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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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IMPACT FACTOR 2016 (Open Mathematics): 0.682
IMPACT FACTOR 2016 (Central European Journal of Mathematics): 0.489

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Volume 13, Issue 1 (Sep 2015)


A finite difference method for fractional diffusion equations with Neumann boundary conditions

Béla J. Szekeres
  • Department of Applied Analysis and Computational Mathematics, Eötvös Loránd University, H-1117, Budapest, Pázmány P. s. 1/C, Hungary
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Ferenc Izsák
  • Department of Applied Analysis and Computational Mathematics, Eötvös Loránd University, H-1117, Budapest, Pázmány P. s. 1/C, Hungary
  • MTA-ELTE NumNet Research Group, Eötvös Loránd University, 1117 Budapest, Pázmány P. stny. 1C, Hungary
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2015-09-25 | DOI: https://doi.org/10.1515/math-2015-0056


A finite difference numerical method is investigated for fractional order diffusion problems in one space dimension. The basis of the mathematical model and the numerical approximation is an appropriate extension of the initial values, which incorporates homogeneous Dirichlet or Neumann type boundary conditions. The wellposedness of the obtained initial value problem is proved and it is pointed out that each extension is compatible with the original boundary conditions. Accordingly, a finite difference scheme is constructed for the Neumann problem using the shifted Grünwald–Letnikov approximation of the fractional order derivatives, which is based on infinite many basis points. The corresponding matrix is expressed in a closed form and the convergence of an appropriate implicit Euler scheme is proved.

Keywords: Fractional order diffusion; Grünwald–Letnikov formula; Non-local derivative; Neumann boundary conditions; Implicit Euler scheme


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About the article

Received: 2015-03-05

Accepted: 2015-05-15

Published Online: 2015-09-25

Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2015-0056.

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©2015 Béla J. Szekeres and Ferenc Izsák. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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