Reconstruction of a spacewise-dependent heat source in a time-dependent heat diffusion process

摘要

We investigate the inverse problem of determining a spacewise-dependent heat source in the parabolic heat equation, where the governing heat operator has coefficients that depend both on space and time. The aim is to recover a spacewise-dependent heat source, given the usual conditions of the direct problem and additional information from a supplementary temperature measurement at a given single instant of time. We show that this inverse problem has a unique solution, and for this inverse problem we propose a stable iterative procedure to find the heat source. This method is based on a sequence of well-posed direct problems, which are numerically solved at each iteration step using the finite element method. The instability of this inverse source problem is overcome by stopping the iterations using the discrepancy principle. Numerical results are included, showing that an accurate and stable reconstruction can be obtained.