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Type: | Artigo |

Title: | Mittag-Leffler Functions and the Truncated V-fractional Derivative |

Author: | Sousa, J. Vanterler da C. Capelas de Oliveira, E. |

Abstract: | In this paper, we introduce a new type of fractional derivative, which we called truncated V-fractional derivative, for α-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 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 V-fractional derivative of the two-parameter Mittag–Leffler function. Finally, we present the 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 V-fractional integral of the two-parameter Mittag–Leffler function. Further, we were able to establish the relation between the truncated V-fractional derivative and the truncated V-fractional integral and the fractional derivative and fractional integral in the Riemann–Liouville sense when the order parameter α lies between 0 and 1 (0 < α < 1) |

Subject: | Mittag-Leffler Funções de Riemann-Liouville Derivada fracionária de Riemann-Liouville Integral de Mittag-Leffler functions Riemann-Liouville fractional derivative Riemann-Liouville integral |

Country: | Suíça |

Editor: | Springer |

Rights: | fechado |

Identifier DOI: | 10.1007/s00009-017-1046-z |

Address: | https://link.springer.com/article/10.1007%2Fs00009-017-1046-z |

Date Issue: | 2017 |

Appears in Collections: | IMECC - Artigos e Outros Documentos |

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