Advertisement

Numerical analysis for the fractional diffusion and fractional Buckmaster equation by the two-step Laplace Adam-Bashforth method

  • 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.

In this paper, we aim to use the alternative numerical scheme given by Gnitchogna and Atangana for solving partial differential equations with integer and non-integer differential operators. We applied this method to fractional diffusion model and fractional Buckmaster models with non-local fading memory. The method yields a powerful numerical algorithm for fractional order derivative to implement. Also we present in detail the stability analysis of the numerical method for solving the diffusion equation. This proof shows that this method is very stable and also converges very quickly to exact solution and finally some numerical simulation is presented.

References

  1. 1.
    A. Atangana, Appl. Math. Comput. 273, 948 (2016)MathSciNetGoogle Scholar
  2. 2.
    M. Caputo, M. Fabrizio, Progr. Fract. Differ. Appl. 1, 73 (2015)Google Scholar
  3. 3.
    E.F. Doungmo Goufo, M.K. Pene, N. Jeanine, Open Math. 13, 839 (2015)MathSciNetGoogle Scholar
  4. 4.
    J.F. Gómez-Aguilar, H. Ypez-Martnez, C. Caldern-Ramn, I. Cruz-Ordua, R.F. Escobar-Jimnez, R.F. Escobar-Jimnez, Entropy 17, 6289 (2015)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    J. Losada, J.J. Nieto, Progr. Fract. Differ. Appl. 1, 87 (2015)Google Scholar
  6. 6.
    A. Atangana, B. Dumitru, Therm. Sci. 20, 763 (2016)CrossRefGoogle Scholar
  7. 7.
    R. Gnitchogna, A. Atangana, Numer. Methods Partial Differ. Equ. (2017)  https://doi.org/10.1002/num.22216
  8. 8.
    D.W. Brzezinski, Appl. Math. Nonlinear Sci. 1, 23 (2016)CrossRefGoogle Scholar
  9. 9.
    R.B. Gnitchogna, A. Atangana, Int. J. Math. Models Methods Appl. Sci. 9, 105 (2015)Google Scholar
  10. 10.
    J. Jiang, D. Cao, H. Chen, Appl. Math. Nonlinear Sci. 1, 11 (2016)CrossRefGoogle Scholar
  11. 11.
    H. Jordan, Therm. Sci. 20, 757 (2016)CrossRefGoogle Scholar
  12. 12.
    K.M. Owolabi, Chaos, Solitons Fractals 93, 89 (2016)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    K.M. Owolabi, Commun. Nonlinear Sci. Numer. Simul. 44, 304 (2017)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    K.M. Owolabi, Numer. Methods Partial Differ. Equ. 34, 274 (2018)CrossRefGoogle Scholar
  15. 15.
    S. Kumar, X.B. Yin, D. Kumar, Adv. Mech. Eng. (2015)  https://doi.org/10.1177/1687814015620330
  16. 16.
    S. Kumar, Appl. Math. Model. 38, 3154 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    A. Atangana, K.M. Owolabi, Chaos, Solitons Fractals 105, 111 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    K.M. Owolabi, Chaos, Solitons Fractals 103, 544 (2017)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    K.M. Owolabi, A. Atangana, Eur. Phys. J. Plus 131, 335 (2017)CrossRefGoogle Scholar
  20. 20.
    K.M. Owolabi, A. Atangana, Adv. Differ. Equ. 2017, 223 (2017)CrossRefGoogle Scholar
  21. 21.
    A. Atangana, I. Koca, Chaos, Solitons Fractals 89, 447 (2016)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    A. Atangana, K.M. Owolabi, New numerical approach for fractional differential equations, arXiv:1707.08177 (2017)Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Science & TechnologyThe ICFAI UniversityJaipurIndia

Personalised recommendations