Simultaneous Identification of Thermal Conductivity and Heat Source in the Heat Equation

M J Huntul,M.S Hussein

doi:10.24996/ijs.2021.62.6.22

Abstract

This paper presents a numerical solution to the inverse problem consisting of recovering time-dependent thermal conductivity and heat source coefficients in the one-dimensional parabolic heat equation. This mathematical formulation ensures that the inverse problem has a unique solution. However, the problem is still ill-posed since small errors in the input data lead to a drastic amount of errors in the output coefficients. The finite difference method with the Crank-Nicolson scheme is adopted as a direct solver of the problem in a fixed domain. The inverse problem is solved subjected to both exact and noisy measurements by using the MATLAB optimization toolbox routine lsqnonlin , which is also applied to minimize the nonlinear Tikhonov regularization functional. The thermal conductivity and heat source coefficients are reconstructed using heat flux measurements. The root mean squares error is used to assess the accuracy of the approximate solutions of the problem. A couple of numerical examples are presented to verify the accuracy and stability of the solutions.

Highlights

1 Introduction Inverse problems for the parabolic heat equation consisting of determining the unknown coefficients and heat source depending on time or space variable, have recently received some attention
In other studies [7, 8, 9], one of the time-dependent unknowns is allowed to be in the free term heat source
We focus on solving numerically the unknown coefficients a(t) and f(t) together with the unknown temperature satisfying the inverse problem, using the measurements of heat flux instead of integral conditions

Summary

Introduction

Inverse problems for the parabolic heat equation consisting of determining the unknown coefficients and heat source depending on time or space variable, have recently received some attention. An example of coefficient identification problem is to determine a single unknown time- dependent property, such as heat capacity, thermal conductivity, or diffusivity, from additional local or non-local measurements of the main dependent variable at the boundary or inside the domain [1, 2, 3]. In previous papers [4, 5, 6], multiple time-dependent coefficient identifications were considered, while they were recently solved numerically In these studies, the unknowns were mainly coefficients multiplying the temperature and its partial derivatives. The inverse problems investigatedinthispaperhave already been proved to be locally uniquely solvable by Bereznyts’ka [12], but no numerical reconstruction has been attempted so far It is the purpose of this paper to undertake the simultaneous numerical solution of these unknowns.

Mathematical formulation of inverse problem

Direct problem

Results and discussion

Example 1