Regularization Methods And Algorithms For Ill-posed Problems Of Several Kinds Time-fractional Diffusion Equation

In recent years,people pay close attention to the abnormal diffusion phenomenon.The fractional diffusion equations can describe the abnormal diffusion phenomenon accurately.Therefore,the research about fractional diffusion equation is getting more and more attention,especially the research on the inverse problem of fractional diffusion equation.This paper is divided into four parts to consider the inverse problems of the time-fractional diffusion equation.In the second chapter,we consider the problem of identifying the spacial source term of the time-space fractional nonhomogeneous diffusion heat equation in a general bounded region.This problem is ill-posed.In this paper,we use the fractional Landweber iterative regularization method and Landweber iterative regularization method to solve this problem.We give the error estimates between the exact solution and the regularized solutions,respectively,and compare this two regularization methods by numerical results.In the third chapter,we consider the problem of identifying the initial value of the time-fractional nonhomogeneous diffusion heat equation in a columnar symmetric region.This problem is ill-posed.In this paper,we use two kinds of fractional Tikhonov regularization methods to solve this problem.We give the error estimates between the exact solutions and the regularized solutions,respectively,and compare this two regularization methods by numerical results and error estimates.In the fourth chapter,we consider the problem of identifying the initial value of the time-fractional nonhomogeneous diffusion wave equation in a special bounded domain.This problem is ill-posed.In this paper,we use truncated regularization method to solve this problem.We give the error estimates between the exact solution and the regularized solution and numerical examples.In the fifth chapter,we consider the problem of identifying the initial value of the time-space fractional non-linear diffusion heat equation in a special bounded domain.In this paper,we give the uniqueness and the ill-posedness of the problem.We use truncated regularization method to solve this problem.We give the error estimates between the exact solution and the regularized solution and numerical example.