Explicit Methods for Inverse Source Identification in Transient Diffusion via Truncated Eigenfunction Regularization of Temperature Measurements

This work presents and compares two explicit approaches for estimating space‐ and time‐dependent source terms in parabolic diffusion equations, derived from analytical solutions obtained through the Classical Integral Transform Technique (CITT). The formulation yields a decoupled system in the modal domain, enabling independent estimation of the transformed source coefficients. Two methodologies are considered: the time‐derivative‐based method (TDBM), which approximates the temporal derivatives of the transformed potentials using finite differences, yielding an explicit differential formulation; and the kernel‐deconvolution‐based method (KDBM), in which the decoupled modal equations obtained through CITT are recast as independent first‐kind Volterra integral equations with known exponential kernels, enabling explicit recovery of the transformed source coefficients through causal kernel deconvolution. A regularization strategy based on the discrepancy principle is introduced, in which the measured temperature data are projected onto a truncated eigenfunction basis, providing spectral noise filtering and selecting the admissible modal content before the source coefficients are estimated. Theoretical error‐propagation analysis shows that both methods originate from equivalent differential and integral representations of the same modal equation. For a fixed time step, the rectangular quadrature adopted in the KDBM recursion bounds the direct noise amplification uniformly with respect to the modal index, while introducing a deterministic attenuation of higher‐order modal coefficients. Numerical results for one‐ and two‐dimensional problems, considering different noise levels and material properties, demonstrate that both methods provide physically consistent reconstructions. However, KDBM exhibits greater robustness in problems with higher modal content, in which the effects of noise amplification become more significant.