Abstract
This paper is motivated by some papers treating the fractional derivatives. We introduce a new definition of fractional derivative which obeys classical properties including linearity, product rule, quotient rule, power rule, chain rule, Rolle’s theorem, and the mean value theorem. The definition D α f t = lim h ⟶ 0 f t + h e α − 1 t − f t / h , for all t > 0 , and α ∈ 0,1 . If α = 0 , this definition coincides to the classical definition of the first order of the function f .
Highlights
E theory of fractional derivative is a very old theory, which dates back to a conversation on September 30, 1695, between Hopital and Leibniz concerning the definition of the operator dn/dxn for n 1/2. us, as the time progresses, certain approaches have been given in the literature such as the definition of Riemann–Liouville and that of Caputo
Many dynamic systems are best characterized by a dynamic fractional-order model, generally based on the notion of differentiation or integration of noninteger order. e study of fractional order systems is more delicate than for their whole order counterparts
Fractional systems are, on the one hand, considered as memory systems, in particular to take into account the initial conditions, and on the other hand, they present a much more complex dynamic system
Summary
E theory of fractional derivative is a very old theory, which dates back to a conversation on September 30, 1695, between Hopital and Leibniz concerning the definition of the operator dn/dxn for n 1/2. us, as the time progresses, certain approaches have been given in the literature such as the definition of Riemann–Liouville and that of Caputo. Ey proved the product rule, and the fractional mean value theorem solved some (conformable) fractional differential equations where the fractional exponential function etα/α played an important rule. International Journal of Differential Equations e purpose of this work is to further generalize the results obtained in [5] and in [6] and introduce a new conformable fractional derivative as the most natural extension of the familiar limit definition of the derivative of a function f at a point. Given a function f: [0, ∞) ⟶ R, and the conformable fractional derivative of f order α is defined by
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