Numerical method for fractional advection-diffusion equation with heredity

We propose a method of construction of difference schemes for fractional partial differential equations with delay in time. For the fractional equation with two-sided diffusion, fractional transfer in time, and a functional aftereffect, we construct an imokicit difference scheme. We use the shifted Grünwald–Letnikov formulas for the approximation of fractional derivatives with respect to spatial variables and the $L1$-algorithm for the approximation of fractional derivatives in time. Also we use the piecewise constant interpolation and extrapolation by extending of the discrete prehistory of model in time. The algorithm is a fractional analog of a purely implicit method; on each time step, it is reduced to the solution of linear algebraic systems. We prove the stability of the method and find its order of convergence.