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

In this paper, we investigate the fractional backward differential formulas (FBDF) and Grünwald 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 < α< 1 and the shifted Grünwald difference operators for 1 < β< 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 O (h2+ κ2+ σ2+ ρ2). 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. © 2019, Sociedad Española de Matemática Aplicada.