General fractional integrals and derivatives of arbitrary order

TL;DR

This paper introduces a comprehensive framework for fractional integrals and derivatives of any order, exploring their properties, kernel classes, and fundamental theorems in fractional calculus.

Contribution

It presents a new generalization of fractional integrals and derivatives with novel kernels satisfying a generalized Sonine condition.

Findings

Established fundamental theorems for the new fractional operators.

Characterized kernels with singular and continuous behaviors at zero.

Provided a unified approach to fractional calculus of arbitrary order.

Abstract

In this paper, we introduce the general fractional integrals and derivatives of arbitrary order and study some of their basic properties and particular cases. First, a suitable generalization of the Sonine condition is presented and some important classes of the kernels that satisfy this condition are introduced. Whereas the kernels of the general fractional derivatives with these kernels possess the integrable singularities at the point zero, the kernels of the general fractional integrals can be - depending on their order - both singular and continuous at the origin. For the general fractional integrals and derivatives of arbitrary order with the kernels introduced in this paper, two fundamental theorems of fractional calculus are formulated and proved.