Convergence of the matrix transformation method for the finite difference approximation of fractional order diffusion problems

Béla J. Szekeres; Ferenc Izsák

Applications of Mathematics (2017)

  • Volume: 62, Issue: 1, page 15-36
  • ISSN: 0862-7940

Abstract

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Numerical solution of fractional order diffusion problems with homogeneous Dirichlet boundary conditions is investigated on a square domain. An appropriate extension is applied to have a well-posed problem on ℝ 2 and the solution on the square is regarded as a localization. For the numerical approximation a finite difference method is applied combined with the matrix transformation method. Here the discrete fractional Laplacian is approximated with a matrix power instead of computing the complicated approximations of fractional order derivatives. The spatial convergence of this method is proved and demonstrated by some numerical experiments.

How to cite

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Szekeres, Béla J., and Izsák, Ferenc. "Convergence of the matrix transformation method for the finite difference approximation of fractional order diffusion problems." Applications of Mathematics 62.1 (2017): 15-36. <http://eudml.org/doc/287590>.

@article{Szekeres2017,
abstract = {Numerical solution of fractional order diffusion problems with homogeneous Dirichlet boundary conditions is investigated on a square domain. An appropriate extension is applied to have a well-posed problem on $\mathbb \{R\}^2$ and the solution on the square is regarded as a localization. For the numerical approximation a finite difference method is applied combined with the matrix transformation method. Here the discrete fractional Laplacian is approximated with a matrix power instead of computing the complicated approximations of fractional order derivatives. The spatial convergence of this method is proved and demonstrated by some numerical experiments.},
author = {Szekeres, Béla J., Izsák, Ferenc},
journal = {Applications of Mathematics},
keywords = {fractional diffusion problem; finite differences; matrix transformation method},
language = {eng},
number = {1},
pages = {15-36},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Convergence of the matrix transformation method for the finite difference approximation of fractional order diffusion problems},
url = {http://eudml.org/doc/287590},
volume = {62},
year = {2017},
}

TY - JOUR
AU - Szekeres, Béla J.
AU - Izsák, Ferenc
TI - Convergence of the matrix transformation method for the finite difference approximation of fractional order diffusion problems
JO - Applications of Mathematics
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 1
SP - 15
EP - 36
AB - Numerical solution of fractional order diffusion problems with homogeneous Dirichlet boundary conditions is investigated on a square domain. An appropriate extension is applied to have a well-posed problem on $\mathbb {R}^2$ and the solution on the square is regarded as a localization. For the numerical approximation a finite difference method is applied combined with the matrix transformation method. Here the discrete fractional Laplacian is approximated with a matrix power instead of computing the complicated approximations of fractional order derivatives. The spatial convergence of this method is proved and demonstrated by some numerical experiments.
LA - eng
KW - fractional diffusion problem; finite differences; matrix transformation method
UR - http://eudml.org/doc/287590
ER -

References

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