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

  • Mahdi Saedshoar Heris
  • Mohammad JavidiEmail author
Original Paper


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.


Fractional backward differential formulas Distributed-order equation Riesz fractional derivatives 

Mathematics Subject Classification

Primary 34A30 Secondary 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.


  1. 1.
    Diethelm, K., Freed, A. D.: On the solution of nonlinear fractional-order dierential equations used in the modeling of viscoplasticity. In: Scientic Computing in Chemical Engineering II, pp. 217–224. Springer, Berlin, Heidelberg (1999)Google Scholar
  2. 2.
    Gaul, L., Klein, P., Kemple, S.: Damping description involving fractional operators. Mech. Syst. Signal Process. 5(2), 81–88 (1992)CrossRefGoogle Scholar
  3. 3.
    Glöckle, W.G., Nonnenmacher, T.F.: A fractional calculus approach to self-similar protein dynamics. Biophys. J. 68(1), 46 (1995)CrossRefGoogle Scholar
  4. 4.
    Magin, R.L.: Fractional Calculus in Bioengineering. Begell House, Redding (2006)Google Scholar
  5. 5.
    Garrappa, R.: Trapezoidal methods for fractional differential equations: theoretical and computational aspects. Math. Comput. Simul. 110, 96–112 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17(3), 704–719 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Diethelm, K., Ford, N.J.: Multi-order fractional differential equations and their numerical solution. Appl. Math. Comput. 154(3), 621–640 (2004)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Edwards, T., Ford, N.J., Simpson, A.C.: The numerical solution of linear multi-term fractional differential equations: systems of equations. J. Comput. Appl. Math. 148(2), 401–418 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Gorenflo, R.: Fractional Calculus: Some Numerical Methods, Fractals and Fractional Calculus in Continuum Mechanics, CISM Lecture Notes, pp. 277–290. Springer, Wien (1997)CrossRefGoogle Scholar
  10. 10.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Heris, M.S., Javidi, M.: On fractional backward differential formulas methods for fractional differential equations with delay. Int. J. Appl. Comput. Math. 4(2), 72 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Čermák, J., Kisela, Horníček T.: Stability regions for fractional differential systems with a time delay. Commun. Nonlinear Sci. Numer. Simul. 31(1), 108–123 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lazarević, M.P., Spasić, A.M.: Finite-time stability analysis of fractional order time-delay systems: Gronwall’s approach. Math. Comput. Model. 49(3), 475–481 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Wu, J.: Theory and Applications of Partial Functional Differential Equations, vol. 119. Springer Science and Business Media, Berlin (2012)Google Scholar
  18. 18.
    Tadjeran, C., Meerschaert, M.M., Scheffler, H.P.: A second-order accurate numerical approximation for the fractional diffusion equation. J. Comput. Phys. 213(1), 205–213 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Liu, Q., Zeng, F., Li, C.: Finite difference method for time-space-fractional Schrodinger equation. Int. J. Comput. Math. 92(7), 1439–1451 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Ding, H., Li, C.: Numerical algorithms for the fractional diffusion-wave equation with reaction term. In: Abstract and Applied Analysis (2013)Google Scholar
  21. 21.
    Bagley, R., Torvik, P.: On the existence of the order domain and the solution of distributed order equations-part I. Int. J. Appl. Math. 2(7), 865–882 (2000)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Bagley, R., Torvik, P.: On the existence of the order domain and the solution of distributed order equations-part II. Int. J. Appl. Math. 2(8), 965–988 (2000)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Caputo, M.: Distributed order differential equations modelling dielectric induction and diffusion. Fract. Calc. Appl. Anal. 4(4), 421–442 (2001)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Caputo, M.: Diffusion with space memory modelled with distributed order space fractional differential equations. Ann. Geophys. 46(2) (2003)Google Scholar
  25. 25.
    Hartley, T.T., Lorenzo, C.F.: Fractional system identification: an approach using continuous order-distributions. (1999)Google Scholar
  26. 26.
    Sokolov, I., Chechkin, A., Klafter, J.: Distributed-order fractional kinetics (2004). arXiv:cond-mat/0401146
  27. 27.
    Umarov, S., Gorenflo, R.: Cauchy and nonlocal multi-point problems for distributed order pseudo-differential equations: part one. J. Anal. Appl. 254(3), 449–466 (2005)zbMATHGoogle Scholar
  28. 28.
    Meerschaert, M.M., Nane, E., Vellaisamy, P.: Distributed-order fractional diffusions on bounded domains. J. Math. Anal. Appl. 379(1), 216–228 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Gorenflo, R., Luchko, Y., Stojanović, M.: Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density. Fract. Calc. Appl. Anal. 16(2), 297–316 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Luchko, Y.: Boundary value problems for the generalized time-fractional diffusion equation of distributed order. Fract. Calc. Appl. Anal. 12(4), 409–422 (2009)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Ye, H., Liu, F., Anh, V., Turner, I.: Numerical analysis for the time distributed-order and riesz space fractional diffusions on bounded domains. IMA J. Appl. Math. 80(3), 825–838 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Ye, H., Liu, F.: Compact difference scheme for distributed-order time-fractional diffusion-wave equation on bounded domains. J. Comput. Phys. 298, 652–660 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Gao, G.H., Sun, H.W., Sun, Z.Z.: Some high-order difference schemes for the distributed-order differential equations. J. Comput. Phys. 298, 337–359 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Yang, Q., Liu, F., Turner, I.: Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Appl. Math. Model. 34(1), 200–218 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    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 (1998)Google Scholar
  36. 36.
    Kilbas, A.A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, vol. 24. Elsevier Science Limited, Amsterdam (2006)zbMATHGoogle Scholar
  37. 37.
    Heris, M.S., Javidi, M.: Second order difference approximation for a class of Riesz space fractional advection-dispersion equations with delay (2018). arXiv:1811.10513 [math.NA]
  38. 38.
    Thomas, J.W.: Numerical Partial Differential Equations: Finite Difference Methods, vol. 22. Springer Science and Business Media, Berlin (2013)Google Scholar
  39. 39.
    Varga, R.S.: Geršgorin and his Circles, vol. 36. Springer Science and Business Media, Berlin (2010)zbMATHGoogle Scholar

Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  1. 1.Faculty of Mathematical SciencesUniversity of TabrizTabrizIran

Personalised recommendations