Inverse Problem of Identifying a Time-Dependent Source Term in a Fractional Degenerate Semi-Linear Parabolic Equation
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 methodol...
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MDPI AG
2025-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/13/9/1486 |
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| author | Maroua Nouar Chattouh Abdeledjalil Omar Mossa Alsalhi Hamed Ould Sidi |
| author_facet | Maroua Nouar Chattouh Abdeledjalil Omar Mossa Alsalhi Hamed Ould Sidi |
| author_sort | Maroua Nouar |
| collection | DOAJ |
| description | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>1</mn></msup></semantics></math></inline-formula>-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. |
| format | Article |
| id | doaj-art-e3359041f0be415985bd1e66babcb5eb |
| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-e3359041f0be415985bd1e66babcb5eb2025-08-20T02:30:46ZengMDPI AGMathematics2227-73902025-04-01139148610.3390/math13091486Inverse Problem of Identifying a Time-Dependent Source Term in a Fractional Degenerate Semi-Linear Parabolic EquationMaroua Nouar0Chattouh Abdeledjalil1Omar Mossa Alsalhi2Hamed Ould Sidi3Departement of Mathematics, Abbes Laghrour University Khenchela, Khenchela 40004, AlgeriaDepartement of Mathematics, Abbes Laghrour University Khenchela, Khenchela 40004, AlgeriaDepartment of Mathematics, Al-Leith University College, Umm Al-Qura University, Mecca 21961, Saudi ArabiaDépartement des Méthodes Quantitatives et Informatiques, Institut Supérieur de Comptabilité et d’Administration des Entreprises (ISCAE), Nouakchott 6093, MauritaniaThis 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>1</mn></msup></semantics></math></inline-formula>-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.https://www.mdpi.com/2227-7390/13/9/1486inverse source problemfractional derivativedegenerate parabolic equationweak solution |
| spellingShingle | Maroua Nouar Chattouh Abdeledjalil Omar Mossa Alsalhi Hamed Ould Sidi Inverse Problem of Identifying a Time-Dependent Source Term in a Fractional Degenerate Semi-Linear Parabolic Equation Mathematics inverse source problem fractional derivative degenerate parabolic equation weak solution |
| title | Inverse Problem of Identifying a Time-Dependent Source Term in a Fractional Degenerate Semi-Linear Parabolic Equation |
| title_full | Inverse Problem of Identifying a Time-Dependent Source Term in a Fractional Degenerate Semi-Linear Parabolic Equation |
| title_fullStr | Inverse Problem of Identifying a Time-Dependent Source Term in a Fractional Degenerate Semi-Linear Parabolic Equation |
| title_full_unstemmed | Inverse Problem of Identifying a Time-Dependent Source Term in a Fractional Degenerate Semi-Linear Parabolic Equation |
| title_short | Inverse Problem of Identifying a Time-Dependent Source Term in a Fractional Degenerate Semi-Linear Parabolic Equation |
| title_sort | inverse problem of identifying a time dependent source term in a fractional degenerate semi linear parabolic equation |
| topic | inverse source problem fractional derivative degenerate parabolic equation weak solution |
| url | https://www.mdpi.com/2227-7390/13/9/1486 |
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