Generalized Fractional Derivative, Fractional differential ring
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 which must be properly defined in its algebra. We introduce a generalized version of the fractional derivative that extends the existing ones in the literature. To those extensions, it is associated with a differentiable operator and a differential ring and applications that show the advantages of the generalization. We also review the different definitions of fractional derivatives proposed by Michele Caputo in GJI:GJI529, Khalil, Al Horani, Yousef, Sababheh in khalil2014new, Anderson and Ulness in anderson2015newly, Guebbai and Ghiat in guebbai2016new, Udita N. Katugampola in katugampola2014new, Camrud in camrud2016conformable and it is shown how the generalized version contains the previous ones as a particular case.