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Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia


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Volume 19, Issue 3

Issues

General time-fractional diffusion equation: some uniqueness and existence results for the initial-boundary-value problems

Yuri Luchko
  • Dept. of Mathematics, Physics, and Chemistry, Beuth Technical University of Applied Sciences, Luxemburger Str. 10, Berlin - 13353, GERMANY
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/ Masahiro Yamamoto
Published Online: 2016-06-28 | DOI: https://doi.org/10.1515/fca-2016-0036

Abstract

In this paper, we deal with the initial-boundary-value problems for a general time-fractional diffusion equation which generalizes the single- and the multi-term time-fractional diffusion equations as well as the time-fractional diffusion equation of the distributed order. First, important estimates for the general time-fractional derivatives of the Riemann-Liouville and the Caputo type of a function at its maximum point are derived. These estimates are applied to prove a weak maximum principle for the general time-fractional diffusion equation. As an application of the maximum principle, the uniqueness of both the strong and the weak solutions to the initial-boundary-value problem for this equation with the Dirichlet boundary conditions is established. Finally, the existence of a suitably defined generalized solution to the the initial-boundary-value problem with the homogeneous boundary conditions is proved.

MSC 2010: Primary 26A33; Secondary 35A05; 35B30; 35B50; 35C05; 35E05; 35L05; 45K05; 60E99

Key Words and Phrases: general fractional derivative; general time-fractional diffusion equation; initial-boundary-value problems; maximum principle; a priori estimates; Fourier method of variables separation; generalized solution

References

  • 1

    Al-Refai M. Luchko Yu. 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 DOI: ;CrossrefGoogle Scholar

  • 2

    Chechkin A.V. Gorenflo R. Sokolov I.M. Fractional diffusion in inhomogeneous media J. Phys. A, Math. Gen. 38 2005 679 684Google Scholar

  • 3

    Chechkin A.V. Gorenflo R. Sokolov I.M. Gonchar V.Yu. Distributed order time fractional diffusion equation Fract. Calc. Appl. Anal. 6 2003 259 279Google Scholar

  • 4

    Chechkin A.V. Gorenflo R. Sokolov I.M. Retarding subdiffusion and accelerating superdiffusion governed by distributed order fractional diffusion equations Phys. Rev. E 66 2002 1 7Google Scholar

  • 5

    Daftardar-Gejji V. Bhalekar S. Boundary value problems for multi-term fractional differential equations J. Math. Anal. Appl. 345 2008 754 765Google Scholar

  • 6

    Feller W. An Introduction to Probability Theory and its Applications Vol. 2 Wiley New York 1966Google Scholar

  • 7

    Jiang H. Liu F. Turner I.W. Burrage K. Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain Computers and Math. with Appl. 64 2012 3377 3388Google Scholar

  • 8

    Kochubei A.N. General fractional calculus, evolution equations, and renewal processes Integr. Equa. Operator Theory 71 2011 583 600Google Scholar

  • 9

    Kochubei A.N. Distributed order calculus and equations of ultraslow diffusion J. Math. Anal. Appl. 340 2008 252 281Google Scholar

  • 10

    Li Z. Liu Y. Yamamoto M. Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients Appl. Math. and Computation 257 2015 381 397Google Scholar

  • 11

    Li Z. Luchko Yu. Yamamoto M. Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations Fract. Calc. Appl. Anal. 17 No 4 2014 1114 1136 DOI: ;CrossrefWeb of ScienceGoogle Scholar

  • 12

    Luchko Yu. Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation Fract. Calc. Appl. Anal. 15 No 1 2012 141 160 DOI: ;CrossrefGoogle Scholar

  • 13

    Luchko Yu. Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation J. Math. Anal. Appl. 374 2011 538 548Google Scholar

  • 14

    Luchko Yu. 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 1772Google Scholar

  • 15

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

  • 16

    Luchko Yu. Boundary value problems for the generalized time-fractional diffusion equation of distributed order Fract. Calc. Appl. Anal. 12 2009 409 422Google Scholar

  • 17

    Luchko Yu. Operational method in fractional calculus Fract. Calc. Appl. Anal. 2 1999 463 489Google Scholar

  • 18

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

  • 19

    Meerschaert M.M. Scheffler H.-P. Stochastic model for ultraslow diffusion Stochastic Process. Appl. 116 2006 1215 1235Google Scholar

  • 20

    Metzler R. Jeon J.-H. Cherstvy A. G. Barkai E. Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking Phys. Chem. Chem. Phys. 16 201424128Web of ScienceGoogle Scholar

  • 21

    Naber M. Distributed order fractional subdiffusion Fractals 12 2004 23 32Google Scholar

  • 22

    Protter M.H. Weinberger H.F. Maximum Principles in Differential Equations Springer Berlin 1999Google Scholar

  • 23

    Sakamoto K. Yamamoto M. 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 447Google Scholar

  • 24

    Samko S.G. Kilbas A.A. Marichev O.I. Fractional Integrals and Derivatives: Theory and Applications Gordon and Breach Yverdon 1993Google Scholar

  • 25

    Schilling R.L. Song R. Vondracek Z. Bernstein Functions. Theory and Application De Gruyter Berlin 2010Google Scholar

  • 26

    Schneider W.R. Wyss W. Fractional diffusion and wave equations J. Math. Phys. 30 1989 134 144Google Scholar

  • 27

    Sokolov I.M. Chechkin A.V. Klafter J. Distributed-order fractional kinetics Acta Phys. Polon. B 35 2004 1323 1341Google Scholar

  • 28

    Suzuki A. Niibori Y. Fomin S.A. Chugunov V.A. Hashida T. Prediction of reinejction effects in fault-related subsidiary structures by using fractional derivative-based mathematical models for sustainable design of geothermal reservoirs Geothermics 57 2015 196 204Google Scholar

  • 29

    Suzuki A. Niibori Y. Fomin S.A. Chugunov V.A. Hashida T. Analysis of water injection in fractured reservoirs using a fractional-derivative-based mass and heat transfer model Mathematical Geosciences 47 2014 31 49Google Scholar

  • 30

    Umarov S. Gorenflo R. Cauchy and nonlocal multi-point problems for distributed order pseudo-differential equations Z. Anal. Anwend. 24 2005 449 466Google Scholar

  • 31

    Vladimirov V.S. Equations of Mathematical Physics Nauka Moscow 1971Google Scholar

  • 32

    Walter W. On the strong maximum principle for parabolic differential equations Proc. Edinb. Math. Soc. 29 1986 93 96Google Scholar

About the article

Received: 2016-01-30

Revised: 2016-03-01

Published Online: 2016-06-28

Published in Print: 2016-06-01


Citation Information: Fractional Calculus and Applied Analysis, Volume 19, Issue 3, Pages 676–695, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.1515/fca-2016-0036.

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