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

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 boundar...

Full description

Saved in:
Bibliographic Details
Published inOpen mathematics (Warsaw, Poland) Vol. 13; no. 1
Main Authors Szekeres, Béla J., Izsák, Ferenc
Format Journal Article
LanguageEnglish
Published Warsaw De Gruyter Open 25.09.2015
De Gruyter Poland
De Gruyter
Subjects
Approximation
Boundary conditions
Boundary value problems
Dirichlet problem
Finite difference method
Fractional order diffusion
Grünwald–Letnikov formula
Implicit Euler scheme
Mathematical analysis
Neumann boundary conditions
Non-local derivative
Numerical methods
Online AccessGet full text

Cover

More Information
Summary: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.
ISSN:2391-5455
2391-5455
DOI:10.1515/math-2015-0056