This paper addresses the optimal recovery of spatial sources in a dynamical problem governed by a space-time fractional diffusion-wave model. Specifically, we estimate the spatial component of the source term from final-time observations in a bounded domain. Such inverse source problems are inherently ill-posed and require additional information here provided by the final-time measurements to guarantee uniqueness. To this end, we propose a nonlinear optimization formulation that provides a stable and effective framework for source identification. Given the sensitivity of the problem to data perturbations, we employ Tikhonov regularization to stabilize the reconstruction. The proposed methodology minimizes a least-squares cost functional that quantifies the mismatch between the model output and the measured final-time data, augmented with a regularization term to control instability and ensure well-posedness. We prove the Fréchet differentiability of the resulting functional and derive an adjoint-based gradient expression, enabling efficient implementation. The numerical solution is obtained using a conjugate gradient algorithm, with the Morozov discrepancy principle used to guide the stopping criterion. A series of one-dimensional numerical experiments is performed to demonstrate the effectiveness of the proposed method. The results confirm the robustness, accuracy, and computational efficiency of the proposed strategy for source reconstruction in fractional-order dynamical systems.
Citation: Maawiya Ould Sidi, Mofareh Alhazmi, Mustapha Benoudi, Kareem Alanazi, Abdeldjalil Chattouh, Hamed Ould Sidi. Optimal reconstruction of spatial sources for a dynamical problem characterized by a space–time fractional diffusion–wave model[J]. AIMS Mathematics, 2026, 11(7): 22004-22033. doi: 10.3934/math.2026890
This paper addresses the optimal recovery of spatial sources in a dynamical problem governed by a space-time fractional diffusion-wave model. Specifically, we estimate the spatial component of the source term from final-time observations in a bounded domain. Such inverse source problems are inherently ill-posed and require additional information here provided by the final-time measurements to guarantee uniqueness. To this end, we propose a nonlinear optimization formulation that provides a stable and effective framework for source identification. Given the sensitivity of the problem to data perturbations, we employ Tikhonov regularization to stabilize the reconstruction. The proposed methodology minimizes a least-squares cost functional that quantifies the mismatch between the model output and the measured final-time data, augmented with a regularization term to control instability and ensure well-posedness. We prove the Fréchet differentiability of the resulting functional and derive an adjoint-based gradient expression, enabling efficient implementation. The numerical solution is obtained using a conjugate gradient algorithm, with the Morozov discrepancy principle used to guide the stopping criterion. A series of one-dimensional numerical experiments is performed to demonstrate the effectiveness of the proposed method. The results confirm the robustness, accuracy, and computational efficiency of the proposed strategy for source reconstruction in fractional-order dynamical systems.
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Maawiya Ould Sidi, Mofareh Alhazmi, Mustapha Benoudi, Kareem Alanazi, Abdeldjalil Chattouh, Hamed Ould Sidi. Optimal reconstruction of spatial sources for a dynamical problem characterized by a space–time fractional diffusion–wave model[J]. AIMS Mathematics, 2026, 11(7): 22004-22033. doi: 10.3934/math.2026890
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