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The General Fractional Derivative and Related Fractional Differential Equations

1
Department of Mathematics, Physics, and Chemistry, Beuth Technical University of Applied Sciences Berlin, Luxemburger Str. 10, 13353 Berlin, Germany
2
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(12), 2115; https://doi.org/10.3390/math8122115
Received: 29 September 2020 / Revised: 9 November 2020 / Accepted: 23 November 2020 / Published: 26 November 2020
(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False”)
In this survey paper, we start with a discussion of the general fractional derivative (GFD) introduced by A. Kochubei in his recent publications. In particular, a connection of this derivative to the corresponding fractional integral and the Sonine relation for their kernels are presented. Then we consider some fractional ordinary differential equations (ODEs) with the GFD including the relaxation equation and the growth equation. The main part of the paper is devoted to the fractional partial differential equations (PDEs) with the GFD. We discuss both the Cauchy problems and the initial-boundary-value problems for the time-fractional diffusion equations with the GFD. In the final part of the paper, some results regarding the inverse problems for the differential equations with the GFD are presented. View Full-Text
Keywords: general fractional derivative; general fractional integral; Sonine condition; fractional relaxation equation; fractional diffusion equation; Cauchy problem; initial-boundary-value problem; inverse problem general fractional derivative; general fractional integral; Sonine condition; fractional relaxation equation; fractional diffusion equation; Cauchy problem; initial-boundary-value problem; inverse problem
MDPI and ACS Style

Luchko, Y.; Yamamoto, M. The General Fractional Derivative and Related Fractional Differential Equations. Mathematics 2020, 8, 2115. https://doi.org/10.3390/math8122115

AMA Style

Luchko Y, Yamamoto M. The General Fractional Derivative and Related Fractional Differential Equations. Mathematics. 2020; 8(12):2115. https://doi.org/10.3390/math8122115

Chicago/Turabian Style

Luchko, Yuri; Yamamoto, Masahiro. 2020. "The General Fractional Derivative and Related Fractional Differential Equations" Mathematics 8, no. 12: 2115. https://doi.org/10.3390/math8122115

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