Eur. Phys. J. Plus (2018) 133: 403
https://doi.org/10.1140/epjp/i2018-12193-8

Regular Article

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

¹ Department of Mathematics, University of Science and Technology of Mazandaran, Behshahr, Iran
² Resson Aerospace Corporation, 46 Dineen Drive, E3B9W4, Fredericton, NB, Canada

^* e-mail: k.rashedi@mazust.ac.ir

Received: 10 May 2018
Accepted: 1 August 2018
Published online: 5 October 2018

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.