A stability result for the determination of order in time-fractional diffusion equations
-
Zhiyuan Li
,
Xinchi Huang
Abstract
This paper deals with an inverse problem of the determination of the fractional order in time-fractional diffusion equations from one interior point observation. We give a representation of the solution via the Mittag-Leffler function and eigenfunction expansion, from which the Lipschitz stability of the fractional order with respect to the measured data at the interior point is established.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11801326
Award Identifier / Grant number: 11771270
Award Identifier / Grant number: 91730303
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: 15H05740
Funding statement: The first author thanks National Natural Science Foundation of China 11801326. The second and third authors are supported by Grant-in-Aid for Scientific Research (S) 15H05740 of Japan Society for the Promotion of Science. The third author is supported by NSFC (No. 11771270, 91730303) and the ‘RUDN University Program 5-100’. This work was also supported by A3 Foresight Program ‘Modeling and Computation of Applied Inverse Problems’ of Japan Society for the Promotion of Science.
Acknowledgements
The second author thanks the Leading Graduate Course for Frontiers of Mathematical Sciences and Physics (FMSP, the University of Tokyo).
References
[1] D. A. Benson, S. W. Wheatcraft and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resources Res. 36 (2000), no. 6, 1403–1412. 10.1029/2000WR900031Suche in Google Scholar
[2] J. Cheng, J. Nakagawa, M. Yamamoto and T. Yamazaki, Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Problems 25 (2009), no. 11, Article ID 115002. 10.1088/0266-5611/25/11/115002Suche in Google Scholar
[3] Y. Hatano, J. Nakagawa, S. Wang and M. Yamamoto, Determination of order in fractional diffusion equation, J. Math-for-Ind. 5A (2013), 51–57. Suche in Google Scholar
[4] B. Jin and W. Rundell, A tutorial on inverse problems for anomalous diffusion processes, Inverse Problems 31 (2015), no. 3, Article ID 035003. 10.1088/0266-5611/31/3/035003Suche in Google Scholar
[5] Y. Kian, E. Soccorsi and M. Yamamoto, On time-fractional diffusion equations with space-dependent variable order, Ann. Henri Poincaré 19 (2018), no. 12, 3855–3881. 10.1007/s00023-018-0734-ySuche in Google Scholar
[6] M. Levy and B. Berkowitz, Measurement and analysis of non-Fickian dispersion in heterogeneous porous media, J. Contaminant Hydrol. 64 (2003), no. 3, 203–226. 10.1016/S0169-7722(02)00204-8Suche in Google Scholar
[7] G. Li, D. Zhang, X. Jia and M. Yamamoto, Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time-fractional diffusion equation, Inverse Problems 29 (2013), no. 6, Article ID 065014. 10.1088/0266-5611/29/6/065014Suche in Google Scholar
[8] Z. Li, O. Y. Imanuvilov and M. Yamamoto, Uniqueness in inverse boundary value problems for fractional diffusion equations, Inverse Problems 32 (2016), no. 1, Article ID 015004. 10.1088/0266-5611/32/1/015004Suche in Google Scholar
[9] Z. Li, Y. Luchko and M. Yamamoto, Analyticity of solutions to a distributed order time-fractional diffusion equation and its application to an inverse problem, Comput. Math. Appl. 73 (2017), no. 6, 1041–1052. 10.1016/j.camwa.2016.06.030Suche in Google Scholar
[10] Z. Li and M. Yamamoto, Uniqueness for inverse problems of determining orders of multi-term time-fractional derivatives of diffusion equation, Appl. Anal. 94 (2015), no. 3, 570–579. 10.1080/00036811.2014.926335Suche in Google Scholar
[11] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. Suche in Google Scholar
[12] K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl. 382 (2011), no. 1, 426–447. 10.1016/j.jmaa.2011.04.058Suche in Google Scholar
[13] J. L. Schiff, The Laplace Transform. Theory and Applications, Undergrad. Texts Math., Springer, New York, 1999. 10.1007/978-0-387-22757-3Suche in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Identification of the interface between acoustic and elastic waves from internal measurements
- A numerical method for an inverse source problem for parabolic equations and its application to a coefficient inverse problem
- Determination of the impulsive Sturm–Liouville operator from a set of eigenvalues
- The enclosure method for inverse obstacle scattering over a finite time interval: VI. Using shell-type initial data
- Identifying space-time dependent force on the vibrating Euler–Bernoulli beam by a boundary functional method
- A stability result for the determination of order in time-fractional diffusion equations
- Recovery of non-smooth radiative coefficient from nonlocal observation by diffusion system
- An inverse problem of triple-thickness parameters determination for thermal protective clothing with Stephan–Boltzmann interface conditions
- Direct and inverse source problems for degenerate parabolic equations
- Partial inverse problems for quadratic differential pencils on a graph with a loop
Artikel in diesem Heft
- Frontmatter
- Identification of the interface between acoustic and elastic waves from internal measurements
- A numerical method for an inverse source problem for parabolic equations and its application to a coefficient inverse problem
- Determination of the impulsive Sturm–Liouville operator from a set of eigenvalues
- The enclosure method for inverse obstacle scattering over a finite time interval: VI. Using shell-type initial data
- Identifying space-time dependent force on the vibrating Euler–Bernoulli beam by a boundary functional method
- A stability result for the determination of order in time-fractional diffusion equations
- Recovery of non-smooth radiative coefficient from nonlocal observation by diffusion system
- An inverse problem of triple-thickness parameters determination for thermal protective clothing with Stephan–Boltzmann interface conditions
- Direct and inverse source problems for degenerate parabolic equations
- Partial inverse problems for quadratic differential pencils on a graph with a loop
