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A new numerical approximation of the fractal ordinary differential equation

  • Abdon Atangana
  • Sonal JainEmail author
Regular Article
Part of the following topical collections:
  1. Focus Point on Modelling Complex Real-World Problems with Fractal and New Trends of Fractional Differentiation

Abstract.

The concept of fractal medium is present in several real-world problems, for instance, in the geological formation that constitutes the well-known subsurface water called aquifers. However, attention has not been quite devoted to modeling for instance, the flow of a fluid within these media. We deem it important to remind the reader that the concept of fractal derivative is not to represent the fractal sharps but to describe the movement of the fluid within these media. Since this class of ordinary differential equations is highly complex to solve analytically, we present a novel numerical scheme that allows to solve fractal ordinary differential equations. Error analysis of the method is also presented. Application of the method and numerical approximation are presented for fractal order differential equation. The stability and the convergence of the numerical schemes are investigated in detail. Also some exact solutions of fractal order differential equations are presented and finally some numerical simulations are presented.

References

  1. 1.
    F. Ali, N.A. Sheikh, I. Khan, M. Saqib, J. Magn. & Magn. Mater. 423, 327 (2017)CrossRefADSGoogle Scholar
  2. 2.
    F. Ali, M. Saqib, I. Khan, N.A. Sheikh, Eur. Phys. J. Plus 131, 377 (2016)CrossRefGoogle Scholar
  3. 3.
    F. Ali, S.A.A. Jan, I. Khan, M. Gohar, N.A. Sheikh, Eur. Phys. J. Plus 131, 310 (2016)CrossRefGoogle Scholar
  4. 4.
    M. Al-Refai, Y. Luchko, Appl. Math. Comput. 257, 40 (2015)MathSciNetGoogle Scholar
  5. 5.
    A. Atangana, B.S.T. Alkahtani, Entropy 17, 4439 (2015)MathSciNetCrossRefADSGoogle Scholar
  6. 6.
    A. Atangana, B. Dumitru, Therm. Sci. 20, 763 (2016)CrossRefGoogle Scholar
  7. 7.
    A. Atangana, I. Koca, Chaos, Soliton Fractals 89, 447 (2016)CrossRefADSGoogle Scholar
  8. 8.
    A. Atangana, Appl. Math. Comput. 273, 948 (2016)MathSciNetGoogle Scholar
  9. 9.
    M. Caputo, M. Fabrizio, Progr. Fract. Differ. Appl. 1, 73 (2015)Google Scholar
  10. 10.
    E.F. Doungmo Goufo, M.K. Pene, N. Jeanine, Open Math. 13, 839 (2015)MathSciNetGoogle Scholar
  11. 11.
    R. Gnitchogna, A. Atangana, Numer. Methods Part. Differ. Equ. (2017)  https://doi.org/10.1002/num.22216
  12. 12.
    D.W. Brzezinski, Appl. Math. Nonlinear Sci. 1, 23 (2016)CrossRefGoogle Scholar
  13. 13.
    R.B. Gnitchogna, A. Atangana, Int. J. Math. Mod. Methods Appl. Sci. 9, 105 (2015)Google Scholar
  14. 14.
    M.T. Gencoglu, H.M. Baskonus, H. Bulut, AIP Conf. Proc. 1798, 020103 (2017)CrossRefGoogle Scholar
  15. 15.
    S. Kumar, X.B. Yin, D. Kumar, Adv. Mech. Eng. 7, 1 (2015)Google Scholar
  16. 16.
    K.M. Owolabi, Chaos, Solitons Fractals 93, 89 (2016)MathSciNetCrossRefADSGoogle Scholar
  17. 17.
    K.M. Owolabi, Commun. Nonlinear Sci. Numer. Simul. 44, 304 (2017)MathSciNetCrossRefADSGoogle Scholar
  18. 18.
    W. Chen, Chaos, Soliton Fractals 28, 9239 (2016)Google Scholar
  19. 19.
    R. Kanno, Physica A 248, 165 (1998)CrossRefADSGoogle Scholar
  20. 20.
    W. Chen, H.G. Sun, X. Zhang, D. Korosak, Comput. Math. Appl. 59, 1754 (2010)MathSciNetCrossRefGoogle Scholar
  21. 21.
    J.H. Cushman, D.O.’ Malley, M. Park, Phys. Rev. E 79, 032101 (2009)CrossRefADSGoogle Scholar
  22. 22.
    F. Mainardi, A. Mura, G. Pagnini, Int. J. Differ. Equ. 29, 104505 (2010)Google Scholar
  23. 23.
    W. Chen, X.D. Zhang, D. Korosak, Int. J. Nonlinear Sci. Numer. 11, 3 (2010)Google Scholar
  24. 24.
    A. Atangana, Chaos, Soliton Fractals 102, 396 (2017)CrossRefADSGoogle Scholar
  25. 25.
    A. Latif, Fixed Point Theor. Appl. 2009, 170140 (2009)MathSciNetGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute for Groundwater Studies, Faculty for Natural and Agricultural SciencesUniversity of the Free StateBloemfonteinSouth Africa
  2. 2.Department of Mathematics, Faculty of Science & TechnologyThe ICFAI UniversityJaipurIndia

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