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Mittag–Leffler Functions and the Truncated \({\mathcal {V}}\)-fractional Derivative

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Abstract

In this paper, we introduce a new type of fractional derivative, which we called truncated \({\mathcal {V}}\)-fractional derivative, for \(\alpha \)-differentiable functions, by means of the six-parameter truncated Mittag–Leffler function. One remarkable characteristic of this new derivative is that it generalizes several different fractional derivatives, recently introduced: conformable fractional derivative, alternative fractional derivative, truncated alternative fractional derivative, M-fractional derivative and truncated M-fractional derivative. This new truncated \({\mathcal {V}}\)-fractional derivative satisfies several important properties of the classical derivatives of integer order calculus: linearity, product rule, quotient rule, function composition and the chain rule. Also, as in the case of the Caputo derivative, the derivative of a constant is zero. Since the six parameters Mittag–Leffler function is a generalization of Mittag–Leffler functions of one, two, three, four and five parameters, we were able to extend some of the classical results of the integer-order calculus, namely: Rolle’s theorem, the mean value theorem and its extension. In addition, we present a theorem on the law of exponents for derivatives and as an application we calculate the truncated \({\mathcal {V}}\)-fractional derivative of the two-parameter Mittag–Leffler function. Finally, we present the \({\mathcal {V}}\)-fractional integral from which, as a natural consequence, new results appear as applications. Specifically, we generalize the inverse property, the fundamental theorem of calculus, a theorem associated with classical integration by parts, and the mean value theorem for integrals. We also calculate the \({\mathcal {V}}\)-fractional integral of the two-parameter Mittag–Leffler function. Further, we were able to establish the relation between the truncated \({\mathcal {V}}\)-fractional derivative and the truncated \({\mathcal {V}}\)-fractional integral and the fractional derivative and fractional integral in the Riemann–Liouville sense when the order parameter \(\alpha \) lies between 0 and 1 (\(0<\alpha <1\)).

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Authors and Affiliations

  1. Department of Applied Mathematics, Institute of Mathematics, Statistics and Scientific Computation, University of Campinas-UNICAMP, Rua Sérgio Buarque de Holanda 651, Campinas, SP, 13083-859, Brazil

    J. Vanterler da C. Sousa & E. Capelas de Oliveira

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  1. J. Vanterler da C. Sousa
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  2. E. Capelas de Oliveira
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Correspondence to J. Vanterler da C. Sousa.

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Sousa, J.V.d.C., Oliveira, E.C.d. Mittag–Leffler Functions and the Truncated \({\mathcal {V}}\)-fractional Derivative. Mediterr. J. Math. 14, 244 (2017). https://doi.org/10.1007/s00009-017-1046-z

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  • DOI: https://doi.org/10.1007/s00009-017-1046-z

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