The General Fractional Derivative and Related Fractional Differential Equations
By Yuri Luchko, Masahiro Yamamoto. Mathematics, 2020
83 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis
View Paper: https://www.semanticscholar.org/paper/76248477e6a0270788a72a2990de361c6f699d7f
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Citations in the corpus
- General Fractional Integrals and Derivatives with the Sonine Kernels (2021, methodology): “The ordinary fractional differential equations with the Caputo type GFDs are addressed in [1,5–7] and the time-fractional diffusion equations with these derivatives are considered in [1, 3, 5, 8–10].”
- Fractional Differential Equations with the General Fractional Derivatives of Arbitrary Order in the Riemann–Liouville Sense (2022, background): “, [7, 8, 9, 11, 12, 13, 14, 21]), to the best of author’s knowledge, only one publication dealing with the fractional differential equations with the GFDs in the Riemann-Liouville sense has been published until now.”
- 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]).”