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Fractional Backward Differential Formulas for the Distributed-Order Differential Equation with Time Delay

  • Mahdi Saedshoar Heris
  • Mohammad Javidi
Original Paper
  • 8 Downloads

Abstract

In this paper, we investigate the fractional backward differential formulas for the distributed-order differential equation with time delay. The mid-point 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 and in time by using the Crank–Nicolson scheme. We prove that proposed scheme is stable and convergent for \(\beta < \frac{5}{8}\) with the accuracy \(\mathrm{O}({h^2} + {\kappa ^2} + {\sigma ^2})\). Finally, we give some examples to show the effectiveness of the numerical method and the results are in excellent agreement with the theoretical analysis.

Keywords

Fractional backward differential formulas Distributed-order equation Riesz fractional derivatives 

Mathematics Subject Classification

Primary 34A30 Secondary 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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Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  1. 1.Faculty of Mathematical SciencesUniversity of TabrizTabrizIran

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