicine and geography, etc. The difficulty of studying inverse problems of mathematical physics is that most of the inverse problems are ill-posed in Hadamard's sense, while the corresponding forward problems arc well-posed. In this thesis, we will discuss the following three kinds of inverse problems of mathematical physics: inverse problems in the heat transfer, numerical differentiation and inverse problems in MRE(Magnetic Resonance Elastography).In the first part of the thesis, we deal with the mathematical model of the heat transfer in the multi-layers mediums, which models one practical problems in the industry. Firstly, we introduce the background of the problem and then, based on some physical laws, we formulate the problem as a problem of the heat equation with nonlinear transmission conditions on the interfaces. The existence and uniqueness of the global solution for the forward problem are proved. We also propose the numerical algorithms. The purpose of this study is to determine the corrosion of the domain by the measurements outside. This will lead to the inverse problem. We prove the uniqueness, which means that the measurements are enough for determine the corrosion of the domain. Based on [70], one improved stable algorithm is proposed and the numerical results show that our method is effective and can be applied to the practical problem.In the second part of this thesis, we discuss the numerical differentiation for the period functions by Tikhonov regularization. The regularized solutions based on the spline functions are constructed. We prove the uniqueness of the regularized solutions and the error estimates are also given. An application of this method is presented at the end this part. We show that, by the numerical differentiation for the period functions, it can calculate the compliance of artery in the neck. The mathematical model and two numerical examples are presented.In the third part of this thesis, the inverse problems in Magnetic Resonance Elastography are discussed. We first introduce the background of the inverse problems in the elastography. Then a mathematical model with piecewise continues coefficients is introduced base on the practical assumptions. We prove that the space dependent coefficients of the wave equation can be uniquely determined in the domain where the compression (shear) waves reach. Finally, the generalized variational data assimilation method, a kind of variational approach method, is applied to recovery the coefficients of the wave equation numerically, and we add regularized item to the variation equation for overcoming the ill-posedness of the inverse problems.