A comment on a controversial issue: A generalized fractional derivative cannot have a regular kernel

By A. Hanyga. 2020

95 citations recorded by the source corpus.

Abstract The problem whether a given pair of functions can be used as the kernels of a generalized fractional derivative and the associated generalized fractional integral is reduced to the problem of existence of a solution to the Sonine equation. It is shown for some selected classes of functions that a necessary condition for a function to be the kernel of a fractional derivative is an integrable singularity at 0. It is shown that locally integrable completely monotone functions satisfy the Sonine equation if and only if they are singular at 0.

Subfields: Applied Mathematics, Mathematical Physics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/a66bb224b12c5b2a7ea9c7d5baf7a4760ec100b1

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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 Mathematical Physics

1. Yuri Luchko
2. Masahiro Yamamoto
3. Oleg Yu. Imanuvilov
4. Sylvain Ervedoza
5. Gunther Uhlmann
6. Enrique Zuazua
7. Michael V. Klibanov
8. Bangti Jin

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, methodology): “on thatneeds ajustification for the given sets of the kernels and the spaces of functions. In our case, we do this in the following theorem (see [8] for the case of the GFDs with the kernels k ∈ K and [3] for the case k ∈ H). Theorem 4.1 (1st Fundamental Theorem for the GFD). Let k ∈ S−1 be a Sonine ”
• Fractional derivatives and the fundamental theorem of fractional calculus (2020, background): “These discussions mainly concerned the ”new fractional integrals and derivatives” ( [6, 8, 20, 24–26]), the integro-differential operators of convolution type with some general kernels ( [8, 14, 15, 24, 27, 28]), and the abstract axioms of FC ( [11, 19, 22]).”
• The General Fractional Derivative and Related Fractional Differential Equations (2020, background): “As shown in [22], the functions that satisfy the Sonine condition (25) cannot be continuous at the point t = 0 and thus the “new fractional derivatives” with the continuous kernels introduced recently in the FC literature do not belong to the class of the GFDs that are discussed in this paper.”
• Fractional Differential Equations with the General Fractional Derivatives of Arbitrary Order in the Riemann–Liouville Sense (2022, background): “However, merely condition (15) does not ensure an essential requirement for the kernel k of the GFD, namely, the condition that this kernel should have a singularity at the origin, see [43, 44, 46].”
• Operational Calculus for the General Fractional Derivative and Its Applications (2021, unknown): “The starting point for involving the general integro-differential operators of convolution type into FC research was the paper [15] (see also [10, 23, 28]).”