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Symmetry
Volume 13
Issue 5
10.3390/sym13050755
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Article

General Fractional Integrals and Derivatives of Arbitrary Order

by
Yuri Luchko
Department of Mathematics, Physics, and Chemistry, Beuth Technical University of Applied Sciences Berlin, Luxemburger Str. 10, 13353 Berlin, Germany
Symmetry 2021, 13(5), 755; https://doi.org/10.3390/sym13050755
Submission received: 3 April 2021 / Revised: 21 April 2021 / Accepted: 23 April 2021 / Published: 27 April 2021
(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus)
Versions Notes

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 of arbitrary order possess integrable singularities at the point zero, the kernels of the general fractional integrals can—depending on their order—be 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.
Keywords: Sonine kernel; general fractional derivative of arbitrary order; general fractional integral of arbitrary order; first fundamental theorem of fractional calculus; second fundamental theorem of fractional calculus Sonine kernel; general fractional derivative of arbitrary order; general fractional integral of arbitrary order; first fundamental theorem of fractional calculus; second fundamental theorem of fractional calculus

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MDPI and ACS Style

Luchko, Y. General Fractional Integrals and Derivatives of Arbitrary Order. Symmetry 2021, 13, 755. https://doi.org/10.3390/sym13050755

AMA Style

Luchko Y. General Fractional Integrals and Derivatives of Arbitrary Order. Symmetry. 2021; 13(5):755. https://doi.org/10.3390/sym13050755

Chicago/Turabian Style

Luchko, Yuri. 2021. "General Fractional Integrals and Derivatives of Arbitrary Order" Symmetry 13, no. 5: 755. https://doi.org/10.3390/sym13050755

APA Style

Luchko, Y. (2021). General Fractional Integrals and Derivatives of Arbitrary Order. Symmetry, 13(5), 755. https://doi.org/10.3390/sym13050755

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