Stable numerical solution to a Cauchy problem for a time fractional diffusion equation

In this paper, we consider a Cauchy problem of one-dimensional time fractional diffusion equation for determining the Cauchy data at x=1 from the Cauchy data at x=0. Based on the separation of variables and Duhamelʹs principle, we transform the Cauchy problem into a first kind Volterra integral equation with the Neumann data as an 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. The generalized cross validation choice rule is applied to find a suitable regularization parameter. Three numerical examples are provided to show the effectiveness and robustness of the proposed method.