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

The generalized Grunwald–Letnikov fractional derivative is analyzed in this paper. Its Laplace and Fourier transforms are computed, and some current results are criticized. It is shown that only the forward derivative of a sinusoid exists. This result is used to define the frequency response of a fractional linear system.

Generalized Grünwald–Letnikov Fractional Derivative and Its Laplace and Fourier Transforms

Computation of fractional integrals via functions of hypergeometric and Bessel type

The generalized incremental ratio fractional derivative is revised and its main properties deduced. It is shown that in the case of analytic functions, it enjoys some interesting properties like: linearity and causality and has a semi-group structure. Some simple examples are presented. The enlargement of the set of functions for which the group properties of the fractional derivative are valid is done. With this, it is shown that some well-known results are valid in a more general set-up. Some examples are presented.

The incremental ratio based causal fractional calculus

Theory of System of Nonlinear Fractional Differential Equations

In this paper, we establish sufficient conditions for the global relative controllability of nonlinear neutral fractional Volterra integro-differential systems with distributed delays in control. The results are obtained by using the Mittag– Leffler functions and the Schauder fixed-point theorem. Examples are presented to illustrate the results. Copyright © 2015 John Wiley & Sons, Ltd.

Juan J. Trujillo

Papers

Generalized Grünwald–Letnikov Fractional Derivative and Its Laplace and Fourier Transforms

Computation of fractional integrals via functions of hypergeometric and Bessel type

The incremental ratio based causal fractional calculus

Theory of System of Nonlinear Fractional Differential Equations

Relative controllability of nonlinear neutral fractional integro-differential systems with distributed delays in control