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

In this paper,we consider two kinds inverse problems for time-fractional diffusion equation,which are the unknown source identification problem and the inverse initial value problem.They are ill-posed in the sense that its solution(if exists)does not depend continuously on the measure data.In the second chapter of this paper,we consider an identifying unknown source problem for time-fractional diffusion equation with variable coefficients in a general bounded domain.This is an ill-posed problem.We apply Landweber iterative method to recovery the ill-posedness of the problem.Moreover,the convergence estimates are given under an a priori and an a posteriori regularization parameter choice rules,respectively.Finally,numerical examples show that the effectiveness of the regularization method.In the third chapter,we discuss an inverse initial value problem for timefractional diffusion equation with variable coefficients in a general bounded domain.This is an ill-posed problem.We use Landweber iterative method to deal with the problem.Moreover,the convergence estimates under an a priori and an a posteriori regularization parameter choice rules are proved,respectively.Finally,the numerical results show that Landweber iterative method is effective and stable.The forth chapter consider a multi-dimensional inverse initial value problem for time-fractional diffusion equation with inhomogeneous source.This problem is ill-posed.We obtain regularization solution by quasi-reversibility regularization method.And the convergence estimates between regularization solution and the exact solution are given under an a priori and an a posteriori regularization parameter choice rules,respectively.Finally,the numerical examples in one-dimensional and two-dimensional cases are presented which demonstrate the effectiveness of the regularization method.In this paper,we solve the problem of inverse initial value problem for inhomogeneous time-fractional diffusion equation with variable coefficients.It is a novel ill-posed problem.The theoretical results and numerical results in this paper can fully and effectively show that the regularization method can solve the ill-posed problem well.