Heat source identification of some parabolic equations based on the method of fundamental solutions

Abstract

In this paper, we consider inverse problems of the heat conduction process in one and two-dimensional homogeneous bodies of finite size, subject to the given initial and boundary conditions. The considered inverse problems are severely ill-posed since their solutions; if they exist, they do not depend continuously on the input data. We obtain stable solutions for several inverse problems by proposing a meshless regularization technique based on the combination of the method of fundamental solutions and the Tikhonov’s regularization method. In particular, we use the given information at the terminal state to estimate the space-dependent heat source in the one-dimensional case and the space- and time-dependent heat source in the two-dimensional case. Numerical results demonstrate high accuracy and low computational cost.