Numerical solution of fractional order diffusion problems with Neumann boundary conditions

2014/11/06 by Szekeres, Béla J., Izsák, Ferenc
#35R11 #65M06 #65M12 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1411.1596

Abstract

A finite difference numerical method is investigated for fractional order diffusion problems in one space dimension. For this, a mathematical model is developed to incorporate homogeneous Dirichlet and Neumann type boundary conditions. The models are based on an appropriate extension of the initial values. The well-posedness of the obtained initial value problems is proved and it is pointed out that the extensions are compatible with the above 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.

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