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|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)|
Derivada fracionária de
Riemann-Liouville fractional derivative
|Appears in Collections:||IMECC - Artigos e Outros Documentos|
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