Explicit scheme based on integral transforms for estimation of source terms in diffusion problems in heterogeneous media

Abstract

The estimation of source terms present in differential equations has various applications, ranging from structural assessment, industrial process monitoring, equipment failure detection, environmental pollution source detection to identification applications in medicine. Significant progress has been made in recent years in methodologies capable of estimating this parameter. This work employs a methodology based on an explicit formulation of the integral transformation to characterize the unknown source term, reconstructing it through the expansion in known eigenfunctions of the Sturm-Liouville eigenvalue problem. To achieve this, a linear model is considered in a heterogeneous medium with known and spatially varying physical properties and two heat sources, with both temporal and spatial dependencies, and only spatial dependence. The eigenvalue problem contains information about the heterogeneous properties and is solved using the generalized integral transformation technique. Additionally, an initial interpolation of the sensor data is proposed for each observation time, making the inverse problem computationally lighter. The solutions of the inverse problem exhibit optimal performance, even with noisy input data and sources with abrupt discontinuities. The temperatures recovered by the direct problem considering the recovered source closely match synthetic experimental data, showing errors less than 1%, ensuring the robustness and reliability of the technique for the proposed application.