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New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models

  • Mekkaoui Toufik
  • Abdon AtanganaEmail author
Regular Article

Abstract.

Recently a new concept of fractional differentiation with non-local and non-singular kernel was introduced in order to extend the limitations of the conventional Riemann-Liouville and Caputo fractional derivatives. A new numerical scheme has been developed, in this paper, for the newly established fractional differentiation. We present in general the error analysis. The new numerical scheme was applied to solve linear and non-linear fractional differential equations. We do not need a predictor-corrector to have an efficient algorithm, in this method. The comparison of approximate and exact solutions leaves no doubt believing that, the new numerical scheme is very efficient and converges toward exact solution very rapidly.

References

  1. 1.
    Jürg Nievergelt, Commun. ACM. 7, 731 (1964)CrossRefGoogle Scholar
  2. 2.
    Markus Kunze, Tassilo Kupper, Non-smooth dynamical systems: An overview, in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, edited by Bernold Fiedler (Springer Science & Business Media, 2001) p. 431Google Scholar
  3. 3.
    Thao Dang, Model-Based Testing of Hybrid Systems, in Model-Based Testing for Embedded Systems, edited by Justyna Zander, Ina Schieferdecker, Pieter J. Mosterman (CRC Press, 2011) p. 411Google Scholar
  4. 4.
    Ernst Hairer, Syvert Paul Nørsett, Gerhard Wanner, Solving ordinary differential equations I: Nonstiff problems, second edition (Springer Verlag, Berlin, 1993)Google Scholar
  5. 5.
    John Denholm Lambert, Numerical Methods for Ordinary Differential Systems (John Wiley & Sons, Chichester, 1991)Google Scholar
  6. 6.
    W. Kahan, A survey of error-analysis, in Proceedings of IFIP Congress 71, edited by C.V. Freiman, John E. Griffith, J.L. Rosenfeld (North-Holland, Amsterdam, 1972) pp. 1214--1239Google Scholar
  7. 7.
    Michele Caputo, Geophys. J. Int. 13, 529 (1967)CrossRefGoogle Scholar
  8. 8.
    Obaid Jefain Julaighim Algahtani, Chaos, Solitons Fractals 89, 552 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Badr Saad T. Alkahtani, Chaos, Solitons Fractals 89, 547 (2016)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    A. Atangana, B. Dumitru, Therm. Sci. 20, 763 (2016)CrossRefGoogle Scholar
  11. 11.
    A. Atangana, I. Koca, Chaos, Solitons Fractals 89, 447 (2016)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    G.G. Dahlquist, BIT Numer. Math. 3, 27 (1963)CrossRefGoogle Scholar
  13. 13.
    W.E. Milne, Numerical integration of ordinary differential equations, in American Mathematical Monthly, Mathematical Association of America, Vol. 33 (1926) pp. 455--460Google Scholar
  14. 14.
    Germund Dahlquist, Math. Scandin. 4, 33 (1956)CrossRefGoogle Scholar
  15. 15.
    Florian A. Potra, Stephen J. Wright, J. Comput. Appl. Math. 124, 281 (2000)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Wimol San-Um, Banlue Srisuchinwong, J. Comput. 7, 1041 (2012)Google Scholar
  17. 17.
    Tao Yang, Leon O. Chua, J. Bifurc. Chaos 10, 2015 (2015)Google Scholar
  18. 18.
    V. Carmona, E. Freire, E. Ponce, F. Torres, IEEE Trans. Circ. Syst. I 49, 609 (2002)CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of ScienceUniversité Moulay IsmailZitoune MeknèsMorocco
  2. 2.Institute for groundwater studied, Faculty of Natural and Agricultural ScienceUniversity of Free StateBloemfonteinSouth Africa

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