SeMA Journal

, Volume 76, Issue 4, pp 533–551 | Cite as

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

  • Mohammad Javidi
  • Mahdi Saedshoar HerisEmail author


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 \(0<\alpha <1\) and the shifted Grünwald 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.


Fractional backward differential formulas Distributed-order equation Delay Riesz fractional derivatives Stable and convergent 

Mathematics Subject Classification

34A30 35R11 65L06 65L20 65N06 



The authors would like to express special thanks to the referees for carefully reading, constructive comments and valuable remarks which significantly improved the quality of this paper.


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Copyright information

© Sociedad Española de Matemática Aplicada 2019

Authors and Affiliations

  1. 1.Faculty of Mathematical SciencesUniversity of TabrizTabrizIran

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