Analytical solutions and numerical schemes of certain generalized fractional diffusion models

  • Ndolane SeneEmail author
Regular Article


We give the analytical solutions of the fractional diffusion equations in one-, and two-dimensional space described by the Caputo left generalized fractional derivative. We introduce the forward Euler method for fractional diffusion equations represented by the Caputo left generalized fractional derivative. The contribution of this paper is to evaluate the impact of the second parameter of the Caputo left generalized fractional derivative in the behavior of the analytical solutions of the fractional diffusion equations, and to propose a numerical method for the generalized fractional diffusion equations. We will present the difference existing between the classical diffusion equation, the fractional diffusion equation described by Caputo fractional derivative and the fractional diffusion equation expressed by the Caputo left generalized fractional derivative. The Fourier-sine-Laplace-transform method is used to determine the analytical solutions of the fractional diffusion equations described by the Caputo left generalized fractional derivative. Some particular cases of diffusion equations are discussed, and the numerical simulations of their analytical solutions are presented and analyzed.


  1. 1.
    Enrique Nadal, Emmanuelle Abisset-Chavanne, Elias Cueto, Francisco Chinesta, C. R. Méc. 346, 581 (2018)CrossRefGoogle Scholar
  2. 2.
    M.M. Khader, Commun. Nonlinear Sci. Numer. Simul. 16, 2535 (2011)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    J.Y. Yang, J.F. Huang, D.M. Liang, Y.F. Tang, Appl. Math. Model. 38, 3652 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Boris Baeumer, Mihály Kovács, Mark M. Meerschaert, Harish Sankaranarayanan, J. Comput. Appl. Math. 339, 414 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    N.H. Sweilam, A.M. Nagy, Adel A. El-Sayed, J. King Saud Univ. Sci. 28, 41 (2016)CrossRefGoogle Scholar
  6. 6.
    Charles Tadjeran, Mark M. Meerschaert, Hans-Peter Scheffler, J. Comput. Phys. 213, 205 (2006)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Aysegul Cetinkaya, Onur Kiymaz, Math. Comput. Model. 57, 2349 (2013)CrossRefGoogle Scholar
  8. 8.
    Mojtaba Hajipour, Amin Jajarmi, Dumitru Baleanu, HongGuang Sun, Commun. Nonlinear Sci. Numer. Simul. 69, 119 (2019)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Igor Podlubny, J. Comput. Phys. 228, 3137 (2009)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Mohammed Al-Refai, Thabet Abdeljawad, Adv. Differ. Equ. 2017, 315 (2017)CrossRefGoogle Scholar
  11. 11.
    Ndolane Sene, Chaos, Solitons Fractals 117, 68 (2018)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Fahd Jarad, Thabet Abdeljawad, Results Nonlinear Anal. 2018, 88 (2018)Google Scholar
  13. 13.
    Dumitru Baleanu, Guo-Cheng Wu, Sheng-Da Zeng, Chaos, Solitons Fractals 102, 99 (2017)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Zeng Shengda, Baleanu Dumitru, Bai Yunru, Wu Guo cheng, Appl. Math. Comput. 315, 549 (2017)MathSciNetGoogle Scholar
  15. 15.
    S. Priyadharsini, J. Fractional Calculus Appl. 7, 87 (2016)MathSciNetGoogle Scholar
  16. 16.
    Michele Caputo, Mauro Fabrizio, Progr. Fract. Differ. Appl. 1, 73 (2015)Google Scholar
  17. 17.
    Jordan Hristov, On the Atangana-Baleanu derivative and its relation to the fading memory concept: The diffusion equation formulation, in Trends in Theory and Applications of Fractional Derivatives with Mittag-Leffler Kernel, edited by José Francisco Gómez, Lizeth Torres, Ricardo Escobar (Springer, 2019)Google Scholar
  18. 18.
    Ndolane Sene, Progr. Fract. Differ. Appl. 4, 493 (2018)Google Scholar
  19. 19.
    Ndolane Sene, J. Math. Comput. Sci. 18, 388 (2018)CrossRefGoogle Scholar
  20. 20.
    Fahd Jarad, Thabet Abdeljawad, Dumitru Baleanu, J. Nonlinear Sci. Appl. 10, 2607 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ndolane Sene, Fractal Fractional 2, 17 (2018)CrossRefGoogle Scholar
  22. 22.
    Yassine Adjabi, Fahd Jarad, Thabet Abdeljawad, Filomat 31, 5457 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Fahd Jarad, Ekin Uğurlu, Thabet Abdeljawad, Dumitru Baleanu, Adv. Differ. Equ. 2017, 247 (2017)CrossRefGoogle Scholar
  24. 24.
    Udita N. Katugampola, Appl. Math. Comput. 218, 860 (2011)MathSciNetGoogle Scholar
  25. 25.
    Emile Franc Doungmo Goufo, Chaos 26, 084305 (2016)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Yan Li, YangQuan Chen, Igor Podlubny, Automatica 45, 1965 (2009)CrossRefGoogle Scholar
  27. 27.
    Santos B. Yuste, Luis Acedo, SIAM J. Numer. Anal. 42, 1862 (2005)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Chang-Ming Chen, Fawang Liu, I. Turner, Vo Anh, J. Comput. Phys. 227, 886 (2007)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Kolade M. Owolabi, Abdon Atangana, Chaos, Solitons Fractals 111, 119 (2018)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Laura Portero, Juan Carlos Jorge, J. Comput. Appl. Math. 189, 676 (2006)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratoire Lmdan, Département de Mathématiques de la DécisionUniversité Cheikh Anta Diop de Dakar, Faculté des Sciences Economiques et GestionDakar FannSenegal

Personalised recommendations