Approaches to Determining an Object’s Temperature from Boundary Measurements

Abstract

we conducted a comparative analysis of two approaches for solving an inverse boundary problem for the heat conduction equation. These problems allow reconstruction of the temperature distribution in regions of an object that cannot be measured directly, using noisy data recorded on or near the object’s surface. This class of problems is highly relevant due to its wide range of applications, including thermal monitoring and diagnostics during material heat treatment, nondestructive thermal testing, and the diagnostics, monitoring, forecasting, and optimization of thermal modes in technological equipment.
The mathematical model uses the heat conduction equation, a known initial temperature distribution, a time-dependent temperature specified on one boundary, and an additional condition describing the temperature near that boundary. Our objective is to determine the unknown temperature function on the opposite boundary of a one-dimensional object, which can then be used to predict the temperature at all internal points.
Two approaches to the numerical solution of this inverse problem are considered. The first approach is based on applying the Laplace integral transform, which reduces the problem to a Volterra integral equation of the first kind. A numerical algorithm with regularization is proposed to solve this equation. The second approach relies on finite-difference approximations, enabling direct numerical simulation of the original problem. A comparative analysis of the proposed approaches is presented in terms of accuracy in reconstructing the temperature distribution and robustness with respect to errors in the input data. The methods discussed can be applied to problems of predicting the thermal state of objects subjected to external thermal effects.