Inverse Problem of Identifying a Time-Dependent Source Term in a Fractional Degenerate Semi-Linear Parabolic Equation

Maroua Nouar, Chattouh Abdeledjalil, Omar Mossa Alsalhi and Hamed Ould Sidi ()
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Maroua Nouar: Departement of Mathematics, Abbes Laghrour University Khenchela, Khenchela 40004, Algeria
Chattouh Abdeledjalil: Departement of Mathematics, Abbes Laghrour University Khenchela, Khenchela 40004, Algeria
Omar Mossa Alsalhi: Department of Mathematics, Al-Leith University College, Umm Al-Qura University, Mecca 21961, Saudi Arabia
Hamed Ould Sidi: Département des Méthodes Quantitatives et Informatiques, Institut Supérieur de Comptabilité et d’Administration des Entreprises (ISCAE), Nouakchott 6093, Mauritania

Mathematics, 2025, vol. 13, issue 9, 1-25

Abstract: This work investigates the inverse problem of identifying a time-dependent source term in a time-fractional semi-linear degenerate parabolic equation using integral measurement data. We establish the unique solvability of the inverse problem within a suitable functional framework. The proof methodology is based on the Rothe method, where the variational formulation is discretized in time, and a priori estimates for discrete solutions are derived. These estimates are then utilized to demonstrate the convergence of Rothe approximations to a unique weak solution. Additionally, we develop a numerical scheme based on the L 1 -Galerkin finite element method, combined with iterative refinement, to reconstruct the unknown source term. The numerical performance of the proposed method is validated through a series of computational experiments, demonstrating its stability and robustness against noisy data.

Keywords: inverse source problem; fractional derivative; degenerate parabolic equation; weak solution (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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