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Title: Reconstruction of an orthotropic thermal conductivity from non-local heat flux measurements

Authors: M.J. Huntul; M.S. Hussein; D. Lesnic; M.I. Ivanchov; N. Kinash

Addresses: Department of Mathematics, College of Science, Jazan University, Jazan, Saudi Arabia ' Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq ' Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK ' Deceased; Formerly of: Department of Differential Equations, Faculty of Mechanics and Mathematics, Ivan Franko National University of Lviv, 1, Universytetska str., Lviv, 79000, Ukraine ' Department of Cybernetics, Tallinn University of Technology, Ehitajate Tee 5, 19086 Tallinn, Estonia

Abstract: Raw materials are anisotropic and heterogeneous in nature, and recovering their conductivity is of utmost importance to the oil, aerospace and medical industries concerned with the identification of soils, reinforced fibre composites and organs. Due to the ill-posedness of the anisotropic inverse conductivity problem certain simplifications are required to make the model tracktable. Herein, we consider such a model reduction in which the conductivity tensor is orthotropic with the main diagonal components independent of one space variable. Then, the conductivity components can be taken outside the divergence operator and the inverse problem requires reconstructing one or two components of the orthotropic conductivity tensor of a two-dimensional rectangular conductor using initial and Dirichlet boundary conditions, as well as non-local heat flux over-specifications on two adjacent sides of the boundary. We prove the unique solvability of this inverse coefficient problem. Afterwards, numerical results indicate that accurate and stable solutions are obtained.

Keywords: inverse problem; orthotropic thermal conductivity; two-dimensional heat equation; nonlinear optimisation.

DOI: 10.1504/IJMMNO.2020.104327

International Journal of Mathematical Modelling and Numerical Optimisation, 2020 Vol.10 No.1, pp.102 - 122

Received: 25 Oct 2018
Accepted: 10 Jun 2019

Published online: 26 Dec 2019 *

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