Abstract
There are many possible definitions of derivatives, here we present some and present one that we have called generalized that allows us to put some of the others as a particular case of this but, what interests us is to determine that there is an infinite number of possible definitions of fractional derivatives, all are correct as differential operators each of them must be properly defined its algebra. We introduce a generalized version of fractional derivative that extends the existing ones in the literature. To those extensions it is associated a differentiable operator and a differential ring and applications that shows the advantages of the generalization. We also review the different definitions of fractional derivatives and it is shown how the generalized version contains the previous ones as a particular cases.
Highlights
Introduction and PreliminariesFractional derivative was defined for responding to a question ’what does it mean dα f dtα if α = ’ inFollowing that, finding the right definition of fractional derivative has attracted significant attention of researcher and in the last few years it has seen significantly progress in mathematical and non-mathematical journals
Since some of previous definitions do not satisfy the classical formulas of the usual derivative, it has been proposed an ad hoc algebra associated to each definition
We propose a version of fractional derivative that has the advantages that generalized the already existing in the literature and where the different algebras are unified under the notion of fractional differential ring
Summary
Fractional derivative was defined for responding to a question ’what does it mean dα f dtα if α. Following that, finding the right definition of fractional derivative has attracted significant attention of researcher and in the last few years it has seen significantly progress in mathematical and non-mathematical journals (see [1–10]). In particular in past three years several definitions of fractional derivative have been proposed (see [3,4,10–23]). Since some of previous definitions do not satisfy the classical formulas of the usual derivative, it has been proposed an ad hoc algebra associated to each definition. We propose a version of fractional derivative that has the advantages that generalized the already existing in the literature and where the different algebras are unified under the notion of fractional differential ring. The present paper is organized as follows: In the Section 2 we give the previous definitions of fractional derivative and our generalized fractional derivative (GFD) definition, in the Section 3 we introduce a fractional differential ring, in the Section 4 we give some result of GFD, in the Section 5 we give a definition of partial fractional differential derivative, in the Section 6 we give a definition of GFD when α ∈
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