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.

On a Fractional Operator Combining Proportional and Classical Differintegrals

“…There are many different ways of defining fractional derivatives and fractional integrals: Riemann-Liouville, Caputo, Marchaud, tempered, Hilfer, and Atangana-Baleanu, to name but a few [8][9][10]. These diverse definitions may be categorised into general classes according to their structure and properties [11].…”

Nonlinear generalized fractional differential equations with generalized fractional integral conditions

“…Fractional operators used to illustrate better the reality of real-world phenomena with the hereditary property. For instance, various applications and comprehensive strategy of the fractional calculus are addressed in the works of Baleanu et al [1,2], Abd-Elhameed et al [3,4], Jarad et al [5], Hafez et al [6], and Youssri et al [7]. A good review of different fractional operators can be found in [8].…”

On the conformable fractional logistic models

“…These differential and integral operators are frequently used to construct mathematical models in several scientific areas. In particular, they have been successfully applied to the study of the logistics model (see previous studies [1][2][3][4][5][6] ).…”

“…For a complementary study on the recent developments in the field of the fractional calculus as well as its applications, see the literature. [1][2][3][4][5][6] It is important to note that the global fractional derivatives (eg, Caputo and Riemann-Liouville) are not collecting mere local information. By contrast, fractional operators keep track of the history of the process being studied; this feature allows modeling the nonlocal and distributed responses that commonly appear in natural and physical phenomena.…”