Fractional derivatives and the fundamental theorem of fractional calculus

By Yuri Luchko. 2020

95 citations recorded by the source corpus.

Abstract In this paper, we address the one-parameter families of the fractional integrals and derivatives defined on a finite interval. First we remind the reader of the known fact that under some reasonable conditions, there exists precisely one unique family of the fractional integrals, namely, the well-known Riemann-Liouville fractional integrals. As to the fractional derivatives, their natural definition follows from the fundamental theorem of the Fractional Calculus, i.e., they are introduced as the left-inverse operators to the Riemann-Liouville fractional integrals. Until now, three families of such derivatives were suggested in the literature: the Riemann-Liouville fractional derivatives, the Caputo fractional derivatives, and the Hilfer fractional derivatives. We clarify the interconnections between these derivatives on different spaces of functions and provide some of their properties including the formulas for their projectors and the Laplace transforms. However, it turns out that there exist infinitely many other families of the fractional derivatives that are the left-inverse operators to the Riemann-Liouville fractional integrals. In this paper, we focus on an important class of these fractional derivatives and discuss some of their properties.

Subfields: Applied Mathematics, Geometry and Topology, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/948586c6590bff7fc3b04296604d8e51b8d7b7d2

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Yuri Luchko
2. Masahiro Yamamoto
3. Enrique Zuazua
4. Ibtissem Ben Aicha
5. C. Sin
6. Sylvain Ervedoza
7. Xu Liu

In Geometry and Topology

1. Masahiro Yamamoto

In Modeling and Simulation

1. Yuri Luchko
2. Masahiro Yamamoto
3. Bangti Jin
4. C. Sin

Citations in the corpus

• General Fractional Integrals and Derivatives with the Sonine Kernels (2021, background): “In particular, these derivatives and the appropriate defined fractional integrals have to satisfy the fundamental theorem of FC (see [4] for a discussion of this theorem).”
• Fractional Differential Equations with the General Fractional Derivatives of Arbitrary Order in the Riemann–Liouville Sense (2022, background): “1 is usually referred to as the 1st Fundamental Theorem of FC for the GFD (19) of arbitrary order (see [45] for a discussion of the 1st and the 2nd Fundamental Theorems of FC for several different kinds of the fractional derivatives).”
• Operational Calculus for the General Fractional Derivative and Its Applications (2021, unknown): “, they satisfy the 1st Fundamental Theorem of FC ( [15, 22, 23]).”
• The General Fractional Derivative and Related Fractional Differential Equations (2020, background): “holds true on a nontrivial space of functions (see [6] for details).”