Advertisement

International Journal of Dynamics and Control

, Volume 7, Issue 4, pp 1350–1357 | Cite as

A numerical technique for variable-order fractional functional nonlinear dynamic systems

  • F. Khane Keshi
  • B. P. MoghaddamEmail author
  • A. Aghili
Article

Abstract

This paper provides an efficient technique to discretize the variable-order fractional functional nonlinear differential equations. The proposed technique is based on a piecewise integro quadratic spline interpolation and finite difference approximation. To reveal the performance and accuracy of the proposed method, the behavioral responses of the suitcase with two wheels and pantograph models with variable-order fractional order are analyzed.

Keywords

Fractional calculus Functional differential equation Finite difference Integro quadratic spline interpolation 

Mathematics Subject Classification

26A33 34A08 34K28 34K37 65L12 

References

  1. 1.
    David S, Fischer C, Machado JT (2018) Fractional electronic circuit simulation of a nonlinear macroeconomic model. AEU-Int J Electron Commun 84:210–220.  https://doi.org/10.1016/j.aeue.2017.11.019 CrossRefGoogle Scholar
  2. 2.
    Mostaghim ZS, Moghaddam BP, Haghgozar HS (2018) Numerical simulation of fractional-order dynamical systems in noisy environments. Comput Appl Math 133:1–15.  https://doi.org/10.1007/s40314-018-0698-z MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Coimbra C (2003) Mechanics with variable-order differential operators. Ann der Phys 12(1112):692–703.  https://doi.org/10.1002/andp.200310032 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dabiri A, Butcher EA, Nazari M (2017) Coefficient of restitution in fractional viscoelastic compliant impacts using fractional Chebyshev collocation. J Sound Vib 388:230–244.  https://doi.org/10.1016/j.jsv.2016.10.013 CrossRefGoogle Scholar
  5. 5.
    Caponetto R, Dongola G, Fortuna L, Gallo A (2010) New results on the synthesis of FO-PID controllers. Commun Nonlinear Sci Numer Simul 15(4):997–1007.  https://doi.org/10.1016/j.cnsns.2009.05.040 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Caponetto R, Dongola G, Fortuna L, Petráš I (2010) Fractional order systems. World Scientific Publishing Co. Pte. Ltd., Singapore.  https://doi.org/10.1142/9789814304207 CrossRefzbMATHGoogle Scholar
  7. 7.
    Machado JAT, Moghaddam BP (2018) A robust algorithm for nonlinear variable-order fractional control systems with delay. Int J Nonlinear Sci Numer Simul 19(3–4):231–238.  https://doi.org/10.1515/ijnsns-2016-0094 CrossRefGoogle Scholar
  8. 8.
    Moghaddam BP, Mostaghim ZS (2014) A novel matrix approach to fractional finite difference for solving models based on nonlinear fractional delay differential equations. Ain Shams Eng J 5(2):585–594.  https://doi.org/10.1016/j.asej.2013.11.007 CrossRefGoogle Scholar
  9. 9.
    Hilfer R (2000) Applications of fractional calculus in physics. World Scientific, SingaporeCrossRefGoogle Scholar
  10. 10.
    Moghaddam BP, Machado JAT, Babaei A (2017) A computationally efficient method for tempered fractional differential equations with application. Comput Appl Math 37(3):3657–3671.  https://doi.org/10.1007/s40314-017-0522-1 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bhrawy AH, Zaky MA (2014) Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation. Nonlinear Dyn 80(1–2):101–116.  https://doi.org/10.1007/s11071-014-1854-7 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dabiri A, Nazari M, Butcher EA (2016) The spectral parameter estimation method for parameter identification of linear fractional order systems. In: American Control Conference (ACC), IEEE, pp 2772–2777.  https://doi.org/10.1109/acc.2016.7525338
  13. 13.
    Bhrawy A, Zaky M (2017) An improved collocation method for multi-dimensional space–time variable-order fractional Schrödinger equations. Appl Numer Math 111:197–218.  https://doi.org/10.1016/j.apnum.2016.09.009 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dehghan R (2018) A numerical solution of variable order fractional functional differential equation based on the shifted Legendre polynomials. SeMA J.  https://doi.org/10.1007/s40324-018-0173-1 CrossRefGoogle Scholar
  15. 15.
    Chen C-M, Liu F, Anh V, Turner I (2012) Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation. Math Comput 81(277):345–366.  https://doi.org/10.1090/s0025-5718-2011-02447-6 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Sun H, Chen W, Li C, Chen Y (2012) Finite difference schemes for variable-order time fractional diffusion equation. Int J Bifurc Chaos 22(04):1250085.  https://doi.org/10.1142/s021812741250085x MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Moghaddam BP, Machado JAT (2016) Extended algorithms for approximating variable order fractional derivatives with applications. J Sci Comput 71(3):1351–1374.  https://doi.org/10.1007/s10915-016-0343-1 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mostaghim ZS, Moghaddam BP, Haghgozar HS (2018) Computational technique for simulating variable-order fractional Heston model with application in US stock market. Math Sci.  https://doi.org/10.1007/s40096-018-0267-z MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Zhang H, Liu F, Phanikumar MS, Meerschaert MM (2013) A novel numerical method for the time variable fractional order mobile–immobile advection–dispersion model. Comput Math Appl 66(5):693–701.  https://doi.org/10.1016/j.camwa.2013.01.031 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Shen S, Liu F, Anh V, Turner I, Chen J (2013) A characteristic difference method for the variable-order fractional advection–diffusion equation. J Appl Math Comput 42(1–2):371–386.  https://doi.org/10.1007/s12190-012-0642-0 MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Yaghoobi S, Moghaddam BP, Ivaz K (2016) An efficient cubic spline approximation for variable-order fractional differential equations with time delay. Nonlinear Dyn 87(2):815–826.  https://doi.org/10.1007/s11071-016-3079-4 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Keshi FK, Moghaddam BP, Aghili A (2018) A numerical approach for solving a class of variable-order fractional functional integral equations. Comput Appl Math 37(4):4821–4834.  https://doi.org/10.1007/s40314-018-0604-8 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Moghaddam BP, Machado JAT (2017) SM-algorithms for approximating the variable-order fractional derivative of high order. Fundam Inform 151:293–311.  https://doi.org/10.3233/FI-2017-1493 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Moghaddam BP, Machado JAT, Behforooz H (2017) An integro quadratic spline approach for a class of variable-order fractional initial value problems. Chaos, Solitons Fractals 102:354–360.  https://doi.org/10.1016/j.chaos.2017.03.065 MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Bhrawy AH, Zaky MA (2016) Numerical algorithm for the variable-order Caputo fractional functional differential equation. Nonlinear Dyn 85(3):1815–1823.  https://doi.org/10.1007/s11071-016-2797-y MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Li X, Wu B (2015) A numerical technique for variable fractional functional boundary value problems. Appl Math Lett 43:108–113.  https://doi.org/10.1016/j.aml.2014.12.012 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Li X, Li H, Wu B (2017) A new numerical method for variable order fractional functional differential equations. Appl Math Lett 68:80–86.  https://doi.org/10.1016/j.aml.2017.01.001 MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Jia Y-T, Xu M-Q, Lin Y-Z (2017) A numerical solution for variable order fractional functional differential equation. Appl Math Lett 64:125–130.  https://doi.org/10.1016/j.aml.2016.08.018 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Samko S (2012) Fractional integration and differentiation of variable order: an overview. Nonlinear Dyn 71(4):653–662.  https://doi.org/10.1007/s11071-012-0485-0 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wu J, Zhang X (2015) Integro quadratic spline interpolation. Appl Math Model 39(10–11):2973–2980.  https://doi.org/10.1016/j.apm.2014.11.015 MathSciNetCrossRefGoogle Scholar
  31. 31.
    Plaut RH (1996) Rocking instability of a pulled suitcase with two wheels. Acta Mech 117(1–4):165–179.  https://doi.org/10.1007/bf01181045 CrossRefzbMATHGoogle Scholar
  32. 32.
    Suherman S, Plaut R, Watson L, Thompson S (1997) Effect of human response time on rocking instability of a two-wheeled suitcase. J Sound Vib 207(5):617–625.  https://doi.org/10.1006/jsvi.1997.1141 CrossRefGoogle Scholar
  33. 33.
    Horvath HZ, Takacs D (2018) Modelling and simulation of rocking suitcases. Acta Polytech CTU Proc 18:61.  https://doi.org/10.14311/app.2018.18.0061 CrossRefGoogle Scholar
  34. 34.
    Ockendon JR, Tayler AB (1971) The dynamics of a current collection system for an electric locomotive. Proc R Soc A: Math, Phys Eng Sci 322(1551):447–468.  https://doi.org/10.1098/rspa.1971.0078 CrossRefGoogle Scholar
  35. 35.
    Dehghan M, Shakeri F (2008) The use of the decomposition procedure of Adomian for solving a delay differential equation arising in electrodynamics. Phys Scr 78(6):065004.  https://doi.org/10.1088/0031-8949/78/06/065004 CrossRefzbMATHGoogle Scholar
  36. 36.
    Ambartsumian VA (1944) On the fluctuation of the brightness of the Milky way, Doklady Akad. Nauk USSR 44:223–226Google Scholar
  37. 37.
    Gaver DP (1964) An absorption probability problem. J Math Anal Appl 9(3):384–393.  https://doi.org/10.1016/0022-247x(64)90024-1 MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Ahmad I, Mukhtar A (2015) Stochastic approach for the solution of multi-pantograph differential equation arising in cell-growth model. Appl Math Comput 261:360–372.  https://doi.org/10.1016/j.amc.2015.04.001 MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Yaghoobi S, Moghaddam BP, Ivaz K (2018) A numerical approach for variable-order fractional unified chaotic systems with time-delay. Comput Methods Differ Equ 6(4):396–410MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Lahijan BranchIslamic Azad UniversityLahijanIran
  2. 2.Department of Applied Mathematics, Faculty of Mathematical SciencesUniversity of GuilanRashtIran

Personalised recommendations