Studies On Several Kinds Of Inverse Problems For Fractional Diffusion Equations

Motivated by practical problems, fractional diffusion equations have attracted wide attentions in recent years and the direct problems have been studied extensively. However, in some practical situations, part of boundary data, or initial data, or source term, or diffusion coefficients may not be given and we want to find them by additional measured data which will yield to some fractional diffusion inverse problems. This thesis discusses the following inverse problems for fractional diffusion equations.Part1discusses the Cauchy problem for the time fractional diffusion equa-tion. Based on the separation of variables and Duhamel’s principle, we transform the Cauchy problem into a first kind Volterra integral equation with the Neumann data as unknown function and then show the ill-posedness of problem. Further, we use a boundary element method combined with a generalized Tikhonov regularization to solve the first kind integral equation. Numerical examples are provided to show the effectiveness and robustness of the proposed method.Part2studies the inverse source problem for the time fractional diffusion equa-tion. For the source term depending on the time variable, based on the separation of variables and Duhamel’s principle, we also transform the inverse source problem into a first kind Volterra integral equation with the source term as the unknown function and then show the ill-posedness of the problem. Further, we use a boundary element method combined with a generalized Tikhonov regularization to solve the Volterra integral equation of the first kind. For source term depending on the space variable, we transform the inverse source problem into a first kind Fredholm integral equation. A truncation method is presented to deal with the ill-posedness of the problem and error estimates are obtained with an a priori choice rule and an a posteriori choice rule to find the regularization parameter.Part3, we consider a nonlinear inverse problem, i.e., identifying a Robin coef-ficient for time fractional diffusion equation from part of the boundary data. Based on the separation of variables, we transform the problem into a nonlinear integral equations. By using the boundary element method, we finally obtain an optimization problem in finite dimensional space and the conjugate gradient method is used to solve it. Numerical examples are provided to show the effectiveness of our method.Part4, a space-fractional backward diffusion problem (SFBDP) is considered. By the Fourier transform, we propose an optimal modified method to solve this problem in the frequency domain. The convergence estimates for the approximate solutions with the regularization parameter selected by an a priori and an a posteriori strategy are provided, respectively. Numerical experiments show that the proposed methods are effective and stable.