On a Caputo-type fractional derivative

Abstract

Abstract In this paper, we present a new differential operator of arbitrary order defined by means of a Caputo-type modification of the generalized fractional derivative recently proposed by Katugampola. The generalized fractional derivative, when convenient limits are considered, recovers the Riemann–Liouville and the Hadamard derivatives of arbitrary order. Our differential operator recovers as limiting cases the arbitrary order derivatives proposed by Caputo and by Caputo–Hadamard. Some properties are presented as well as the relation between this differential operator of arbitrary order and the Katugampola generalized fractional operator. As an application we prove the fundamental theorem of fractional calculus associated with our operator.