Numerical solution of fractional order diffusion problems with Neumann boundary conditions

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 Gr"unwald–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.