Identification and stability analysis for inverse diffusion-wave fractional problems

Ghaziyah Alsahli; Mustapha Benoudi; Maawiya Ould Sidi; Eid Sayed Kamel; Ibrahim Omer Ahmed; MedYahya Ould-MedSalem; Hamed Ould Sidi; Ghaziyah Alsahli; Mustapha Benoudi; Maawiya Ould Sidi; Eid Sayed Kamel; Ibrahim Omer Ahmed; MedYahya Ould-MedSalem; Hamed Ould Sidi

doi:10.3934/math.2026046

AIMS Mathematics

2026, Volume 11, Issue 1: 1050-1070. doi: 10.3934/math.2026046

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Research article

Identification and stability analysis for inverse diffusion-wave fractional problems

1.
Department of Mathematics, College of Science, Jouf University, Sakaka, Aljouf 72341, Saudi Arabia
2.
National School of Applied Sciences at Université Sultan Moulay Slimane Beni Mellal, Morocco
3.
FST, University of Nouakchott, Mauritania
4.
Institut Supérieur de Comptabilité et d'Administration des Entreprises-ISCAE Nouakchott, Mauritania

Received: 25 October 2025 Revised: 08 December 2025 Accepted: 07 January 2026 Published: 14 January 2026
MSC : 35R11, 35R30, 47A52, 49M41, 49N45

Inverse problems for fractional diffusion-wave equations are vital in various scientific fields, but are inherently ill-posed due to the non-local and memory effects of fractional derivatives. Resolving these challenges, particularly in reconstructing multiple unknown initial values from final time data, remains a significant mathematical and computational hurdle. This work addressed this gap by establishing the well-posedness of such inverse problems involving space-time fractional operators, specifically the fractional Caputo derivative and the fractional Laplacian. We developed a stable regularization framework based on Tikhonov's method to ensure the stability and uniqueness of the solution, despite the poorly posed nature of the problem. An efficient conjugate gradient algorithm was proposed to numerically reconstruct the unknowns, with specialized techniques to handle fractional operators effectively. Numerical experiments with exact and noisy data confirmed the robustness, accuracy, and practicality of our approach, demonstrating its potential for real-world applications in modeling anomalous diffusion, heat conduction, and structural dynamics. Our results contributed both theoretical insights and computational tools for tackling complex inverse problems in fractional systems.
- fractional equation of time and space,
- fractional laplacian,
- inverse problem,
- regularization method,
- conjugate gradient method
Citation: Ghaziyah Alsahli, Mustapha Benoudi, Maawiya Ould Sidi, Eid Sayed Kamel, Ibrahim Omer Ahmed, MedYahya Ould-MedSalem, Hamed Ould Sidi. Identification and stability analysis for inverse diffusion-wave fractional problems[J]. AIMS Mathematics, 2026, 11(1): 1050-1070. doi: 10.3934/math.2026046

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Abstract

Inverse problems for fractional diffusion-wave equations are vital in various scientific fields, but are inherently ill-posed due to the non-local and memory effects of fractional derivatives. Resolving these challenges, particularly in reconstructing multiple unknown initial values from final time data, remains a significant mathematical and computational hurdle. This work addressed this gap by establishing the well-posedness of such inverse problems involving space-time fractional operators, specifically the fractional Caputo derivative and the fractional Laplacian. We developed a stable regularization framework based on Tikhonov's method to ensure the stability and uniqueness of the solution, despite the poorly posed nature of the problem. An efficient conjugate gradient algorithm was proposed to numerically reconstruct the unknowns, with specialized techniques to handle fractional operators effectively. Numerical experiments with exact and noisy data confirmed the robustness, accuracy, and practicality of our approach, demonstrating its potential for real-world applications in modeling anomalous diffusion, heat conduction, and structural dynamics. Our results contributed both theoretical insights and computational tools for tackling complex inverse problems in fractional systems.

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