Recovering boundary data in planar heat conduction using a boundary integral equation method

Abstract

We consider a Cauchy problem for the heat equation, where the temperature field is to be reconstructed from the temperature and heat flux given on a part of the boundary of the solution domain. We employ a Landweber type method proposed in ~\cite{Bast}, where a sequence of mixed well-posed problems are solved at each iteration step to obtain a stable approximation to the original Cauchy problem. We develop an efficient boundary integral equation method for the numerical solution of these mixed problems, based on the method of Rothe. Numerical experiments are presented both with exact and noisy data, showing the efficiency and stability of the proposed procedure and approximations.