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

Mittag–Leffler Functions and the Truncated $${\mathcal {V}}$$ V -fractional Derivative In this paper, we introduce a new type of fractional derivative, which we called truncated $${\mathcal {V}}$$ V -fractional derivative, for $$\alpha $$ 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 $${\mathcal {V}}$$ 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 $${\mathcal {V}}$$ V -fractional derivative of the two-parameter Mittag–Leffler function. Finally, we present the $${\mathcal {V}}$$ 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 $${\mathcal {V}}$$ V -fractional integral of the two-parameter Mittag–Leffler function. Further, we were able to establish the relation between the truncated $${\mathcal {V}}$$ V -fractional derivative and the truncated $${\mathcal {V}}$$ V -fractional integral and the fractional derivative and fractional integral in the Riemann–Liouville sense when the order parameter $$\alpha $$ α lies between 0 and 1 ( $$0<\alpha <1$$ 0 < α < 1 ). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mediterranean Journal of Mathematics Springer Journals

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

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Publisher
Springer International Publishing
Copyright
Copyright © 2017 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Mathematics, general
ISSN
1660-5446
eISSN
1660-5454
D.O.I.
10.1007/s00009-017-1046-z
Publisher site
See Article on Publisher Site

Abstract

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

Journal

Mediterranean Journal of MathematicsSpringer Journals

Published: Nov 29, 2017

References

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