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On the 1st Level General Fractional Derivatives of Arbitrary Order
Version 1
: Received: 14 January 2023 / Approved: 17 January 2023 / Online: 17 January 2023 (01:34:12 CET)
A peer-reviewed article of this Preprint also exists.
Luchko, Y. On the 1st-Level General Fractional Derivatives of Arbitrary Order. Fractal Fract. 2023, 7, 183. Luchko, Y. On the 1st-Level General Fractional Derivatives of Arbitrary Order. Fractal Fract. 2023, 7, 183.
Abstract
In this paper, the 1st level general fractional derivatives of arbitrary order are defined and investigated for the first time. We start with a generalization of the Sonin condition for the kernels of the general fractional integrals and derivatives and then specify a set of the kernels that satisfy this condition and posses an integrable singularity of power law type at the origin. The 1st level general fractional derivatives of arbitrary order are integro-differential operators of convolution type with the kernels from this set. They contain both the general fractional derivatives of arbitrary order of the Riemann-Liouville type and the regularized general fractional derivatives of arbitrary order considered in the literature so far. For the 1st level general fractional derivatives of arbitrary order, some important properties including the 1st and the 2nd fundamental theorems of Fractional Calculus are formulated and proved.
Keywords
generalized Sonin condition; general fractional integral; general fractional derivative of arbitrary order; regularized general fractional derivative of arbitrary order; 1st level general fractional derivative; 1st fundamental theorem of fractional calculus; 2nd fundamental theorem of fractional calculus
Subject
Computer Science and Mathematics, Analysis
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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