Recovering of dual time-varying source terms in a system of coupled time-fractional diffusion equations

Maroua Nouar; Maged Z. Youssef; Hamed Ould Sidi; Abdeldjalil Chattouh; Maroua Nouar; Maged Z. Youssef; Hamed Ould Sidi; Abdeldjalil Chattouh

doi:10.3934/math.20251287

AIMS Mathematics

2025, Volume 10, Issue 12: 29285-29318. doi: 10.3934/math.20251287

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Recovering of dual time-varying source terms in a system of coupled time-fractional diffusion equations

1.
Department of Mathematics, University of Khenchela, 40004, Khenchela, Algeria
2.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia
3.
Département des Méthodes Quantitatives et Informatiques, Institut Supérieur de Comptabilité et d'Administration des Entreprises (ISCAE), Nouakchott 6093, Mauritania
4.
Laboratoire de Mathématiques et d'Intelligence Artificielle, Univerity of Khenchela

Received: 19 October 2025 Revised: 20 November 2025 Accepted: 03 December 2025 Published: 12 December 2025
MSC : 35R30, 35R11, 35K57, 65M32

The present study is devoted to investigating the inverse problem of simultaneously reconstructing two source terms that depend solely on time in a system of coupled fractional reaction–diffusion equations. Such coupled systems are fundamental for modelling multispecies anomalous diffusion processes, where the evolution of each state variable is governed by unknown time-varying sources. The inverse problem is tackled using supplementary measurements of the state variables over the spatial domain. The coupling between the two equations presents a distinct complexity, making the simultaneous reconstruction problem notably challenging and important. Results on the unique solvability, are established by employing the Rothe method, whereby the problem is first discretized in time and then stability results for the semi-discrete approximations are derived. These estimates, together with compactness arguments, are then used to rigorously prove the convergence of Rothe approximations to a unique weak solution. Finally, the proposed method's effectiveness and stability are further validated through a series of numerical simulations.
- inverse source problem,
- coupled fractional system,
- temporal source identification,
- Rothe method,
- numerical reconstruction
Citation: Maroua Nouar, Maged Z. Youssef, Hamed Ould Sidi, Abdeldjalil Chattouh. Recovering of dual time-varying source terms in a system of coupled time-fractional diffusion equations[J]. AIMS Mathematics, 2025, 10(12): 29285-29318. doi: 10.3934/math.20251287

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Abstract

The present study is devoted to investigating the inverse problem of simultaneously reconstructing two source terms that depend solely on time in a system of coupled fractional reaction–diffusion equations. Such coupled systems are fundamental for modelling multispecies anomalous diffusion processes, where the evolution of each state variable is governed by unknown time-varying sources. The inverse problem is tackled using supplementary measurements of the state variables over the spatial domain. The coupling between the two equations presents a distinct complexity, making the simultaneous reconstruction problem notably challenging and important. Results on the unique solvability, are established by employing the Rothe method, whereby the problem is first discretized in time and then stability results for the semi-discrete approximations are derived. These estimates, together with compactness arguments, are then used to rigorously prove the convergence of Rothe approximations to a unique weak solution. Finally, the proposed method's effectiveness and stability are further validated through a series of numerical simulations.

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