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02- Progress in Fractional Differentiation and Applications
An International Journal
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Vol. 4
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No. 4
On the Local M-Derivative
PP: 479-492
doi:10.18576/pfda/040403
Author(s)
Jose Vanterler da C. Sousa
,
Edmundo Capelas de Oliveira
,
Abstract
We denote a new differential operator by Da,b M (·), where the parameter a, associated with the order, is such that 0
0 and M is used to denote that the function to be derived involves a Mittag-Leffler function with one parameter. This new derivative satisfies some properties of integer-order calculus, e.g. linearity, product rule, quotient rule, function composition and the chain rule. Besides as in the case of the Caputo derivative, the derivative of a constant is zero. Because Mittag-Leffler function is a natural generalization of the exponential function, we can extend some of the classical results, namely: Rolle’s theorem, the mean-value theorem and its extension. We present the corresponding M-integral from which, as a natural consequence, new results emerge which can be interpreted as applications. Specifically, we generalize the inversion property of the fundamental theorem of calculus and prove a theorem associated with the classical integration by parts. Finally, we present an application involving linear differential equations by means of local M-derivative with some graphs.
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