# On fractional calculus with general analytic kernels

## TL;DR

This paper introduces a unifying model for fractional calculus with general analytic kernels, connecting various definitions and providing new properties and solutions for fractional differential equations.

## Contribution

It presents a general framework that encompasses many fractional derivatives and integrals, establishing their connection to classical operators and deriving fundamental properties.

## Key findings

- Unified fractional calculus model with general kernels
- Inversion properties and analogues of Leibniz and chain rules established
- Solutions to fractional differential equations using the new operators

## Abstract

Many possible definitions have been proposed for fractional derivatives and integrals, starting from the classical Riemann-Liouville formula and its generalisations and modifying it by replacing the power function kernel with other kernel functions. We demonstrate, under some assumptions, how all of these modifications can be considered as special cases of a single, unifying, model of fractional calculus. We provide a fundamental connection with classical fractional calculus by writing these general fractional operators in terms of the original Riemann-Liouville fractional integral operator. We also consider inversion properties of the new operators, prove analogues of the Leibniz and chain rules in this model of fractional calculus, and solve some fractional differential equations using the new operators.