On Fractional Backward Differential Formulas Methods for Fractional Differential Equations with Delay

  • Mahdi Saedshoar Heris
  • Mohammad Javidi
Original Paper


In this paper, fractional backward differential formulas are presented for the numerical solution of fractional delay differential equations in Caputo sense. We focus on linear equations. Our investigation is focused on stability properties of the numerical methods and we determine stability regions for the fractional delay differential equations. Also we proposed the dependence of the solution on the parameters of the equation and conclude with a numerical treatment and examples based on the adaptation of a fractional backward differential formulas methods to the delay case. Numerical experiments are provided to illustrate potential and limitations of the different methods under investigation.


Fractional backward differential formulas Linear delay differential equations Stability 

Mathematics Subject Classification

34A30 65L06 65L20 



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.


  1. 1.
    Podlubny, I.: Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Math. Sci. Eng. 198 (1999)Google Scholar
  2. 2.
    Bagley, R.L., Calico, R.: Fractional order state equations for the control of viscoelasticallydamped structures. J. Guid. Control Dyn. 14(2), 304–311 (1991)CrossRefGoogle Scholar
  3. 3.
    Marks, R.J., Hall, M.W.: Differintegral interpolation from a bandlimited signals samples. IEEE Trans. Acoust. Speech Signal Process. 29(4), 872–877 (1981)CrossRefzbMATHGoogle Scholar
  4. 4.
    Ichise, M., Nagayanagi, Y., Kojima, T.: An analog simulation of non-integer order transfer functions for analysis of electrode processes. J. Electroanal. Chem. Interfacial Electrochem. 33(2), 253–265 (1971)CrossRefGoogle Scholar
  5. 5.
    Garrappa, R., Popolizio, M.: On accurate product integration rules for linear fractional differential equations. J. Comput. Appl. Math. 235(5), 1085–1097 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Galeone, L., Garrappa, R.: On multistep methods for differential equations of fractional order. Mediterr. J. Math. 3(3–4), 565–580 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Garrappa, R., Moret, I., Popolizio, M.: Solving the time-fractional Schrödinger equation by Krylov projection methods. J. Comput. Phys. 293, 115–134 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Li, C., Chen, A., Ye, J.: Numerical approaches to fractional calculus and fractional ordinary differential equation. J. Comput. Phys. 230(9), 3352–3368 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Garrappa, R.: On linear stability of predictor–corrector algorithms for fractional differential equations. Int. J. Comput. Math. 87(10), 2281–2290 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Diethelm, K., Ford, N.J.: Multi-order fractional differential equations and their numerical solution. Appl. Math. Comput. 154(3), 621–640 (2004)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
  12. 12.
    Driver, R.D.: Ordinary and Delay Differential Equations. Springer, Berlin (2012)zbMATHGoogle Scholar
  13. 13.
    Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, Berlin (2013)zbMATHGoogle Scholar
  14. 14.
    Wang, Z., Huang, X., Shi, G.: Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay. Comput. Math. Appl. 62(3), 1531–1539 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Morgado, M.L., Ford, N.J., Lima, P.: Analysis and numerical methods for fractional differential equations with delay. J. Comput. Appl. Math. 252, 159–168 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Čermák, J., Horníček, J., Kisela, T.: Stability regions for fractional differential systems with a time delay. Commun. Nonlinear Sci. Numer. Simul. 31(1), 108–123 (2016)MathSciNetGoogle Scholar
  17. 17.
    Lazarević, M.P., Spasić, A.M.: Finite-time stability analysis of fractional order time-delay systems: Gronwalls approach. Math. Comput. Modell. 49(3), 475–481 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Heris, M.S., Javidi, M.: On FBDF5 method for delay differential equations of fractional order with periodic and anti-periodic conditions. Mediterr. J. Math. 14(3), 134 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Heris, M.S., Javidi, M.: On fractional backward differential formulas for fractional delay differential equations with periodic and anti-periodic conditions. Appl. Numer. Math. 118, 203–220 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Moghaddam, B.P., Yaghoobi, Sh, Machado, J.A.T.: An extended predictorcorrector algorithm for variable-order fractional delay differential equations. J. Comput. Nonlinear Dyn. 11(6), 061001 (2016)CrossRefGoogle Scholar
  21. 21.
    Zaky, M.A., Machado, J.A.T.: On the formulation and numerical simulation of distributed-order fractional optimal control problems. Commun. Nonlinear Sci. Numer. Simul. 52, 177–189 (2017)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Moghaddam, B.P., Machado, J.A.T., Babaei, A.: A computationally efficient method for tempered fractional differential equations with application. Comput. Appl. Math. 2017, 1–15 (2017). Google Scholar
  23. 23.
    Zaky M.A.: An improved tau method for the multi-dimensional fractional Rayleigh–Stokes problem for a heated generalized second grade fluid. Comput. Math. Appl. (2017).
  24. 24.
    Zaky, M.A.: A Legendre collocation method for distributed-order fractional optimal control problems. Nonlinear Dyn. 91(4), 2667–2681 (2018)CrossRefGoogle Scholar
  25. 25.
    Zaky M.A.: A Legendre spectral quadrature tau method for the multi-term time-fractional diffusion equations. Comput. Appl. Math. 1–14 (2017).
  26. 26.
    Heris, M.S., Javidi, M.: Second order difference approximation for a class of Riesz space fractional advection-dispersion equations with delay (Submitted)Google Scholar
  27. 27.
    Heris, M.S., Javidi, M.: On fractional backward differential formulas method for differential equations of fractional order: applied to fractional order Rikitake system (Submitted)Google Scholar
  28. 28.
    Galeone, L., Garrappa, R.: On multistep methods for differential equations of fractional order. Mediterr. J. Math. 3(3), 565–580 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer (India) Private Ltd., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Mathematical SciencesUniversity of TabrizTabrizIran

Personalised recommendations