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.
Digital Object Identifier (DOI)
Vanterler da C. Sousa, Jose and Capelas de Oliveira, Edmundo
"On the Local M-Derivative,"
Progress in Fractional Differentiation & Applications: Vol. 4
, Article 3.
Available at: https://dc.naturalspublishing.com/pfda/vol4/iss4/3