Numerical Methods for a Class of Fractional Advection-Diffusion Models with Functional Delay

In this paper, we consider a technique of creation of difference schemes for time and space fractional partial differential equations with effect of delay on time. For two sided space fractional diffusion equation and fractional advection equations with time functional after-effect, an implicit numerical method is constructed. We use shifted Grunwald-Letnikov formulae to approximate space fractional derivatives and L1-algorithm to approximate time fractional derivatives. We also use piece-wise constant interpolation and extrapolation by continuation for the prehistory of model with respect to time. The algorithm is a fractional analogue of the pure implicit numerical method in which the model is reduced on each time step to the solution of linear algebraic system. The order of convergence is obtained. Numerical experiments are carried out to support the obtained theoretical results.