Title: The fully-implicit finite difference method for solving nonlinear inverse parabolic problems with unknown source term

Authors: Hassan Dana Mazraeh; Reza Pourgholi; Sahar Tavana

Addresses: School of Mathematics and Computer Science, Damghan University, Damghan 36715-364, Iran ' School of Mathematics and Computer Science, Damghan University, Damghan 36715-364, Iran ' School of Mathematics and Computer Science, Damghan University, Damghan 36715-364, Iran

Abstract: A numerical procedure based on a fully implicit finite difference method for an inverse problem of identification of an unknown source in a heat equation is presented. The approach of the proposed method is to approximate unknown function from the solution of the minimisation problem based on the overspecified data. This problem is ill-posed, in the sense that the solution (if it exist) does not depend continuously on the data. To regularise this ill-conditioned, we apply the Tikhonov regularisation 0^th, 1^st and 2^nd method to obtain the stable numerical approximation to the solution. A stability analysis shows that this numerical scheme approximation is unconditionally stable. Numerical results for two inverse source identification problems show that the proposed numerical algorithm is simple, accurate, stable and computationally efficient.

Keywords: inverse problems; ill-posed problem; fully implicit; unknown source; Tikhonov regularisation method; least square; noisy data.

DOI: 10.1504/IJCSM.2018.094652

International Journal of Computing Science and Mathematics, 2018 Vol.9 No.4, pp.405 - 418

Received: 12 Apr 2016
Accepted: 19 Jul 2017
Published online: 11 Sep 2018 *

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