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 paper is devoted to study the Cauchy-type problem for the nonlinear differential equation of fractional order a 6 C, Re(a) > 0, (D°+y)(x) = f[x,y(x)] (n 1 < Re(a) < n, n = [ R e ( a ) ] ) , (.D«-ky)(a+) = bk! bk e C (k = 1,2,..., n), containing the Riemann-Liouville fractional derivative D\"+y, on a finite interval [a, 6] of the real axis R = (—oo, oo) in the space of summable functions L(a, b). The equivalence of this problem and the nonlinear Volterra integral equation is established. The existence and uniqueness of the solution y(x) of the above Cauchy-type problem is proved by using the method of successive approximations. The corresponding assertions for the ordinary differential equations are presented. Examples are given.

Existence and uniqueness theorems for nonlinear fractional differential equations

In this article, we obtain the solutions of some fractional differential systems with different orders using the inverse operator method and Mittag–Leffler function. Moreover, the controllability of time-invariant linear fractional system is investigated. Some examples are provided to illustrate the theory.

Juan J. Trujillo

Papers

Existence and uniqueness theorems for nonlinear fractional differential equations

Note on controllability of linear fractional dynamical systems

Exponentials and Laplace transforms on nonuniform time scales

Existence results for an impulsive fractional integro-differential equation with state-dependent delay

α-Analytic solutions of some linear fractional differential equations with variable coefficients