An inverse source time-fractional diffusion problem via an input-output mapping

Authors

  • Rahima Atmania University of Badji Mokhtar Annaba.
  • Loubna Settara University of 20 august 1955.

DOI:

https://doi.org/10.22199/issn.0717-6279-4401

Keywords:

inverse source problem, time-fractional diffusion equation, Riemann-Liouville fractional derivative, input-output mapping, distinguishability

Abstract

In this paper, we investigate an inverse source problem involving a one-dimensional diffusion equation of a time-fractional RiemannLiouville derivative with 0 < α < 1. First, results on the existence and regularity of the weak solution of the direct problem are obtained. For the determination of the unknown time-dependent source term, we use a monotone and distinguishable input-output mapping defined by the additional over-determination integral data for the considered sub-diffusion problem. Finally, the uniqueness of the solution of the inverse problem is proved.

Author Biographies

Rahima Atmania, University of Badji Mokhtar Annaba.

Applied Mathematics Laboratory, Department of Mathematics, Faculty of Sciences.

Loubna Settara, University of 20 august 1955.

Lamahis Laboratory, Departement of Mathematics.

References

J. R. Cannon and Y. Lin, “Determination of source parameter in parabolic equations”, Meccanica, vol. 27, pp. 85-94, 1992. https://doi.org/10.1007/BF00420586

J. Cheng, J. Nakagawa, M. Yamamoto and T. Yamazaki, “Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation”, Inverse Problems, vol. 25, 2009. ID 115002. https://doi.org/10.1088/0266-5611/25/11/115002

M. Choulli and M. Yamamoto, “Some stability estimates in determining sources and coefficients”, Journal of Inverse and Ill-Posed Problems, vol. 14, no. 4, pp. 355-373, 2006. https://doi.org/10.1515/156939406777570996

A. Erdem, D. Lesnic and A. Hasanov, “Identification of a spacewise dependent heat source”, Applied Mathematical Modelling, vol. 37, pp. 10231-10244, 2013. https://doi.org/10.1016/j.apm.2013.06.006

R. Gorenflo, F. Mainardi, D. Moretti and P. Paradisi, “Time fractional diffusion: a discrete random walk approach”, Nonlinear Dynamics, vol. 29, pp. 129-143, 2002. https://doi.org/10.1023/A:1016547232119

A. Hasanov, A. Demir and A. Erdem, “Monotonicity of input—output mappings in inverse coefficient and source problems for parabolic equations”, Journal of Mathematical Analysis and Applications, vol. 335, pp. 1434-1451, 2007. https://doi.org/10.1016/j.jmaa.2007.01.097

A. Hazanee, D. Lesnic, M. I. Ismailov and N. B. Kerimov, “An inverse timedependent source problem for the heat equation with a non-classical boundary condition”, Applied Mathematical Modelling, vol. 39, no. 20, pp. 6258-6272, 2015. https://doi.org/10.1016/j.apm.2015.01.058

R. Hilfer, Application of fractional in physics. Singapore: World scientific publishing company, 2000.

V. Isakov, Inverse Problems for Partial Differential Equations. New York: Springer, 1998.

B. Jin and W. Rundell, “An inverse problem for a one-dimensional time-fractional diffusion problem”, Inverse Problems, vol. 28, 2012. ID 075010. https://doi.org/10.1088/0266-5611/28/7/075010

B. Kaltenbacher and W. Rundell, “On an inverse potential problem for a fractional reaction-diffusion equation”, Inverse Problems, vol. 35, 2019, ID 065004. https://doi.org/10.1088/1361-6420/ab109e

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier, 2006.

A. A. Kilbas, J. J. Trujillo and A. A. Voroshilov, “Cauchy type problem for diffusion- wave equations with the Riemann-Liouville derivative”, Fractional Calculus and Applied Analysis, vol. 8, no. 4, pp. 403-430, 2005.

Z. Li, Y. Liu and M. Yamamoto, “Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients”, Applied Mathematics and Computation, vol. 257, pp. 381-397, 2015. https://doi.org/10.1016/j.amc.2014.11.073

J. J. Liu and M. Yamamoto, “A backward problem for the time-fractional diffusion equation”, Applicable Analysis, vol. 89, pp. 1769-1788, 2010. https://doi.org/10.1080/00036810903479731

Yu. Luchko, “Some uniqueness and existence results for the initial boundary-value problems for the generalized time-fractional diffusion equation”, Computers & Mathematics with Applications, vol. 59, pp. 1766-1772, 2010. https://doi.org/10.1016/j.camwa.2009.08.015

J. Nakagawa, K. Sakamoto and M. Yamamoto, “Overview to mathematical analysis for fractional diffusion equations-new mathematical aspects motivated by industrial collaboration”, Journal of Math for Industry, vol. 2, pp. 99-108, 2010.

E. Ozbilge and A. Demir, “Inverse problem for a time-fractional parabolic equation”, Journal of Inequalities and Applications, vol. 81, pp. 1-9, 2015. https://doi.org/10.1186/s13660-015-0602-y

B. Ozbilge, A. Demir, F. Kanca and E. Ozbilge, “Determination of the unknown source function in time fractional parabolic equation with Dirichlet boundary conditions”, Applied Mathematics & Information Sciences, vol. 10, no. 1, pp. 283-289, 2016. https://doi.org/10.18576/amis/100129

W. Rundell, X. Xu and L. Zuo, “The determination of an unknown boundary condition in a fractional diffusion equation”, Applicable Analysis, vol. 92, no. 15, pp. 11-26, 2013. https://doi.org/10.1080/00036811.2012.686605

K. Sakamoto and M. Yamamoto, “Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems”, Journal of Mathematical Analysis and Applications, vol. 382, pp. 426-447, 2011. https://doi.org/10.1016/j.jmaa.2011.04.058

K. Sakamoto and M. Yamamoto, “Inverse source problem with a final overdetermination for a fractional diffusion equation”, Mathematical Control and Related Fields, vol. 1, no. 4, pp. 509-518, 2011. https://doi.org/10.3934/mcrf.2011.1.509

S. G. Samko, A. A. Kilbas and D. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, 1993.

W.R. Schneider, Fractional diffusion. In: R. Lima, L. Streit, R. Vilela Mendes (eds) Dynamics and Stochastic Processes Theory and Applications. Lecture Notes in Physics, vol 355. Berlin: Springer, 1990. https://doi.org/10.1007/3-540-52347-2_37

L. Settara and R. Atmania, “An inverse coefficient-source problem for a time-fractional diffusion equation”, International Journal of Applied Mathematics and Statistics, vol. 57, no. 3, pp. 68-78, 2018.

S. Umarov, “On fractional Duhamels principle and its applications”, J. D. Equations, vol. 252, pp. 5217-5234, 2012. https://doi.org/10.48550/arXiv.1004.2098

S. Wang, M. Zhang and X. Li, “Radial anomalous diffusion in an annulus”, Physica A: Statistical Mechanics and its Applications, vol. 390, pp. 3397-3403, 2011. https://doi.org/10.1016/j.physa.2011.05.022

Y. Zhang and X. Xiang, “Inverse source problem for a fractional diffusion equation”, Inverse Problems, vol. 27, 2011, ID 035010. https://doi.org/10.1088/0266-5611/27/3/035010

Published

2023-09-13

How to Cite

[1]
R. Atmania and L. Settara, “An inverse source time-fractional diffusion problem via an input-output mapping”, Proyecciones (Antofagasta, On line), vol. 42, no. 5, pp. 1105-1127, Sep. 2023.

Issue

Vol. 42 No. 5 (2023)

Section

Artículos

Copyright (c) 2023 Rahima Atmania, Loubna Settara

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

  • No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.