publication . Article . Preprint . 2014

Uniqueness for inverse problems of determining orders of multi-term time-fractional derivatives of diffusion equation

Li, Zhiyuan; Yamamoto, Masahiro;
Open Access
  • Published: 07 Mar 2014 Journal: Applicable Analysis, volume 94, pages 570-579 (issn: 0003-6811, eissn: 1563-504X, Copyright policy)
  • Publisher: Informa UK Limited
Abstract
This article proves the uniqueness for two kinds of inverse problems of identifying fractional orders in diffusion equations with multiple time-fractional derivatives by pointwise observation. By means of eigenfunction expansion and Laplace transform, we reduce the uniqueness for our inverse problems to the uniqueness of expansions of some special function and complete the proof.
Subjects
free text keywords: Applied Mathematics, Analysis, Eigenfunction, Mathematical optimization, Mathematics, Term (time), Inverse problem, Fractional calculus, Laplace transform, Diffusion equation, Uniqueness, Pointwise, Mathematical analysis, Mathematics - Analysis of PDEs
17 references, page 1 of 2

[1] E. E. Adams and L. W. Gelhar, Field study of dispersion in a heterogeneous aquifer 2: spatial moments analysis Water Resources Res. 28(1992), 32933307. [OpenAIRE]

[2] S. Beckers, M. Yamamoto, Regularity and unique existence of solution to linear diffusion equation with multiple time-fractional derivatives, in: K. Bredies, C. Clason, K. Kunisch, G. von Winckel (Eds.), Control and Optimization with PDE Constraints, Birkh¨auser, Basel, 2013, pp. 45-56.

[3] M. Cheng, J. Nakagawa, M. Yamamoto and T. Yamazaki, Uniqueness in an inverse problem for a one dimensional fractional diffusion equation. Inverse Problems 25 (2009) 115002.

[4] R. Courant, D. Hilbert, Methods of Mathematical Physics, Wiley-VCH, 1989.

[5] K. Diethelm and Y. Luchko, Numerical solution of linear multi-term initial value problems of fractional order. J. Comput. Anal. Appl. 6 (2004), 243-263. [OpenAIRE]

[6] Y. Hatno, N. Hatano, Dispersive transport of ions in column experiments: an explanation of long-tailed profiles, Water Resources Res. 34(1980) 1027-1033.

[7] Y. Hatano, Junichi Nakagawa, Shengzhang Wang, M. Yamamoto, Determination of order in fractional diffusion equation. J. Math-for-Ind. 5A (2013), 5157.

[8] Gongsheng Li, Dali Zhang, Xianzheng Jia, 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, 065014, 36 pp.

[9] Zhiyuan Li, M. Yamamoto, Initial-boundary value problems for linear diffusion equation with multiple time-fractional derivatives, arXiv:1306.2778v2, 2013.

[10] Zhiyuan Li, Yikan Liu and M. Yamamoto, Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients, arXiv: 1312.2112, 2013. [OpenAIRE]

[11] Y. Luchko, Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation. Computers and Mathematics with Applications 59 (2010), 1766-1772. [OpenAIRE]

[12] Y. Luchko, Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation. J. Math. Anal. Appl. 374 (2011), 538-548. [OpenAIRE]

[13] Y. Luchko, R. Gorenflo, An operational method for solving fractional differential equations with the Caputo derivatives, Acta Math. Vietnam, 24 (1999) 207-233.

[14] F. Mainardi, Antonio Mura, Gianni Pagnini and R. Gorenflo, Time-fractional diffusion of distributed order. J. Vib. Control 14 (2008), no. 9-10, 12671290.

[15] R. Metzler, J, Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Physics Reports 339 (2000), 1-77.

17 references, page 1 of 2
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publication . Article . Preprint . 2014

Uniqueness for inverse problems of determining orders of multi-term time-fractional derivatives of diffusion equation

Li, Zhiyuan; Yamamoto, Masahiro;