The paper by Christian Maxime Steve Oumarou of AIMS Cameroon, Research Chair and Postdoctoral Fellow Dr. Jean-Daniel Djida, Hafiz Muhammad Fahad and Arran Fernandez of Eastern Mediterranean University in Famagusta, Northern Cyprus, was published on August 16th, 2021.
According to these researchers, several types of Fractional Calculus have been proposed and they can be organised into some general classes of operators. For a unified mathematical theory, results should be proved in the most general possible setting. Two important classes of Fractional-Calculus Operators are the Fractional Integrals and Derivatives with respect to Functions, dating back to the 1970s, and those with general analytic kernels, introduced in 2019. To cover both of these settings in a single study, the researchers considered Fractional Integrals and Derivatives with Analytic Kernels with respect to Functions, which they say, have never been studied in detail before.
In this paper, they established the basic properties of these general operators, including series formulae, composition relations, function spaces, and Laplace transforms. For them, the tools of convergent series, from fractional calculus with analytic kernels, and of operational calculus, from fractional calculus with respect to functions, are essential ingredients in the analysis of the general class that covers both.
Nowadays, many scientists are interested in the field due to its applications in engineering, physics, chemistry, biology, and economics, amongst others. These applications arise because fractional calculus is very useful for modelling different types of physical systems: often due to its nonlocality properties, as opposed to classical derivatives which are local, and also the ability of fractional derivatives to capture intermediate behaviours, such as viscoelastic substances which are intermediate between solid and liquid.
After reviewing Fractional Calculus in this paper, the researchers considered Fractional Calculus with Analytic Kernels with Respect to Functions and proved various new results concerning these generalised operators, such as establishing series formulae and composition properties. They further considered functional analysis of these new operators, establishing appropriate function spaces in which these can be applied, before showing how some fractional integro-differential equations using the new operators may be solved using a type of generalized Laplace transform.
The main goal of this paper was to study the general class of Fractional Operators with Analytic Kernels with respect to Functions, in particular, using convergent infinite series and an operational calculus formulation. The researchers achieved the main purpose, establishing various properties of these operators, such as establishing appropriate composition properties and function spaces, by considering each problem using the same methods as used in one of the smaller pre-existing classes of fractional operators. To achieve these goals, several concepts from fractional calculus and its generalizations needed to be pieced together. Key roles were played by the concept of Series Formulae, which comes from the Theory of Fractional Calculus with Analytic Kernels, and by the concept of Conjugation of Operators, which comes from the Theory of Fractional Calculus with respect to Functions.
The main point of this work was to demonstrate the ultimate generality to which such operators of fractional calculus can be taken: combining two already very general classes of operators to obtain a new, even more general class. Even if this class in its full generality is not applicable to solve many real-world problems directly, its advantage lies in the fact that it is broad and general enough to cover a huge number of types of fractional calculus, from Prabhakar to Erdelyi to Hadamard-type, which all have different properties and behaviours, but which can all be covered by the general formulae introduced in the paper.
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