Few-Body Systems

, 58:156 | Cite as

Stable Numerical Approach for Fractional Delay Differential Equations

  • Harendra Singh
  • Rajesh K. Pandey
  • D. Baleanu


In this paper, we present a new stable numerical approach based on the operational matrix of integration of Jacobi polynomials for solving fractional delay differential equations (FDDEs). The operational matrix approach converts the FDDE into a system of linear equations, and hence the numerical solution is obtained by solving the linear system. The error analysis of the proposed method is also established. Further, a comparative study of the approximate solutions is provided for the test examples of the FDDE by varying the values of the parameters in the Jacobi polynomials. As in special case, the Jacobi polynomials reduce to the well-known polynomials such as (1) Legendre polynomial, (2) Chebyshev polynomial of second kind, (3) Chebyshev polynomial of third and (4) Chebyshev polynomial of fourth kind respectively. Maximum absolute error and root mean square error are calculated for the illustrated examples and presented in form of tables for the comparison purpose. Numerical stability of the presented method with respect to all four kind of polynomials are discussed. Further, the obtained numerical results are compared with some known methods from the literature and it is observed that obtained results from the proposed method is better than these methods.


  1. 1.
    H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences (Springer, New York, 2011)CrossRefzbMATHGoogle Scholar
  2. 2.
    V. Lakshmikantham, S. Leela, Differential and Integral Inequalities (Academic Press, New York, 1969)zbMATHGoogle Scholar
  3. 3.
    W.G. Aiello, H.I. Freedman, A time-delay model of single-species growth with stage structure. Math. Biosci. 101, 139 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    A.R. Davis, A. Karageorghis, T.N. Phillips, Spectral Galerkin methods for the primary two-point bour problem in modelling viscoelastic flows. Int. J. Numer. Methods Eng. 26, 647 (1988)CrossRefzbMATHGoogle Scholar
  5. 5.
    S.A. Gourley, Y. Kuang, A stage structured predator-prey model and its dependence on maturation delay and death rate. J. Math. Biol. 49, 188 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    J. Li, Y. Kuang, C. Mason, Modelling the glucose-insulin regulatory system and ultradian insulin secretory oscillations with two time delays. J. Theor. Biol. 242, 722 (2006)CrossRefGoogle Scholar
  7. 7.
    A.D. Robinson, The use of control systems analysis in neurophysiology of eye movements. Ann. Rev. Neurosci. 4, 462 (1981)CrossRefGoogle Scholar
  8. 8.
    R.L. Bagley, P.J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27, 201 (1983)ADSCrossRefzbMATHGoogle Scholar
  9. 9.
    R.L. Bagley, P.J. Torvik, Fractional calculus a differential approach to the analysis of viscoelasticity damped structures. AIAA J. 21, 741 (1983)ADSCrossRefzbMATHGoogle Scholar
  10. 10.
    R.L. Bagley, P.J. Torvik, Fractional calculus in the transient analysis of viscoelasticity damped structures. AIAA J. 23, 918 (1985)ADSCrossRefzbMATHGoogle Scholar
  11. 11.
    R. Panda, M. Dash, Fractional generalized splines and signal processing. Signal Process 86, 2340 (2006)CrossRefzbMATHGoogle Scholar
  12. 12.
    R.L. Magin, Fractional calculus in bioengineering. Crit. Rev. Biomed. Eng. 32, 1 (2004)CrossRefGoogle Scholar
  13. 13.
    G.W. Bohannan, Analog fractional order controller in temperature and motor control applications. J. Vib. Control 14, 1487 (2008)MathSciNetCrossRefGoogle Scholar
  14. 14.
    C.E. Falbo, Some Elementary Methods for Solving Functional Differential Equations.
  15. 15.
    X. Lv, Y. Gao, The RKHSM for solving neutral functional differential equations with proportional delays. Math. Methods Appl. Sci. 36, 642 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    W. Wang, Y. Zhang, S. Li, Stability of continuous Runge–Kutta type methods for nonlinear neutral delay-differential equations. Appl. Math. Model. Simul. Comput. Eng. Environ. Syst. 33(8), 3319 (2009)MathSciNetzbMATHGoogle Scholar
  17. 17.
    U. Saeed, M. ur Rehman, Hermite wavelet method for fractional delay differential equations. J. Differ. Equ. Article ID 359093, 8 (2014)Google Scholar
  18. 18.
    M.A. Iqbal, A. Ali, S.T. Mohyud-Din, Chebyshev wavelets method for fractional delay differential equations. Int. J. Mod. Appl. Phys. 4(1), 49 (2013)Google Scholar
  19. 19.
    D.J. Evans, K.R. Raslan, The Adomian decomposition method for solving delay differential equation. Int. J. Comput. Math. 82(1), 49 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    B.P. Mohaddam, Z.S. Mostaghim, A numerical method based on finite difference for solving fractional delay differential equations. J. Taibah Univ. Sci. 7, 120 (2013)CrossRefGoogle Scholar
  21. 21.
    Z. Wang, A numerical method for delayed fractional-order differential equations. J. Appl. Math. Article ID 256071, 7 (2013)Google Scholar
  22. 22.
    Z. Wang, X. Huang, J. Zhou, A numerical method for delayed fractional-order differential equations: based on GL definition. Appl. Math. 7(2), 525 (2013)MathSciNetGoogle Scholar
  23. 23.
    R.K. Pandey, N. Kumar, N. Mohaptra, An approximate method for solving fractional delay differential equations. Int. J. Appl. Comput. Math. (2016). doi: 10.1007/s40819-016-0186-3 Google Scholar
  24. 24.
    M.A. Iqbal, U. Saeed, S.T. Mohyud-Din, Modified Laguerre wavelets method for delay differential equation of fractional-order. Egypt. J. Basic Appl. Sci. 2, 50 (2015)CrossRefGoogle Scholar
  25. 25.
    E.H. Doha, A.H. Bhrawy, D. Baleanu, S.S. Ezz-Eldien, The operational matrix formulation of the Jacobi tau approximation for space fractional diffusion equation. Adv. Differ. Equ. (2014).
  26. 26.
    A. Ahmadian, M. Suleiman, S. Salahshour, D. Baleanu, A Jacobi operational matrix for solving a fuzzy linear fractional differential equation. Adv. Differ. Equ. (2014).
  27. 27.
    A.H. Bhrawy, M.M. Tharwat, M.A. Alghamdi, A new operational matrix of fractional integration for shifted Jacobi polynomials. Bull. Malays. Math. Sci. Soc. (2) 37(4), 983 (2014)MathSciNetzbMATHGoogle Scholar
  28. 28.
    S. Kazem, An integral operational matrix based on Jacobi polynomials for solving fractional-order differential equations. Appl. Math. Model. 37, 1126 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    R.K. Pandey, S. Suman, K.K. Singh, O.P. Singh, Approximate solution of Abel inversion using Chebshev polynomials. Appl. Math. Comput. 237, 120 (2014)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria 2017

Authors and Affiliations

  • Harendra Singh
    • 1
  • Rajesh K. Pandey
    • 1
  • D. Baleanu
    • 1
    • 2
    • 3
  1. 1.Department of Mathematical Sciences, Indian Institute of TechnologyBanaras Hindu UniversityVaranasiIndia
  2. 2.Department of MathematicsCankaya UniversityAnkaraTurkey
  3. 3.Institute of Space SciencesMăgurele, BucharestRomania

Personalised recommendations