Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia

6 Issues per year

IMPACT FACTOR 2017: 2.865
5-year IMPACT FACTOR: 3.323

CiteScore 2017: 3.06

SCImago Journal Rank (SJR) 2017: 1.967
Source Normalized Impact per Paper (SNIP) 2017: 1.954

Mathematical Citation Quotient (MCQ) 2017: 0.98

See all formats and pricing
More options …

Strong maximum principle for fractional diffusion equations and an application to an inverse source problem

Yikan Liu / William Rundell / Masahiro Yamamoto
Published Online: 2016-08-31 | DOI: https://doi.org/10.1515/fca-2016-0048


The strong maximum principle is a remarkable property of parabolic equations, which is expected to be partly inherited by fractional diffusion equations. Based on the corresponding weak maximum principle, in this paper we establish a strong maximum principle for time-fractional diffusion equations with Caputo derivatives, which is slightly weaker than that for the parabolic case. As a direct application, we give a uniqueness result for a related inverse source problem on the determination of the temporal component of the inhomogeneous term.

MSC 2010: Primary 35R11; Secondary 26A33; 35B50; 35R30

Key Words and Phrases: fractional diffusion equation; Caputo derivative; strong maximum principle; Mittag-Leffler function; inverse source problem; fractional Duhamel's principle


  • [1]

    E.E. Adams, L.W. Gelhar, Field study of dispersion in a heterogeneous aquifer 2. Spatial moments analysis. Water Resources Res. 28, No 12 (1992), 3293–3307.Google Scholar

  • [2]

    R.A. Adams, Sobolev Spaces. Academic Press, New York (1975).Google Scholar

  • [3]

    M. Al-Refai, Y. Luchko, Maximum principles for the fractional diffusion equations with the Riemann-Liouville fractional derivative and their applications. Fract. Calc. Appl. Anal. 17, No 2 (2014), 483–498; ; http://www.degruyter.com/view/j/fca.2014.17.issue-2/issue-files/fca.2014.17.issue-2.xml.CrossrefGoogle Scholar

  • [4]

    J.R. Cannon, S.P. Esteva, An inverse problem for the heat equation. Inverse Problems 2, No 4 (1986), 395–403.Google Scholar

  • [5]

    R. Courant, D. Hilbert, Methods of Mathematical Physics. Interscience, New York (1953).Google Scholar

  • [6]

    L.C. Evans, Partial Differential Equations, 2nd ed. American Mathematical Society, Providence - RI (2010).Google Scholar

  • [7]

    D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001).Google Scholar

  • [8]

    M. Ginoa, S. Cerbelli, H.E. Roman, Fractional diffusion equation and relaxation in complex viscoelastic materials. Phys. A 191, No 1 (1992), 449–453.Google Scholar

  • [9]

    R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications. Springer, Berlin (2014).Google Scholar

  • [10]

    R. Gorenflo, Y. Luchko, M. Yamamoto, Time-fractional diffusion equation in the fractional Sobolev spaces. Fract. Calc. Appl. Anal. 18, No 3 (2015), 799–820; ; http://www.degruyter.com/view/j/fca.2015.18.issue-3/issue-files/fca.2015.18.issue-3.xml.CrossrefGoogle Scholar

  • [11]

    Y. Hatano, N. Hatano, Dispersive transport of ions in column experiments: an explanation of long-tailed profiles. Water Resources Res. 34, No 5 (1998), 1027–1033.Google Scholar

  • [12]

    B. Jin, R. Lazarov, J. Pasciak, Z. Zhou, Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion. IMA J. Numer. Anal. 35, No 2 (2015), 561–582.Google Scholar

  • [13]

    B. Jin, R. Lazarov, Z. Zhou, Error estimates for a semidiscrete finite element method for fractional order parabolic equations. SIAM J. Numer. Anal. 51, No 1 (2013), 445–466.Google Scholar

  • [14]

    B. Jin, W. Rundell, A tutorial on inverse problems for anomalous diffusion processes. Inverse Problems 31, No 3 (2015), 035003.Google Scholar

  • [15]

    F. Liu, P. Zhuang, V. Anh, I. Turner, K. Burrage, Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. Appl. Math. Comput. 191, No 1 (2007), 12–20.Google Scholar

  • [16]

    Y. Luchko, Maximum principle for the generalized time-fractional diffusion equation. J. Math. Anal. Appl. 351, No 1 (2009), 218–223.Google Scholar

  • [17]

    Y. Luchko, Some uniqueness and existence results for the initial-boundary value problems for the generalized time-fractional diffusion equation. Comput. Math. Appl. 59, No 5 (2010), 1766–1772.Google Scholar

  • [18]

    Y. Luchko, R. Gorenflo, An operational method for solving fractional differential equations with the Caputo derivatives. Acta Math. Vietnam 24, No 2 (1999), 207–233.Google Scholar

  • [19]

    Y. Luchko, W. Rundell, M. Yamamoto, L. Zuo, Uniqueness and reconstruction of an unknown semilinear term in a time-fractional reaction-diffusion equation. Inverse Problems 29, No 6 (2013), 065019.Google Scholar

  • [20]

    M.M. Meerschaert, C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, No 1 (2004), 65–77.Google Scholar

  • [21]

    R. Metzler, J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, No 1 (2000), 1–77.Google Scholar

  • [22]

    R.R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry. Phys. Stat. Sol. B 133, No 1 (1986), 425–430.Google Scholar

  • [23]

    I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).Google Scholar

  • [24]

    W. Rundell, X. Xu, L. Zuo, The determination of an unknown boundary condition in a fractional diffusion equation. Appl. Anal. 92, No 7 (2013), 1511–1526.Google Scholar

  • [25]

    S. Saitoh, V.K. Tuan, M. Yamamoto, Reverse convolution inequalities and applications to inverse heat source problems. J. Inequal. Pure Appl. Math. 3, No 5 (2002), 1–11.Google Scholar

  • [26]

    S. Saitoh, V.K. Tuan, M. Yamamoto, Convolution inequalities and applications. J. Inequal. Pure Appl. Math. 4, No 3 (2003), 1–8.Google Scholar

  • [27]

    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, No 1 (2011), 426–447.Google Scholar

  • [28]

    E.C. Titchmarsh, The zeros of certain integral functions. Proc. London Math. Soc. 2, No 1 (1926), 283–302.Google Scholar

About the article

Published Online: 2016-08-31

Published in Print: 2016-08-01

Citation Information: Fractional Calculus and Applied Analysis, Volume 19, Issue 4, Pages 888–906, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.1515/fca-2016-0048.

Export Citation

© 2016 Diogenes Co., Sofia.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

William Rundell and Zhidong Zhang
Journal of Computational Physics, 2018, Volume 368, Page 299
Nguyen Huy Tuan, Tran Bao Ngoc, Salih Tatar, and Le Dinh Long
Journal of Computational and Applied Mathematics, 2018
Chunlong Sun, Gongsheng Li, and Xianzheng Jia
Advances in Mathematical Physics, 2017, Volume 2017, Page 1
Yikan Liu and Zhidong Zhang
Journal of Physics A: Mathematical and Theoretical, 2017, Volume 50, Number 30, Page 305203
Daijun Jiang, Zhiyuan Li, Yikan Liu, and Masahiro Yamamoto
Inverse Problems, 2017, Volume 33, Number 5, Page 055013

Comments (0)

Please log in or register to comment.
Log in