Combination of Integral Transforms and Linear Optimization for Source Reconstruction in Heat and Mass Diffusion Problems

This paper presents a novel methodology for estimating space- and time-dependent source terms in heat and mass diffusion problems. The approach combines classical integral transform techniques (CITTs) with the least squares optimization method, enabling an efficient reconstruction of source terms. The method employs a double expansion framework, using both spatial eigenfunction and temporal expansions. The new presented idea assumes that the source term can be expressed as a spatial expansion in eigenfunctions of the eigenvalue problem, and then each transient function associated with each term of spatial expansion is rewritten as an additional expansion, where the unknown coefficients approximating the transformed source enable the direct use of the solution in the objective function. This, in turn, results in a linear optimization problem that can be quickly minimized. Numerical experiments, including one-dimensional and two-dimensional scenarios, demonstrate the accuracy of the proposed method in the presence of noisy data. The results highlight the method’s robustness and computational efficiency, even with minimal temporal expansion terms.