On fractional biparabolic inverse source problem

Abdelghani Lakhdari; Nadjib Boussetila; Thabet Abdeljawad; Abdellatif Ben Makhlouf

doi:10.3934/dcdss.2024216

Article Contents

2026, Volume 22: 353-371. Doi: 10.3934/dcdss.2024216

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On fractional biparabolic inverse source problem

Abdelghani Lakhdari^1,,
Nadjib Boussetila^{2, 3,},
Thabet Abdeljawad^{4, 5, 6, 7, 8, ,} and
Abdellatif Ben Makhlouf^9,

1.
Laboratory of Energy Systems Technology, Department CPST, National Higher School of Technology and Engineering, Annaba 23005, Algeria
2.
Department of Mathematics, University of 8 May 1945 Guelma, P.O. Box 401, Guelma 24000, Algeria
3.
Applied Mathematics Laboratory, University Badji Mokhtar-Annaba, P.O.Box 12, Annaba 23000, Algeria
4.
Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Saveetha University, Chennai 602105, Tamil Nadu, India
5.
Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia
6.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
7.
Center for Applied Mathematics and Bioinformatics (CAMB), Gulf University for Science and Technology, Hawally, 32093, Kuwait
8.
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Garankuwa, Medusa 0204, South Africa
9.
Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Sfax, Tunisia

^*Corresponding author: Thabet Abdeljawad
^*Corresponding author: Thabet Abdeljawad

Received: September 2024

Revised: December 2024

Early access: December 17, 2024

Published online: December 17, 2024

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Abstract

This paper addresses a fractional biparabolic inverse source problem that integrates fractional derivatives with the biparabolic equation to achieve a more precise characterization of anomalous diffusion. We show that this problem is ill-posed according to Hadamard's sense. To tackle the instability associated with this problem, we apply regularization techniques using the Landweber iterative method. We provide convergence estimates based on a priori information on the exact solution. The paper concludes with numerical experiments that confirm the validity of the proposed theoretical framework.

Keywords:

Mathematics Subject Classification: Primary: 35R30, 65F22; Secondary: 00A71, 80D23.

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Figure 1. Diffusion rates

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Figure 2. Exact and inexact data of Example 4.1: $ \delta = 10 \% $

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Figure 3. The comparison of the exact solution $ f(x) $ and its regularized solution $ f^{k, \delta}(x) $ of Example 4.1: $ r = 0 $, $ \delta = 10 \% $, $ k = 20 $

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Figure 4. The comparison of the exact solution $ f(x) $ and its regularized solution $ f^{k, \delta}(x) $ of Example 4.1: $ r = 0.5 $, $ \delta = 10 \% $, $ k = 20 $

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Figure 5. The comparison of the exact solution $ f(x) $ and its regularized solution $ f^{k, \delta}(x) $ of Example 4.1: $ r = 1 $, $ \delta = 10 \% $, $ k = 20 $

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Figure 6. Exact and inexact data of Example 4.1: $ \delta = 1 \% $

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Figure 7. The comparison of the exact solution $ f(x) $ and its regularized solution $ f^{k, \delta}(x) $ of Example 4.1: $ r = 0 $, $ \delta = 1 \% $, $ k = 20 $

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Figure 8. The comparison of the exact solution $ f(x) $ and its regularized solution $ f^{k, \delta}(x) $ of Example 4.1: $ r = 0.5 $, $ \delta = 1 \% $, $ k = 20 $

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Figure 9. The comparison of the exact solution $ f(x) $ and its regularized solution $ f^{k, \delta}(x) $ of Example 4.1: $ r = 1 $, $ \delta = 1 \% $, $ k = 20 $

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