SeMA Journal

, Volume 76, Issue 4, pp 533–551

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

• Mahdi Saedshoar Heris
Article

## Abstract

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.

## Keywords

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

## Mathematics Subject Classification

34A30 35R11 65L06 65L20 65N06

## Notes

### Acknowledgements

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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