A stability result for the determination of order in time-fractional diffusion equations

  • 1 School of Mathematics and Statistics, Shandong University of Technology, Zibo, Shandong, P. R. China
  • 2 Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, Japan
  • 3 Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, Japan
Zhiyuan LiORCID iD: https://orcid.org/0000-0002-5961-7211, Xinchi Huang
  • Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan
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and Masahiro YamamotoORCID iD: https://orcid.org/0000-0002-4050-871X
  • Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan, and Honorary Member of Academy of Romanian Scientists, Splaiul Independentei Street, no 54, 050094 Bucharest Romania; and Peoples’ Friendship University of Russia (RUDN University) 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation
  • orcid.org/0000-0002-4050-871X
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar

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.

  • [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.

    • Crossref
    • Export Citation
  • [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.

  • [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.

  • [4]

    B. Jin and W. Rundell, A tutorial on inverse problems for anomalous diffusion processes, Inverse Problems 31 (2015), no. 3, Article ID 035003.

  • [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.

    • Crossref
    • Export Citation
  • [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.

    • Crossref
    • Export Citation
  • [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.

  • [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.

  • [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.

    • Crossref
    • Export Citation
  • [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.

    • Crossref
    • Export Citation
  • [11]

    I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.

  • [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.

    • Crossref
    • Export Citation
  • [13]

    J. L. Schiff, The Laplace Transform. Theory and Applications, Undergrad. Texts Math., Springer, New York, 1999.

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