Advertisement

New Fractional Derivative in Colombeau Algebra

  • Said MellianiEmail author
  • A. Chafiki
  • L. S. Chadli
Chapter
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 372)

Abstract

In this paper we introduce an approach to fractional derivatives involving singularities based on the theory of algebras of generalized functions in the Colombeau algebra \(\mathscr {G}\), using new definition of fractional derivative called conformable fractional derivative introduced by the authors Khalil et al. in [1].

References

  1. 1.
    R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    M. Stojanovic, Extension Of Colombeau algebra to derivatives of arbitrary order \(D^{\alpha }\); \(\alpha \in {\mathbb{R}}_{+}\cup \{0\}\): application to ODEs and PDEs with entire and fractional derivatives. Nonlinear Anal. 71, 5458–5475 (2009)Google Scholar
  3. 3.
    D. Rajterc Ciric, M. Stojanovic, Convolution-type derivatives and transforms Of Colombeau generalized stochastic processes. Integr. Transforms Spec. Funct. 22(4–5), 319–326 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    T. Abdeljawad, On conformable fractional calculus (Submitted For Publication)Google Scholar
  5. 5.
    J.F. Colombeau, Elementary Introduction in New Generalized Functions (North Holland, Amsterdam, 1985)zbMATHGoogle Scholar
  6. 6.
    M. Stojanovic, Fondation of the fractional calculus in generalized function algebras. Anal. Appl. 10(4), 439–467 (2012)MathSciNetCrossRefGoogle Scholar
  7. 7.
    M. Oberguggenberger, Generalized functions in nonlinear models—a survey. Nonlinear Anal. 47, 5029–5040 (2001)MathSciNetCrossRefGoogle Scholar
  8. 8.
    D. Rajterc Ciric, M. Stojanovic, Fractional derivatives of multidimensional Colombeau generalized stochastic processes. Fractional Calc. Appl. Anal. 16(4), 949–961 (2013)MathSciNetzbMATHGoogle Scholar
  9. 9.
    D. Rajter-Ciric, A note on fractional derivatives of Colombeau generalized stochastic processes. Novi Sad J. Math. 40(1), 111–121 (2010)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Sultan Moulay Slimane UniversityBeni MellalMorocco

Personalised recommendations