Received: 22 July 2019 Revised: 27 November 2019 Accepted: 21 December 2019 DOI: 10.1002/cmm4.1084 RESEARCH ARTICLE Stability and convergence of difference methods for two-dimensional Riesz space fractional advection-dispersion equations with delay Mahdi Saedshoar Heris 1 Mohammad Javidi 1 Bashir Ahmad 2 1 Department of Applied Mathematics, University of Tabriz, Tabriz, Iran 2 Nonlinear Analysis and Applied Mathematics Research Group, King Abdulaziz University, Jeddah, Saudi Arabia Correspondence Bashir Ahmad, Nonlinear Analysis and Applied Mathematics Research Group, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. Email: bashirahmad_qau@yahoo.com In this article, the Riesz space fractional advection-dispersion equations with delay in two-dimensional (RFADED in 2D) are considered. The Riesz space fractional derivative is approximated with the aid of backward differential for- mulas method of second order and shifted Grünwald difference operators. We develop the Crank-Nicolson scheme using the finite difference method for the RFADED in 2D and show that it is conditionally stable and convergent with the accuracy order O( 2 + h 2 + k 2 ). Finally, some numerical examples are constructed to demonstrate the efficacy and usefulness of the numerical method. KEYWORDS fractional advection-dispersion equation with delay, fractional backward differential formulas method, Riesz fractional derivative, stability and convergence 1 INTRODUCTION The tools of fractional calculus have been successfully applied to model a wide range of real world problems in physics, biology, engineering, economics, and finance. 1-5 In consequence, fractional differential equations (FDEs) are gaining significant importance and have been analyzed by several researchers, for example, see References 6-10. Some of the latest works related with the high-order numerical methods for FDEs have been worked in References 11-13. The concept of time delay may be related with the duration of certain hidden processes like the time between infec- tion of a cell and the production of new viruses. In fact, the evolution of a delay differential system is more complex than the classical one as it relies on its current time as well on its past stages. For some recent works on fractional differential equations with delay, we refer the reader to a series of articles 14-20 and the references cited therein. It is worth mentioning that partial differential equations with delay have useful applications in biology, control systems, medicine, population ecology, and so on. 21 Fractional advection-dispersion equation plays an important role in the field of groundwater hydrology as it is used to model the transport of passive tracers carried by fluid flow in a porous medium. 22-24 In this article, we investigate the following Riesz space fractional advection-dispersion equations with delay in two-dimensional (RFADED in 2D): u(x, y, t) t = [-K  (-Δ)  2 - K  (-Δ)  2 ]u(x, y, t)+ u(x, y, t -  )+ f (x, y, t), (1) Comp and Math Methods. 2020;2:e1084. wileyonlinelibrary.com/journal/cmm4 © 2020 John Wiley & Sons, Ltd. 1 of 19 https://doi.org/10.1002/cmm4.1084