A Unified Block‐Modal Framework for Inverse Source Problems in Heat and Mass Transfer

ABSTRACT This work presents a unified block–modal framework for inverse source identification in linear diffusive problems arising in heat and mass transfer. Starting from a general parabolic model with mixed boundary operators, the Classical Integral Transform Technique is employed to project the dynamics onto an orthonormal eigenbasis, yielding a family of decoupled convolution‐type ordinary differential equations in time. The transformed source in each mode is represented by a parametric temporal expansion, so that the inverse problem becomes a block‐diagonal linear least‐squares estimation problem. The Tikhonov regularization is incorporated either with a single global parameter or with mode‐wise parameters, with automatic parameter selection by generalized cross‐validation, the L‐curve, quasi‐optimality, and the Bazán fixed‐point criterion, while the truncation order of the inverse series is determined by an efficient implementation of the discrepancy principle based on cumulative reconstructions in the transformed space. The methodology is assessed on one‐ and two‐dimensional heat conduction benchmarks with smooth and discontinuous space–time sources under different boundary conditions, using noisy synthetic temperature data. The results show that the relative performance of the mode‐wise and global regularization strategies depends on both the dimensionality of the problem and the adopted parameter‐choice rule: mode‐wise GCV provides the best overall performance in the one‐dimensional benchmarks, whereas global regularization becomes advantageous in the two‐dimensional problem. Spatial and temporal data‐enrichment analyses further show that higher‐order spectral information can be recovered as the amount of measurement information increases. Direct comparisons with another CITT‐based least‐squares formulation and with an adjoint conjugate‐gradient method show that the proposed block‐modal organization preserves comparable reconstruction accuracy while substantially reducing the computational cost. The spectral analysis clarifies the interplay between modal energy content, regularization strategy, and optimal truncation, and supports the computational efficiency and scalability of the proposed framework for inverse source problems in heat and mass transfer.