Abstract
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.
Highlights
In functional analysis, the integral operators with the weakly singular kernels have been an important topic for research for many years
As shown in [22], the functions that satisfy the Sonine condition (25) cannot be continuous at the point t = 0 and the “new fractional derivatives” with the continuous kernels introduced recently in the Fractional Calculus (FC) literature do not belong to the class of the general fractional derivative (GFD) that are discussed in this paper
Following [14], in this subsection, we address a nonlinear fractional differential equation with the GFD of the Caputo type (DCk u)(t) = f (t, u(t)), t > 0 (43)
Summary
The integral operators with the weakly singular kernels have been an important topic for research for many years. In [7] (see the references therein), the abstract Volterra integral equations including the operators (13) with the completely positive kernels k ∈ L1 (0, T ) have been studied This class of the kernels can be characterized as follows: A function k ∈ L1 (0, T ) is completely positive on [0, T ] if and only if there exist a ≥ 0 and l ∈ L1 (0, T ), non-negative and non-increasing, satisfying the relation a k(t) +. No construction of the corresponding fractional integral was presented and no conditions that ensure the physically relevant properties of solutions to the time-fractional differential equations including these derivatives were suggested. Some important results from the recent publications [17,18,19] regarding inverse problems for the fractional differential and integral equations with the GFD are shortly presented
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
