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Fractional Calculus and Applied Analysis

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Volume 17, Issue 2


Maximum principle for the fractional diffusion equations with the Riemann-Liouville fractional derivative and its applications

Mohammed Al-Refai / Yuri Luchko
  • Department of Mathematics, Beuth Technical University of Applied Sciences Berlin, Luxemburger Str. 10, D - 13353, Berlin, Germany
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Published Online: 2014-03-21 | DOI: https://doi.org/10.2478/s13540-014-0181-5


In this paper, the initial-boundary-value problems for the one-dimensional linear and non-linear fractional diffusion equations with the Riemann-Liouville time-fractional derivative are analyzed. First, a weak and a strong maximum principles for solutions of the linear problems are derived. These principles are employed to show uniqueness of solutions of the initial-boundary-value problems for the non-linear fractional diffusion equations under some standard assumptions posed on the non-linear part of the equations. In the linear case and under some additional conditions, these solutions can be represented in form of the Fourier series with respect to the eigenfunctions of the corresponding Sturm-Liouville eigenvalue problems.

MSC: Primary 26A33; Secondary 33E12, 35S10, 45K05

Keywords: Riemann-Liouville fractional derivative; extremum principle for the Riemann-Liouville fractional derivative; maximum principle; linear and non-linear time-fractional diffusion equations; uniqueness and existence of solutions

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About the article

Published Online: 2014-03-21

Published in Print: 2014-06-01

Citation Information: Fractional Calculus and Applied Analysis, Volume 17, Issue 2, Pages 483–498, ISSN (Online) 1314-2224, DOI: https://doi.org/10.2478/s13540-014-0181-5.

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© 2014 Diogenes Co., Sofia. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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