Numerical Methods For Two Inverse Problems On Partial Differential Equation

In this paper, we first combine the boundary control technique with the method of fundamental solution to solve a Cauchy problem of the Laplace equation in an multi-connected domain, and then we proposed a operator equation method cou-pled with the method of fundamental solutions to solve a Cauchy problem of heat equation. The main idea is to transform the Cauchy problem into a abstract operator equation problem and then use the method of fundamental solutions (MFS) to find the missing Dirichlet and Neumann data at the inaccessible end. The difference with the usual MFS is that we use the method of fundamental solutions to solve a sequence of direct problems instead of solving the inverse problem directly. Due to the ill-posedness of Cauchy problem, the discrete Tikhonov regularization method and the generalized cross-validation(GCV) criterion have been used to stabilize a numerical solution. The effectiveness of the proposed approaches to solve both Cauchy problems are illustrated by several numerical examples.