Inverse problem for determining the coefficient in the heat conduction equation

Abstract

Abstract. This paper considers the inverse problem of determining the thermal conductivity coefficient in the heat equation. The objective of this study is to determine the unknown coefficient based on measured boundary temperature data over time. The governing equation is a parabolic partial differential equation that describes the heat transfer process, with the unknown thermal conductivity playing a decisive role in the solution. The inverse problem is formulated as an optimization problem, in which the discrepancy between the simulated temperature distribution and the experimental data is minimized. Numerical modeling methods were used to solve the problem, including the tridiagonal matrix algorithm (Thomas algorithm) for discretizing the heat equation. The optimization process was performed using gradient methods, where the adjoint problem was used to efficiently calculate the gradient of the objective function with respect to the thermal conductivity coefficient. The results demonstrated acceptable accuracy in reconstructing the coefficient. Different parameters for the reduction factor in the gradient method were also considered. These findings are important for applications in fields such as materials science, geophysics, and engineering, where accurate estimation of thermal properties is essential.

Keywords: inverse problem, heat transfer, adjoint problem, thermal conductivity coefficient, numerical modeling.