Regularization Methods For Ill-posed Problems Of Several Partial Differential Equations

Inverse problem of partial differential equation is a new branch of mathematics. It has an important and wide application in geological engineering, medicine, environment,remote sensing and other fields. Due to inverse problem is often ill-posed, the solution can not be obtained by the usual methods, and some special methods must be given.Therefore, it has important theoretical significance and practical value for the study of the mathematical theory of ill-posed problems.Emergence of regularization method indicates that the study of ill-posed problem has entered a new stage. So far, the regularization methods can be roughly divided into two categories: one is the deterministic methods, i.e., Tikhonov regularization method,wavelet regularization method, Landweber iterative method; the other is the stochastic methods, i.e., Bayesian inference method, the spectral stochastic method. The topic of this dissertation is to study the regularization methods for ill-posed problems of several partial differential equations, especially the methods of the instability problems. The main research contents for the subject are as follow:Firstly, the unknown source identification of Poisson equation and modified Helmholtz equation are studied. For the unknown source identification of Poisson equation, we analyze the cause of the ill-posedness, prove the conditional stability estimate, provide three regularization methods to solve the problem, and further obtain the convergence estimates. For the unknown source identification of modified Helmholtz equation, we give the analysis of the ill-posedness, apply three regularization methods to solve the problem, and discuss the convergence order of the methods. Finally, the comparison of three regularization methods is given and the effectiveness of the proposed methods is proved.Secondly, the Cauchy problem of Laplace equation and a class of elliptic equation with variable coe?cient are considered. For the Cauchy problem of Laplace equation, we analyze the ill-posedness, apply the modified equation method to solve the problem, and prove the error estimate. For the Cauchy problem of a class of elliptic equation with variable coe?cient, we discuss several regularization methods for solving Cauchy problem of Laplace equation and give the kernel functions of these regularization methods, then through analysis, we can see that the cause of ill-posed problem is the unbounded kernel function. Based on the idea of modified kernel, we construct a new kernel function and give a regularized solution, and prove the convergence estimate between the regularized solution and the exact solution.Thirdly, the Cauchy problem of two-dimensional heat conduction equation is investigated. We get the expression in the frequency domain by the Fourier transform, and analyze the cause of the ill-posedness. Based on the idea of modified kernel, we construct a new kernel function, and provide a regularized solution. Under a priori parameter choice rule, the convergence estimate between the regularized solution and the exact solution is proved strictly.Finally, the backward problem and inverse advection-dispersion problem of fractional equation are presented, including the backward problem of space-fractional equation and the time fractional inverse advection-dispersion problem. For the backward problem of space-fractional equation, we analyze the ill-posedness of the problem,give the optimality analysis, and provide a simplified Tikhonov regularization method to solve the problem. In addition, the convergence estimate is obtained. Compared with the Tikhonov regularization method, the proposed method is effective. For the time advection-dispersion problem, the analysis of ill-posedness is given in the frequency domain by the Fourier transform, and we give an optimal filtering method and obtain the optimal convergence order.In addition, based on the results of theoretical analysis and numerical examples,the unknown source identification problem of Poisson equation and modified Helmholtz equation, the Cauchy problem of Laplace equation and a class of elliptic equation with variable coe?cient, the Cauchy problem of two-dimensional heat conduction equation,the backward problem and inverse advection-dispersion problem of fractional equation are simulated by Matlab software. The simulation results show that the proposed and designed methods can be well done on the above ill-posed problems of partial differential equation, and also confirm the conclusions of theoretical analysis.