Analysis and numerical methods for the Riesz space distributed-order advection-diffusion equation with time delay

Abstract

In this paper, we investigate the fractional backward differential formulas (FBDF) and Grunwald difference method for the Riesz space distributed-order advection-diffusion equation with delay. The midpoint quadrature rule is used to approximate the distributed-order equation by a multi-term fractional form. Next the transformed multi-term fractional equation is solved by discretizing in space by the fractional backward differential formulas method for $$0<\alpha <1$$ and the shifted Grunwald difference operators for $$1< \beta < 2$$ to approximate the Riesz space fractional derivative and in time by using the Crank-Nicolson scheme. We prove that the Crank-Nicolson scheme is conditionally stable and convergent with second-order accuracy $$\mathrm{O}\left( {h^2} + {\kappa ^2} + {\sigma ^2}+ {\rho ^2}\right) $$ . Finally, we give some examples and compare the results of our method with two works. This results show the effectiveness of the proposed numerical method.