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The European Physical Journal Special Topics

, Volume 226, Issue 16–18, pp 3457–3471 | Cite as

Generalized fractional derivatives generated by a class of local proportional derivatives

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

Abstract

Recently, Anderson and Ulness [Adv. Dyn. Syst. Appl. 10, 109 (2015)] utilized the concept of the proportional derivative controller to modify the conformable derivatives. In parallel to Anderson’s work, Caputo and Fabrizio [Progr. Fract. Differ. Appl. 1, 73 (2015)] introduced a fractional derivative with exponential kernel whose corresponding fractional integral does not have a semi-group property. Inspired by the above works and based on a special case of the proportional-derivative, we generate Caputo and Riemann-Liouville generalized proportional fractional derivatives involving exponential functions in their kernels. The advantage of the newly defined derivatives which makes them distinctive is that their corresponding proportional fractional integrals possess a semi-group property and they provide undeviating generalization to the existing Caputo and Riemann-Liouville fractional derivatives and integrals. The Laplace transform of the generalized proportional fractional derivatives and integrals are calculated and used to solve Cauchy linear fractional type problems.

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References

  1. 1.
    R. Hilfer, Applications of fractional calculus in physics (Word Scientific, Singapore, 2000) Google Scholar
  2. 2.
    L. Debnath, Int. J. Math. Math. Sci. 54, 3413 (2003) CrossRefGoogle Scholar
  3. 3.
    A. Kilbas, H.M. Srivastava, J.J. Trujillo, in Theory and application of fractional differential equations (North Holland mathematics studies, 2006), Vol. 204 Google Scholar
  4. 4.
    R.L. Magin, Fractional calculus in bioengineering (Begell House Publishers, 2006) Google Scholar
  5. 5.
    R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, J. Comput. Appl. Math. 264, 65 (2014) MathSciNetCrossRefGoogle Scholar
  6. 6.
    T. Abdeljawad, J. Comput. Appl. Math. 279, 57 (2013) CrossRefGoogle Scholar
  7. 7.
    U.N. Katugampola, Appl. Math. Comput. 218, 860 (2011) MathSciNetGoogle Scholar
  8. 8.
    U.N. Katugampola, Bull. Math. Anal. Appl. 6, 1 (2014) MathSciNetGoogle Scholar
  9. 9.
    D.R. Anderson, D.J. Ulness, Adv. Dyn. Syst. Appl. 10, 109 (2015) MathSciNetGoogle Scholar
  10. 10.
    D.R. Anderson, Commun. Appl. Nonlinear Anal. 24, 17 (2017) Google Scholar
  11. 11.
    M. Caputo, M. Fabrizio, Progr. Fract. Differ. Appl. 1, 73 (2015) Google Scholar
  12. 12.
    J. Losada, J.J. Nieto, Progr. Fract. Differ. Appl. 1, 87 (2015) Google Scholar
  13. 13.
    T. Abdeljawad, D. Baleanu, Adv. Differ. Equ. 2017, 78 (2017) CrossRefGoogle Scholar
  14. 14.
    T. Abdeljawad, D. Baleanu, Rep. Math. Phys. 80, 11 (2017) ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    A. Atangana, D. Baleanu, Thermal Sci. 20, 757 (2016) CrossRefGoogle Scholar
  16. 16.
    T. Abdeljawad, D. Baleanu, J. Nonlinear Sci. Appl. 10, 1098 (2017) MathSciNetCrossRefGoogle Scholar
  17. 17.
    A.N. Kochubei, Integr. Equ. Oper. Theory 7, 583 (2011) MathSciNetCrossRefGoogle Scholar
  18. 18.
    A.A. Kilbas, M. Saigo, K. Saxena, Integral Transforms Spec. Funct. 15, 31 (2004) MathSciNetCrossRefGoogle Scholar
  19. 19.
    F. Jarad, T. Abdeljawad, D. Baleanu, J. Nonlinear Sci. Appl. 10, 2607 (2017) MathSciNetCrossRefGoogle Scholar
  20. 20.
    I. Podlubny, Fractional differential equations (Academic Press, San Diego, CA, 1999) Google Scholar
  21. 21.
    S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional integrals and derivatives: theory and applications (Gordon and Breach, Yverdon, 1993) Google Scholar

Copyright information

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

Authors and Affiliations

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

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