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Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia


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Volume 17, Issue 1

Issues

The space-fractional diffusion-advection equation: Analytical solutions and critical assessment of numerical solutions

Robin Stern
  • Institut für Theoretische Physik IV, Ruhr-Universität Bochum, Universitätsstrasse 150, D-44780, Bochum, Germany
  • Interdisciplinary Centre for Advanced Materials Simulation (ICAMS), Ruhr-Universität Bochum, Universitätsstr. 150, 44780, Bochum, Germany
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/ Frederic Effenberger
  • Institut für Theoretische Physik IV, Ruhr-Universität Bochum, Universitätsstrasse 150, D-44780, Bochum, Germany
  • Department of Mathematics, University of Waikato, P.B. 3105, Hamilton, New Zealand
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  • De Gruyter OnlineGoogle Scholar
/ Horst Fichtner
  • Institut für Theoretische Physik IV, Ruhr-Universität Bochum, Universitätsstrasse 150, D-44780, Bochum, Germany
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/ Tobias Schäfer
Published Online: 2013-12-28 | DOI: https://doi.org/10.2478/s13540-014-0161-9

Abstract

The present work provides a critical assessment of numerical solutions of the space-fractional diffusion-advection equation, which is of high significance for applications in various natural sciences. In view of the fact that, in contrast to the case of normal (Gaussian) diffusion, no standard methods and corresponding numerical codes for anomalous diffusion problems have been established yet, it is of importance to critically assess the accuracy and practicability of existing approaches. Three numerical methods, namely a finite-difference method, the so-called matrix transfer technique, and a Monte-Carlo method based on the solution of stochastic differential equations, are analyzed and compared by applying them to three selected test problems for which analytical or semi-analytical solutions were known or are newly derived. The differences in accuracy and practicability are critically discussed with the result that the use of stochastic differential equations appears to be advantageous.

MSC: 26A33; 35R11; 33C60; 35R60; 60J60

Keywords: space-fractional diffusion-advection equation; anomalous diffusion; Riesz fractional derivative; series representation of analytical solutions; numerical approximations

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About the article

Published Online: 2013-12-28

Published in Print: 2014-03-01


Citation Information: Fractional Calculus and Applied Analysis, Volume 17, Issue 1, Pages 171–190, ISSN (Online) 1314-2224, DOI: https://doi.org/10.2478/s13540-014-0161-9.

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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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