Classes of operators in fractional calculus: A case study

Abstract

The notion of general classes of operators has recently been proposed as an approach to fractional calculus that respects pure and applied viewpoints equally. Here we demonstrate this approach as it applies to the operators with three-parameter Mittag-Leffler kernels defined by Prabhakar in 1971. By considering the general such operator as a class, we are able to better understand its fundamental nature and the different special cases that emerge. In particular, we show that many other named models of fractional calculus can fit within the class of operators defined by Prabhakar and that this class contains both singular and nonsingular operators together. We characterise completely the cases in which these operators are singular or nonsingular and the cases in which they can be written as finite or infinite sums of Riemann-Liouville differintegrals, to obtain finally a catalogue of subclasses with different types of properties.