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Fractional Calculus and Applied Analysis

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Volume 15, Issue 1


Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation

Yuri Luchko
  • Department of Mathematics, Beuth Technical University of Applied Sciences Berlin, Luxemburger Str. 10, 13353, Berlin, Germany
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Published Online: 2011-12-29 | DOI: https://doi.org/10.2478/s13540-012-0010-7


In this paper, some initial-boundary-value problems for the time-fractional diffusion equation are first considered in open bounded n-dimensional domains. In particular, the maximum principle well-known for the PDEs of elliptic and parabolic types is extended for the time-fractional diffusion equation. In its turn, the maximum principle is used to show the uniqueness of solution to the initial-boundary-value problems for the time-fractional diffusion equation. The generalized solution in the sense of Vladimirov is then constructed in form of a Fourier series with respect to the eigenfunctions of a certain Sturm-Liouville eigenvalue problem. For the onedimensional time-fractional diffusion equation $$(D_t^\alpha u)(t) = \frac{\partial } {{\partial x}}\left( {p(x)\frac{{\partial u}} {{\partial x}}} \right) - q(x)u + F(x,t), x \in (0,l), t \in (0,T)$$ the generalized solution to the initial-boundary-value problem with Dirichlet boundary conditions is shown to be a solution in the classical sense. Properties of this solution are investigated including its smoothness and asymptotics for some special cases of the source function.

MSC: Primary 26A33; Secondary 35B45, 35B50, 45K05

Keywords: Caputo fractional derivative; time-fractional diffusion equation; Mittag-Leffler function; initial-boundary-value problems; maximum principle; generalized solution; spectral method

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About the article

Published Online: 2011-12-29

Published in Print: 2012-03-01

Citation Information: Fractional Calculus and Applied Analysis, Volume 15, Issue 1, Pages 141–160, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.2478/s13540-012-0010-7.

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© 2012 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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