ESTONIAN ACADEMY
PUBLISHERS
eesti teaduste
akadeemia kirjastus
PUBLISHED
SINCE 1952
 
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proceedings
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)
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Inverse problem to identify a space-dependent diffusivity coefficient in a generalized subdiffusion equation from final data; pp. 3–15
PDF | 10.3176/proc.2022.1.01

Authors
Jaan Janno, Kairi Kasemets, Nataliia Kinash
Abstract

An inverse problem to determine a space-dependent diffusivity coefficient in a one-dimensional generalized time fractional diffusion equation from final data is considered. The global uniqueness and local existence and stability of the solution to this problem is proved. Proof of these statements is based on the fixed-point principle and previously obtained results regarding an inverse source problem for a generalized subdiffusion equation.

References

1. Baeumer, B., Kurita, S. and Meerschaert, M. M. Inhomogeneous fractional diffusion equations. Fract. Calc. Appl. Anal., 2005, 8(4), 371–386. 

2. Chechkin, A. V., Gorenflo, R. and Sokolov, I. M. Fractional diffusion in inhomogeneous media. J. Phys. A: Math. Gen., 2005, 38(42), 679–684.
https://doi.org/10.1088/0305-4470/38/42/L03

3. Gajda, J. and Magdziarz, M. Fractional Fokker–Planck equation with tempered α-stable waiting times: Langevin picture and computer simulation. Phys. Rev. E., 2010, 82(1), 011117. 
https://doi.org/10.1103/PhysRevE.82.011117

4. Gripenberg, G. On Volterra equations of the first kind. Integral Equ. Oper. Theory, 1980, 3(4), 473–488.
https://doi.org/10.1007/BF01702311

5. Gripenberg, G., Londen, S.-O. and Staffans, O. J. Volterra Integral and Functional Equations. Cambridge University Press, Cambridge, 1990.
https://doi.org/10.1017/CBO9780511662805

6. Janno, J. and Kasemets, K. Uniqueness for an inverse problem for a semilinear time-fractional diffusion equation. Inverse Probl. Imaging, 2017, 11(1), 125–149.
https://doi.org/10.3934/ipi.2017007

7. Kian, Y., Oksanen, L., Soccorsi, E. and Yamamoto, M. Global uniqueness in an inverse problem for time fractional diffusion equations. J. Differ. Equ., 2018, 264(2), 1146–1170.
https://doi.org/10.1016/j.jde.2017.09.032

8. Kinash, N. and Janno, J. Inverse problems for a perturbed time fractional diffusion equation with final overdetermination. Math. Methods Appl. Sci., 2018, 41(5), 1925–1943.
https://doi.org/10.1002/mma.4719

9. Kinash, N. and Janno, J. Inverse problems for a generalized subdiffusion equation with final overdetermination. Math. Model. Anal., 2019, 24(2), 236–262.
https://doi.org/10.3846/mma.2019.016

10. Kinash, N. Inverse problems for generalized subdiffusion equations. PhD Thesis. Tallinn University of Technology, Estonia, 2020. 
https://digikogu.taltech.ee/et/Item/a3776907-fc6f-42e1-9d7c-5a0e6ae314f6

11. Kirane, M., Samet, B. and Torebek, B. T. Determination of an unknown source term temperature distribution for the sub-diffusion equation at the initial and final data. Electron. J. Differ. Equ., 2017, 217, 1–13.

12. Mainardi, F., Mura, A., Pagnini, G. and Gorenflo, R. Time-fractional diffusion of distributed order. J. Vib. Control, 2008, 14(9–10), 1267–1290.
https://doi.org/10.1177/1077546307087452

13. Metzler, R. and Klafter, J. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep., 2000, 339(1), 1–77.
https://doi.org/10.1016/S0370-1573(00)00070-3

14. Orlovsky, D. Parameter determination in a differential equation of fractional order with Riemann - Liouville fractional derivative in a Hilbert space. J. Sib. Fed. Univ. Math. Phys., 2015, 8(1), 55–63.
https://doi.org/10.17516/1997-1397-2015-8-1-55-63

15. Orlovsky, D. and Piskarev, S. Inverse problem with final overdetermination for time-fractional differential equation in a Banach space. J. Inverse Ill-Posed Probl., to appear.

16. Povstenko, Y. Z. Fractional heat conduction and associated thermal stress. J. Therm. Stresses, 2004, 28(1), 83–102.
https://doi.org/10.1080/014957390523741

17. Ren, C. and Xu, X. Local stability for an inverse coefficient problem of a fractional diffusion equation. Chin. Ann. Math. Ser. B, 2014, 35, 429–446.
https://doi.org/10.1007/s11401-014-0833-0

18. Sakamoto, K. and Yamamoto, M. Inverse source problem with a final overdetermination for a fractional diffusion equation. Math. Control Relat. Fields, 2011, 1(4), 509–518.
https://doi.org/10.3934/mcrf.2011.1.509

19. Samko, S. G. and Cardoso, R. P. Integral equations of the first kind of Sonine type. Int. J. Math. Math. Sci., 2003, 2003, 238394. 20. Sandev, T., Metzler, R. and Chechkin, A. From continuous time random walks to the generalized diffusion equation. Fract. Calc. Appl. Anal., 2018, 21(1), 10–28.
https://doi.org/10.1515/fca-2018-0002

21. Tuan, N. and Dinh, L. Fourier truncation method for an inverse source problem for space-time fractional diffusion equation. Electron. J. Differ. Equ., 2017, 2017(122), 1–16.

22. Wu, X., Deng, W. and Barkai, E. Tempered fractional Feynman–Kac equation: Theory and examples. Phys. Rev. E, 2016, 93(3), 032151.
https://doi.org/10.1103/PhysRevE.93.032151

23. Yamamoto, M. and Zhang, Y. Conditional stability in determining a zeroth-order coefficient in a half-order fractional diffusion equation by a Carleman estimate. Inverse Probl., 2012, 28(10), 105010. 
https://doi.org/10.1088/0266-5611/28/10/105010
 

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