The 1st Level General Fractional Derivatives and some of their Properties

TL;DR

This paper introduces the first level of general fractional derivatives combining Riemann-Liouville and Caputo types, exploring their properties and fundamental theorems within fractional calculus.

Contribution

It constructs the 1st level general fractional derivatives that unify Riemann-Liouville and Caputo derivatives, expanding the theoretical framework of fractional calculus.

Findings

Defined the 1st level general fractional derivatives.

Proved fundamental theorems of fractional calculus for these derivatives.

Analyzed properties of the new derivatives and integrals.

Abstract

In this paper, we first provide a short summary of the main properties of the so-called general fractional derivatives with the Sonin kernels introduced so far. These are integro-differential operators defined as compositions of the first order derivative and an integral operator of convolution type. Depending on succession of these operators, the general fractional derivatives of the Riemann-Liouville and of the Caputo types were defined and studied. The main objective of this paper is a construction of the 1st level general fractional derivatives that comprise both the general fractional derivative of the Riemann-Liouville type and the general fractional derivative of the Caputo type. We also provide some of their properties including the 1st and the 2nd fundamental theorems of Fractional Calculus for these derivatives and the suitably defined general fractional integrals.