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Analytical and Numerical Solutions of Riesz Space Fractional Advection-Dispersion Equations with Delay

Mahdi Saedshoar Heris1, Mohammad Javidi1, Bashir Ahmad2,*
1 Department of Applied Mathematics, University of Tabriz, Tabriz, Iran.
2 Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia.
∗ Corresponding Author: Bashir Ahmad. Email:

Computer Modeling in Engineering & Sciences 2019, 121(1), 249-272.


In this paper, we propose numerical methods for the Riesz space fractional advection-dispersion equations with delay (RFADED). We utilize the fractional backward differential formulas method of second order (FBDF2) and weighted shifted Grünwald difference (WSGD) operators to approximate the Riesz fractional derivative and present the finite difference method for the RFADED. Firstly, the FBDF2 and the shifted Grünwald methods are introduced. Secondly, based on the FBDF2 method and the WSGD operators, the finite difference method is applied to the problem. We also show that our numerical schemes are conditionally stable and convergent with the accuracy of O(k+ h2) and O(k2 + h2) respectively. Thirdly we find the analytical solution for RFDED in terms Mittag- Leffler type functions. Finally, some numerical examples are given to show the efficacy of the numerical methods and the results are found to be in complete agreement with the analytical solution.


Riesz fractional derivative, shifted Grünwald difference operators, fractional advection-dispersion equation, delay differential equations, FBDF method

Cite This Article

Heris, M. S., Javidi, M., Ahmad, B. (2019). Analytical and Numerical Solutions of Riesz Space Fractional Advection-Dispersion Equations with Delay. CMES-Computer Modeling in Engineering & Sciences, 121(1), 249–272.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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