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

Béla J. Szekeres; Ferenc Izsák

Open Mathematics (2015)

  • Volume: 13, Issue: 1, page 553-561
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Béla J. Szekeres, and Ferenc Izsák. "A finite difference method for fractional diffusion equations with Neumann boundary conditions." Open Mathematics 13.1 (2015): 553-561. <http://eudml.org/doc/275973>.

@article{BélaJ2015,
abstract = {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.},
author = {Béla J. Szekeres, Ferenc Izsák},
journal = {Open Mathematics},
keywords = {Fractional order diffusion; Grünwald–Letnikov formula; Non-local derivative; Neumann boundary conditions; Implicit Euler scheme; fractional order Laplacian; matrix transformation method; finite element method; error estimation},
language = {eng},
number = {1},
pages = {553-561},
title = {A finite difference method for fractional diffusion equations with Neumann boundary conditions},
url = {http://eudml.org/doc/275973},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Béla J. Szekeres
AU - Ferenc Izsák
TI - A finite difference method for fractional diffusion equations with Neumann boundary conditions
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - 553
EP - 561
AB - 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.
LA - eng
KW - Fractional order diffusion; Grünwald–Letnikov formula; Non-local derivative; Neumann boundary conditions; Implicit Euler scheme; fractional order Laplacian; matrix transformation method; finite element method; error estimation
UR - http://eudml.org/doc/275973
ER -

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