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Fractional Calculus and Applied Analysis

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Volume 19, Issue 3


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


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


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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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