# 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

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topBéla 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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