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 purpose of this article and companion ones is to present a new approach to generalizations of Stirling numbers of the first and the second kind in terms of fractional calculus analysis by using differences and differentiation operators of fractional order. Such an approach allows us to extend the classical Stirling numbers of the first and the second kind in a natural way not only to any positive, but also to any negative order. Moreover, an application of the fractional approach gives us the opportunity to extend the classical Stirling numbers to more general complex functions. In the present article we extend the classical Stirling numbers of the second kind, S(n, k), for the first parameter from a nonnegative integer number n to any complex α. Such constructions, S(α, k), will be defined for any complex α and by when k >0, while S(0, 0) = 1 and when k = 0. We show that S(α, k) with positive α can be represented by the Liouville and Marchaud fractional derivatives of the exponential functio...

Stirling Functions of the Second Kind in the Setting of Difference and Fractional Calculus

This paper studies the periodic functions in the perspective of fractional calculus application. It is shown that the fractional derivative of a periodic signal is periodic if it is defined on the whole real line. Several common fractional derivative formulations are considered, namely the Grunwald–Letnikov, Liouville and Caputo on $$\mathbb {R}$$
 , and the two-sided fractional derivatives. It is verified that the fractional derivative of a causal periodic signal is never causal periodic. The periodic behaviour of the fractional linear systems is also studied. If such systems are defined with suitable derivatives the output corresponding to periodic input is also periodic. Is is concluded that only the integer order linear systems can have a sinusoidal impulse response.

Fractional derivatives and periodic functions

This manuscript presents the basic general theory for sequential linear fractional differential equations, involving the well known Riemann-Liouville fractional operators, D a+ (a ∈ R, 0 < α ≤ 1) Lnα(y) = " D a+ + n−1 X k=0 ak(x)D a+ # (y) = y + n−1 X k=0 ak(x)y (kα = f(x). (1) where {ak(x)}n−1 k=0 are continuous real functions defined in [a, b] ⊂ R and D a+ = D a+ D a+ = D a+D a+ . (2) We also consider the case where f(x) is a continuous real function in (a, b] ⊂ R and f(a) = o(xα−1). We then introduce the Mittag-Leffler type function e α , which we will call αexponential. This function is the product of a Mittag-Leffler function and a power function. This function allows us to directly obtain the general solution to homogeneous and non-homogeneous linear fractional differential equations with constant coefficients. This method is a variation of the usual one for the ordinary case.

Linear Differential Equations of Fractional Order

Krätzel Function as a Function of Hypergeometric Type

In this paper, we obtain formulas of fractional integration (of Marichev– Saigo–Maeda type) of the generalized multi-index Mittag-Leffler functions Eγ,κ[(αj,βj)m; z] generalizing 2m-parametric Mittag-Leffler functions studied by Saxena and Nishimoto (J. Fract. Calc.37 (2010] 43–52). Some interesting special cases of our main results are considered too.

Juan J. Trujillo

Papers

Stirling Functions of the Second Kind in the Setting of Difference and Fractional Calculus

Fractional derivatives and periodic functions

Linear Differential Equations of Fractional Order

Krätzel Function as a Function of Hypergeometric Type

Certain fractional integral operators and the generalized multi-index Mittag-Leffler functions