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Advances in Pure and Applied Mathematics

Editor-in-Chief: Trimeche, Khalifa

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CiteScore 2017: 1.29

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1869-6090
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On a Caputo-type fractional derivative

Daniela S. Oliveira
  • Corresponding author
  • Department of Applied Mathematics, Imecc – Unicamp, Universidade Estadual de Campinas, Campinas, SP 13083-859, Brazil
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/ Edmundo Capelas de Oliveira
  • Department of Applied Mathematics, Imecc – Unicamp, Universidade Estadual de Campinas, Campinas, SP 13083-859, Brazil
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Published Online: 2018-01-23 | DOI: https://doi.org/10.1515/apam-2017-0068

Abstract

In this paper, we present a new differential operator of arbitrary order defined by means of a Caputo-type modification of the generalized fractional derivative recently proposed by Katugampola. The generalized fractional derivative, when convenient limits are considered, recovers the Riemann–Liouville and the Hadamard derivatives of arbitrary order. Our differential operator recovers as limiting cases the arbitrary order derivatives proposed by Caputo and by Caputo–Hadamard. Some properties are presented as well as the relation between this differential operator of arbitrary order and the Katugampola generalized fractional operator. As an application we prove the fundamental theorem of fractional calculus associated with our operator.

Keywords: Caputo-type modification; generalized fractional derivative; Caputo fractional derivative; Caputo–Hadamard fractional derivative; fundamental theorem of fractional calculus

MSC 2010: 26A33

References

  • [1]

    R. Almeida, A. B. Malinowska and T. Odzijewicz, Fractional differential equations with dependence on the Caputo–Katugampola derivative, J. Comput. Nonlinear Dynam. 11 (2016), Paper No. CND-15-1334. Google Scholar

  • [2]

    E. Capelas de Oliveira and J. A. Tenreiro Machado, A review of definitions for fractional derivatives and integral, Math. Probl. Eng. 2014 (2014), Article ID 238459. Web of ScienceGoogle Scholar

  • [3]

    E. Contharteze Grigoletto and E. Capelas de Oliveira, Fractional version of the fundamental theorem of calculus, Appl. Math. 4 (2013), 23–33. CrossrefGoogle Scholar

  • [4]

    R. Figueiredo Camargo and E. Capelas de Oliveira, Fractional Calculus (in Portuguese), Editora Livraria da Física, São Paulo, 2015. Google Scholar

  • [5]

    Y. Y. Gambo, F. Jarad, D. Baleanu and T. Abdeljawad, On Caputo modification of the Hadamard fractional derivatives, Adv. Difference Equ. 2014 (2014), 10.1186/1687-1847-2014-10. Web of ScienceGoogle Scholar

  • [6]

    J. Hadamard, Essai sur l’étude des fonctions données par leur développement de Taylor, J. Math. Pures Appl. (4) 8 (1892), 101–186. Google Scholar

  • [7]

    F. Jarad, T. Abdeljawad and D. Baleanu, Caputo-type modification of the Hadamard fractional derivatives, Adv. Difference Equ. 2012 (2012), 10.1186/1687-1847-2012-142. Web of ScienceGoogle Scholar

  • [8]

    F. Jarad, T. Abdeljawad and D. Baleanu, On the generalized fractional derivatives and their Caputo modification, J. Nonlinear Sci. Appl. 10 (2017), no. 5, 2607–2619. CrossrefWeb of ScienceGoogle Scholar

  • [9]

    U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput. 218 (2011), no. 3, 860–865. Web of ScienceGoogle Scholar

  • [10]

    U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl. 6 (2014), no. 4, 1–15. Google Scholar

  • [11]

    A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud. 204, Elsevier Science, Amsterdam, 2006. Google Scholar

  • [12]

    A. Laforgia and P. Natalini, On the asymptotic expansion of a ratio of gamma functions, J. Math. Anal. Appl. 389 (2012), no. 2, 833–837. Web of ScienceCrossrefGoogle Scholar

  • [13]

    K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, 1993. Google Scholar

  • [14]

    M. D. Ortigueira and J. A. Tenreiro Machado, What is a fractional derivative?, J. Comput. Phys. 293 (2015), 4–13. CrossrefWeb of ScienceGoogle Scholar

  • [15]

    B. Ross, A brief history and exposition of the fundamental theory of fractional calculus, Fractional Calculus and its Applications, Springer, Berlin (1975), 1–36. Google Scholar

  • [16]

    H. L. Royden and P. Fitzpatrick, Real Analysis, Prentice Hall, Boston, 2010. Google Scholar

  • [17]

    S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon & Breach Science, Yverdon, 1993. Google Scholar

  • [18]

    Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. Google Scholar

About the article

Received: 2017-06-04

Revised: 2017-12-13

Accepted: 2017-12-14

Published Online: 2018-01-23


Citation Information: Advances in Pure and Applied Mathematics, ISSN (Online) 1869-6090, ISSN (Print) 1867-1152, DOI: https://doi.org/10.1515/apam-2017-0068.

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[1]
J. Vanterler da C. Sousa and E. Capelas de Oliveira
Bulletin of the Brazilian Mathematical Society, New Series, 2018
[2]
Y. Y. Gambo, R. Ameen, F. Jarad, and T. Abdeljawad
Advances in Difference Equations, 2018, Volume 2018, Number 1

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