An explicit transform-domain integral method for inverse source identification in diffusion problems

This work presents the Direct Integration Method (DIM), an explicit transform-domain formulation for the identification of space-time sources in transient diffusion problems. The formulation is obtained by directly integrating the modal equations, allowing the cumulative source to be estimated first and its instantaneous coefficients to be subsequently recovered by temporal differentiation. The resulting discrete formulation retains the same central difference as the derivative-based modal estimator, while introducing a three-point weighted average of the nondifferentiated modal term, reducing its contribution to noise variance. Spatial regularization is performed through spectral truncation, whose order is selected prior to the inversion using the discrepancy principle applied to the projected data. The method was evaluated for a two-dimensional heat conduction problem involving four materials, discontinuous and smooth sources, different noise levels, and different spatial and temporal data resolutions. The results showed stable reconstructions, with truncation orders close to the minimum-error region and a progressive loss of spatial resolution as thermal diffusivity increases. Comparisons with the Time-Derivative-Based Method (TDBM), Tikhonov regularization formulations, and the Conjugate Gradient Method (CGM) showed that DIM combines competitive accuracy with low computational cost. As a complementary assessment, the method was applied to experimental data from a sapphire pulsating heat pipe, showing qualitative consistency with reference inverse heat-flux reconstructions.