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 well-
posedness 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.
