Advertisement

The European Physical Journal Special Topics

, Volume 226, Issue 16–18, pp 3333–3354 | Cite as

Fractional proportional differences with memory

  • Thabet Abdeljawad
  • Fahd Jarad
  • Jehad Alzabut
Regular Article
  • 24 Downloads
Part of the following topical collections:
  1. Fractional Dynamical Systems - Recent Trends in Theory and Applications

Abstract

In this paper, we formulate nabla fractional sums and differences and the discrete Laplace transform on the time scale hℤ. Based on a local type h-proportional difference (without memory), we generate new types of fractional sums and differences with memory in two parameters which are generalizations to the formulated fractional sums and differences. The kernel of the newly defined generalized fractional sum and difference operators contain h-discrete exponential functions. The discrete h-Laplace transform and its convolution theorem are then used to study the newly introduced discrete fractional operators and also used to solve Cauchy linear fractional difference type problems with step 0 < h ≤ 1.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, in Theory and application of fractional differential equations (North Holland Mathematics Studies, Amsterdam, 2006), Vol. 204 Google Scholar
  2. 2.
    I. Podlubny, Fractional differential equations (Academic Press, San Diego, CA, 1999) Google Scholar
  3. 3.
    C. Goodrich, A. Peterson, Discrete fractional calculus (Springer, New York, 2015) Google Scholar
  4. 4.
    F.M. At"i"c"i", P.W. Eloe, Electr. J. Qual. Theor. Differ. Equ., Spec. Ed. I 2009, 1 (2009) Google Scholar
  5. 5.
    T. Abdeljawad, F. Atici, Abstr. Appl. Anal. 2012, 406757 (2012) Google Scholar
  6. 6.
    T. Abdeljawad, Discr. Dynam. Nat. Soc. 2013, 406910 (2013) MathSciNetGoogle Scholar
  7. 7.
    T. Abdeljawad, Adv. Differ. Equ. 2013, 36 (2013) CrossRefGoogle Scholar
  8. 8.
    T. Abdeljawad, F. Jarad, D. Baleanu, Adv. Differ. Equ. 2012, 72 (2012) CrossRefGoogle Scholar
  9. 9.
    R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, J. Comput. Appl. Math. 264, 65 (2014) MathSciNetCrossRefGoogle Scholar
  10. 10.
    T. Abdeljawad, J. Comput. Appl. Math. 279, 57 (2013) CrossRefGoogle Scholar
  11. 11.
    D.R. Anderson, D.J. Ulness, Adv. Dyn. Syst. Appl. 10, 109 (2015) MathSciNetGoogle Scholar
  12. 12.
    D.R. Anderson, Commun. Appl. Nonlinear Anal. 24, 17 (2017) Google Scholar
  13. 13.
    U.N. Katugampola, Appl. Math. Comput. 218, 860 (2011) MathSciNetGoogle Scholar
  14. 14.
    U.N. Katugampola, Bull. Math. Anal. Appl. 6, 1 (2014) MathSciNetGoogle Scholar
  15. 15.
    F. Jarad, T. Abdeljawad, D. Baleanu, J. Nonlinear Sci. Appl. 10, 2607 (2017) MathSciNetCrossRefGoogle Scholar
  16. 16.
    F. Jarad, T. Abdeljawad, J. Alzabut, Eur. Phys. J. Special Topics 226, 3457 (2018) Google Scholar
  17. 17.
    A. Atangana, D. Baleanu, Thermal Sci. 20, 757 (2016) CrossRefGoogle Scholar
  18. 18.
    M. Caputo, M. Fabrizio, Progr. Fract. Differ. Appl. 1, 73 (2015) Google Scholar
  19. 19.
    X.J. Yang, F. Gao, J.A.T. Machado, D. Baleanu, arXiv:1701.05590 (2017)
  20. 20.
    T. Abdeljawad, D. Baleanu, J. Nonlinear Sci. Appl. 10, 1098 (2017) MathSciNetCrossRefGoogle Scholar
  21. 21.
    T. Abdeljawad, D. Baleanu, Adv. Differ. Equ. 2017, 78 (2017) CrossRefGoogle Scholar
  22. 22.
    T. Abdeljawad, D. Baleanu, Choas Solitons Fractals 102, 106 (2017) ADSCrossRefGoogle Scholar
  23. 23.
    A.A. Kilbas, M. Saigo, K. Saxena, Integral Transforms Spec. Funct. 15, 31 (2004) MathSciNetCrossRefGoogle Scholar
  24. 24.
    M. Bohner, A. Peterson, Advances in dynamic equations on time scales (Birkhäuser, Boston, 2003) Google Scholar
  25. 25.
    G.E. Andrews, R. Askey, R. Roy, Special functions (Cambridge University Press, Cambridge, 1999) Google Scholar

Copyright information

© EDP Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and General SciencesPrince Sultan UniversityRiyadhSaudi Arabia
  2. 2.Department of MathematicsÇankaya UniversityAnkaraTurkey

Personalised recommendations