Open Access

ARTICLE

Solvability of the Nonlocal Inverse Parabolic Problem and Numerical Results

by M. J. Huntul^1,*, Taki-Eddine Oussaeif²

1 Department of Mathematics, College of Science, Jazan University, Jazan, Saudi Arabia
2 Department of Mathematics and Informatics, Larbi Ben M’hidi University, Oum El Bouaghi, Algeria

* Corresponding Author: M. J. Huntul. Email:

Computer Systems Science and Engineering 2022, 40(3), 1109-1126. https://doi.org/10.32604/csse.2022.020175

Received 11 May 2021; Accepted 12 June 2021; Issue published 24 September 2021

Abstract

In this paper, we consider the unique solvability of the inverse problem of determining the right-hand side of a parabolic equation whose leading coefficient depends on time variable under nonlocal integral overdetermination condition. We obtain sufficient conditions for the unique solvability of the inverse problem. The existence and uniqueness of the solution of the inverse parabolic problem upon the data are established using the fixed point theorem. This inverse problem appears extensively in the modelling of various phenomena in engineering and physics. For example, seismology, medicine, fusion welding, continuous casting, metallurgy, aircraft, oil and gas production during drilling and operation of wells. In addition, the numerical solution of the inverse problem is studied by using the Crank-Nicolson finite difference method together with the Tikhonov regularization to find a stable and accurate approximate solution of finite differences. The resulting nonlinear system of parabolic equation is solved computationally using the MATLAB subroutine lsqnonlin. Both analytical and numerically simulated noisy input data are inverted. The root mean square error values for various noise levels for both continuous and discontinuous time-dependent heat source term are compared. Numerical results presented for two examples show the efficiency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data. Furthermore, the choice of the regularization parameter is also discussed based on the trial and error technique.

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