All linear fractional derivatives with power functions’ convolution kernel and interpolation properties

Abstract

Our attempt is an axiomatic approach to find all classes of possible definitions for fractional derivatives with three axioms. In this paper, we consider a special case of linear integro-differential operators with power functions’ convolution kernel a(α)(t−s)b(α) of order α∈(0,1). We determine analytic functions a(α) and b(α) such that when α→0+, the corresponding operator becomes identity operator, and when α→1− the corresponding operator becomes derivative operator. Then, a sequential operator is used to extend the fractional operator to a higher order. Some properties of the sequential operator in this regard also are studied. The singularity properties, Laplace transform and inverse of the new class of fractional derivatives are investigated. Several examples are provided to confirm theoretical achievements. Finally, the solution of the relaxation equation with diverse fractional derivatives is obtained and compared.