The Cauchy Problem For A Time-fractional Diffusion Equation

In this paper, we consider a Cauchy problem of the time fractional diffusion equation for determining the Cauchy data at one part of the boundary from the Cauchy data at the other part. The variational formulation of the problem is given. And we deduce the gradient of the regularization functional based on a sensitivity problem and an adjoint problem. Then the conjugate gradient method is proposed for solving the variational problem, in which the finite difference method is used to solve the direct problem. Several numerical examples are provided to show the effectiveness and robustness of the proposed method.