Journal of Mathematical Sciences

, Volume 230, Issue 5, pp 737–741

# Numerical Method for Fractional Advection-Diffusion Equation with Heredity

• V. G. Pimenov
Article

## Abstract

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 implicit 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 piecewise constant interpolation and extrapolation by extending the discrete prehistory of the 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.

## Keywords and phrases

equation with fractional derivatives functional delay mesh scheme interpolation extrapolation order of convergence

65N12

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