There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Fractional Differential Equations

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

A new definition of fractional derivative

In the paper, we present a new definition of fractional deriva tive with a smooth kernel which takes on two different representations for the temporal and spatial variable. The first works on the time variables; thus it is suitable to use th e Laplace transform. The second definition is related to the spatial va riables, by a non-local fractional derivative, for which it is more convenient to work with the Fourier transform. The interest for this new approach with a regular kernel was born from the prospect that there is a class of non-local systems, which have the ability to descri be the material heterogeneities and the fluctuations of diff erent scales, which cannot be well described by classical local theories or by fractional models with singular kernel.

http://www.naturalspublishing.com/files/published/0gb83k287mo759.pdf

A new Definition of Fractional Derivative without Singular Kernel

https://theory.sinp.msu.ru/~tarasov/PDF/AP2013.pdf

Fractional power-law spatial dispersion in electrodynamics

Controllability of fractional damped dynamical systems

Editorial: Fractional signals and systems

This paper establishes a set of sufficient conditions for the controllability of nonlinear fractional dynamical system of order 1<α<2 in finite dimensional spaces. The main tools are the Mittag–Leffler matrix function and the Schaefer’s fixed-point theorem. An example is provided to illustrate the theory.

Juan J. Trujillo

Papers

Fractional power-law spatial dispersion in electrodynamics

Controllability of fractional damped dynamical systems

Editorial: Fractional signals and systems

Controllability Results for Nonlinear Fractional-Order Dynamical Systems

From a generalised Helmholtz decomposition theorem to fractional Maxwell equations