Method of fundamental solutions for a Cauchy problem of the Laplace equation in a half-plane

Abstract

This paper is to provide an analysis of an ill-posed Cauchy problem in a half-plane. This problem is novel since the Cauchy data on the accessible boundary is given, whilst the additional temperature is involved on a line. The Dirichlet boundary condition on part of the boundary is an essential condition in the physical meaning. Then we use a redefined method of fundamental solutions (MFS) to determine the temperature and the normal heat flux on the inaccessible boundary. The present approach will give an ill-conditioned system, and this is a feature of the numerical method employed. In order to overcome the ill-posedness of the corresponding system, we use the Tikhonov regularization method combining Morozov’s discrepancy principle to get a stable solution. At last, four numerical examples, including a smooth boundary, a boundary with a corner, and a boundary with a jump, are given to show the effectiveness of the present approach.