Regularization Methods And Algorithms For Three Kinds Of Ill-posed Problems

In this paper,we devote the regularization methods and algorithms for three kinds of ill-posed problems,which are the Cauchy problem for Helmholtz and modified Helmholtz equation,the unknown source identification problem for time-fractional diffusion equation on a columnar symmetric domain.We use the different regularization methods to deal with these ill-posed problems.The Cauchy problem for Helmholtz equation is a classical ill-posed problem.In the second and third chapter of this paper,we consider the Cauchy problem of the inhomogeneous Helmholtz equation in rectangular domain and the modified Helmholtz equation in strip domain respectively.Then the truncation method and the quasi-boundary value method are proposed to recovery the ill-posedness of the problem respectively.Moreover,the error estimates are derived by an a priori and an a posteriori choice rules of regularization parameters.Finally,numerical examples show that the regularization methods work effectively.In the fourth chapter of this thesis,we deal with the inverse problem of identifying the unknown source of time-fractional diffusion equation on a columnar symmetric domain.Firstly,we establish the conditional stability for this inverse problem.Then we use Tikhonov regularization method to recovery the ill-posedness of the problem and the error estimates under an a priori and an a posteriori choice rules of regularization parameters are given.Finally,numerical examples are presented to illustrate the validity and effectiveness of Tikhonov regularization method.