Identification of Time-Wise Thermal Diffusivity, Advection Velocity on the Free-Boundary Inverse Coefficient Problem

M. S. Hussein, Taysir E. Dyhoum (), S. O. Hussein () and Mohammed Qassim
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M. S. Hussein: Department of Mathematics, College of Science, University of Baghdad, Baghdad 10071, Iraq
Taysir E. Dyhoum: Department of Computing and Mathematics, Faculty of Science and Engineering, Manchester Metropolitan University, Manchester M15 6BX, UK
S. O. Hussein: Department of Mathematics, College of Science, University of Sulaymaniyah, Sulaymaniyah 46001, Iraq
Mohammed Qassim: Department of Energy, College of Engineering Al-Musayab, University of Babylon, Babylon 51002, Iraq

Mathematics, 2024, vol. 12, issue 17, 1-21

Abstract: This paper is concerned with finding solutions to free-boundary inverse coefficient problems. Mathematically, we handle a one-dimensional non-homogeneous heat equation subject to initial and boundary conditions as well as non-localized integral observations of zeroth and first-order heat momentum. The direct problem is solved for the temperature distribution and the non-localized integral measurements using the Crank–Nicolson finite difference method. The inverse problem is solved by simultaneously finding the temperature distribution, the time-dependent free-boundary function indicating the location of the moving interface, and the time-wise thermal diffusivity or advection velocities. We reformulate the inverse problem as a non-linear optimization problem and use the l s q n o n l i n non-linear least-square solver from the MATLAB optimization toolbox. Through examples and discussions, we determine the optimal values of the regulation parameters to ensure accurate, convergent, and stable reconstructions. The direct problem is well-posed, and the Crank–Nicolson method provides accurate solutions with relative errors below 0.006 % when the discretization elements are M = N = 80 . The accuracy of the forward solutions helps to obtain sensible solutions for the inverse problem. Although the inverse problem is ill-posed, we determine the optimal regularization parameter values to obtain satisfactory solutions. We also investigate the existence of inverse solutions to the considered problems and verify their uniqueness based on established definitions and theorems.

Keywords: parabolic heat equation; finite-difference method (FDM); Crank–Nicolson method; inverse coefficient identification problem; optimization tool; MATLAB; free-boundary problem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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