On Fractional Operators and Their Classifications

Abstract

Fractional calculus dates its inception to a correspondence between Leibniz and L'Hopital in 1695, when Leibniz described paradoxes and predicted that one day useful consequences will be drawn from them. In today's world, the study of non-integer orders of differentiation has become a thriving field of research, not only in mathematics but also in other parts of science such as physics, biology, and engineering: many of the useful consequences predicted by Leibniz have been discovered. However, the field has grown so far that researchers cannot yet agree on what a fractional derivative can be. In this manuscript, we suggest and justify the idea of classification of fractional calculus into distinct classes of operators.