Generalising the fractional calculus with Sonine kernels via conjugations

Abstract

Many different types of fractional calculus are defined by various kernel functions within the general class of Sonine-type kernels, while many others are given by conjugating the usual fractional calculus with invertible linear operators, such as composition and multiplication operators giving rise to weighted fractional calculus with respect to functions. Here, we combine these two ideas to create a new and very general model of fractional calculus, given by Sonine kernels with conjugations. We prove fundamental theorems of calculus, and other results on function spaces and compositions, in the setting of these very general operators. As special cases, we are able to obtain left-sided and right-sided operators with Sonine kernels on arbitrary intervals in R, as well as operators with Sonine kernels with respect to functions, weighted operators with Sonine kernels, etc. We briefly consider some fractional differential equations, using Laplace transform methods to prove existence-uniqueness results under certain conditions.