Inverse problem of determining diffusion matrix between different structures for time fractional diffusion equation

Feiyang Peng; Yanbin Tang; Feiyang Peng; Yanbin Tang

doi:10.3934/nhm.2024013

Networks and Heterogeneous Media

2024, Volume 19, Issue 1: 291-304. doi: 10.3934/nhm.2024013

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Research article

Inverse problem of determining diffusion matrix between different structures for time fractional diffusion equation

Feiyang Peng ¹,
Yanbin Tang ^{1,2
,
,}

1.
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
2.
Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

Received: 20 December 2023 Revised: 12 March 2024 Accepted: 17 March 2024 Published: 21 March 2024

In this paper we consider some inverse problems of determining the diffusion matrix between different structures for the time fractional diffusion equation featuring a Caputo derivative. We first study an inverse problem of determining the diffusion matrix in the period structure using data from the corresponding homogenized equation, then we investigate an inverse problem of determining the diffusion matrix in the homogenized equation using data from the corresponding period structure of the oscillating equation. Finally, we establish the stability and uniqueness for the first inverse problem, and the asymptotic stability for the second inverse problem.
- time fractional diffusion equation,
- inverse problem,
- homogenized equation,
- periodic structure,
- layer structure
Citation: Feiyang Peng, Yanbin Tang. Inverse problem of determining diffusion matrix between different structures for time fractional diffusion equation[J]. Networks and Heterogeneous Media, 2024, 19(1): 291-304. doi: 10.3934/nhm.2024013

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Abstract

In this paper we consider some inverse problems of determining the diffusion matrix between different structures for the time fractional diffusion equation featuring a Caputo derivative. We first study an inverse problem of determining the diffusion matrix in the period structure using data from the corresponding homogenized equation, then we investigate an inverse problem of determining the diffusion matrix in the homogenized equation using data from the corresponding period structure of the oscillating equation. Finally, we establish the stability and uniqueness for the first inverse problem, and the asymptotic stability for the second inverse problem.

References

[1]	M. A. F. Dos Santos, Analytic approaches of the anomalous diffusion: a review, Chaos Soliton Fract, 124 (2019), 86–96. https://doi.org/10.1016/j.chaos.2019.04.039 doi: 10.1016/j.chaos.2019.04.039
[2]	Y. Zhao, Y. Tang, Critical behavior of a semilinear time fractional diffusion equation with forcing term depending on time and space, Chaos Soliton Fract, 178 (2024), 114309. https://doi.org/10.1016/j.chaos.2023.114309 doi: 10.1016/j.chaos.2023.114309
[3]	A. V. Chechkin, F. Seno, R. Metzler, I. M. Sokolov, Brownian yet non-gaussian diffusion: from superstatistics to subordination of diffusing diffusivities, Phys. Rev. X, 7 (2017), 021002. https://doi.org/10.1103/PhysRevX.7.021002 doi: 10.1103/PhysRevX.7.021002
[4]	Hatano Y, Hatano N, Dispersive transport of ions in column experiments: an explanation of long-tailed profiles, Water Resour Res, 34 (1998), 1027–1033. https://doi.org/10.1029/98WR00214 doi: 10.1029/98WR00214
[5]	Mirko Lukovic, Anomalous diffusion in ecology, (English), Doctoral Thesis of Georg-August University School of Science, Gottingen, 2014.
[6]	S. F. A. Carlos, L. O. Murta, Anomalous diffusion paradigm for image denoising process, The Insight Journal, (2016).
[7]	E. E Adams, L. W Gelhar, Field study of dispersion in a heterogeneous aquifer: 2. Spatial moments analysis, Water Resour Res, 28 (1992), 3293–3307. https://doi.org/10.1029/92WR01757 doi: 10.1029/92WR01757
[8]	R. Metzler, J. Klafter, Boundary value problems for fractional diffusion equations, Physica A, 278 (2000), 107–125. https://doi.org/10.1016/S0378-4371(99)00503-8 doi: 10.1016/S0378-4371(99)00503-8
[9]	H. Ma, Y. Tang, Homogenization of a semilinear elliptic problem in a thin composite domain with an imperfect interface, Math. Methods Appl. Sci., 46 (2023), 19329–19350. https://doi.org/10.1002/mma.9628 doi: 10.1002/mma.9628
[10]	J. Chen, Y. Tang, Homogenization of nonlocal nonlinear $p-$Laplacian equation with variable index and periodic structure, J Math Phys, 64 (2023), 061502. https://doi.org/10.1063/5.0091156 doi: 10.1063/5.0091156
[11]	J. Chen, Y. Tang, Homogenization of nonlinear nonlocal diffusion equation with periodic and stationary structure, Netw. Heterog. Media., 18 (2023), 1118–1177. http://dx.doi.org/10.3934/nhm.2023049 doi: 10.3934/nhm.2023049
[12]	K. Sakamoto, M. Yamamoto, Initial value boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J Math Anal Appl, 382 (2011), 426–447.
[13]	A. Kubica, K. Ryszewska, M. Yamamoto, Time-Fractional Differential Equations: A Theoretical Introduction, Singapore: Springer, 1999.
[14]	J. Hu, G. Li, Homogenization of time-fractional diffusion equations with periodic coefficients, J Comput Phys, 408 (2020), 109231. https://doi.org/10.1016/j.jcp.2020.109231 doi: 10.1016/j.jcp.2020.109231
[15]	A. Kawamoto, M. Machida, M. Yamamoto, Homogenization and inverse problems for fractional diffusion equations, Fract. Calc. Appl. Anal, 26 (2023), 2118–2165. https://doi.org/10.1007/s13540-023-00195-8 doi: 10.1007/s13540-023-00195-8
[16]	R. Gorenflo, Y. Luchko, M. Yamamoto, Time-fractional diffusion equation in the fractional Sobolev spaces, Fract. Calc. Appl. Anal, 18 (2015), 799–820. https://doi.org/10.1515/fca-2015-0048 doi: 10.1515/fca-2015-0048
[17]	R. Gorenflo, A. A. Kilbas, F. Mainardi, S. Rogosin, Mittag-Leffler functions, related topics and applications, Berlin: Springer, 2020.
[18]	Y. Luchko, M. Yamamoto, On the maximum principle for a time-fractional diffusion equation, Fract. Calc. Appl. Anal, 20 (2017), 1131–1145. https://doi.org/10.1515/fca-2017-0060 doi: 10.1515/fca-2017-0060
[19]	Y. Luchko, M. Yamamoto, Maximum principle for the time-fractional PDEs, Volume 2 Fractional Differential Equations, Berlin: De Gruyter, 2019,299–326.
[20]	D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Berlin: Springer-Verlag, 2001.
[21]	X. C. Zheng, H. Wang, Uniquely identifying the variable order of time-fractional partial differential equations on general multi-dimensional domains, Inverse Probl Sci Eng, 29 (2021), 1401–1411. https://doi.org/10.1080/17415977.2020.1849182 doi: 10.1080/17415977.2020.1849182
[22]	J. Cheng, J. Nakagawa, M. Yamamoto, T. Yamazaki, Uniqueness in an inverse problem for a one dimensional fractional diffusion equation, Inverse Probl, 25 (2009), 115002. https://doi.org/10.1088/0266-5611/25/11/115002 doi: 10.1088/0266-5611/25/11/115002

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