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

Abstract The Caputo fractional derivative is one of the most used definitions of a fractional derivative along with the Riemann-Liouville and the GrünwaldLetnikov ones. Whereas the Riemann-Liouville definition of a fractional derivative is usually employed in mathematical texts and not so frequently in applications, and the Grünwald-Letnikov definition – for numerical approximation of both Caputo and Riemann-Liouville fractional derivatives, the Caputo approach appears often while modeling applied problems by means of fractional derivatives and fractional order differential equations. In the mathematical texts and applications, the so called Erdélyi-Kober (E-K) fractional derivative, as a generalization of the Riemann-Liouville fractional derivative, is often used, too. In this paper, we investigate some properties of the Caputo-type modification of the Erdélyi-Kober fractional derivative. The relation between the Caputo-type modification of the E-K fractional derivative and the classical E-K fractional derivative is the same as the relation between the Caputo fractional derivative and the RiemannLiouville fractional derivative, i.e. the operations of integration and differentiation are interchanged in the corresponding definitions. Here, some new properties of the classical Erdélyi-Kober fractional derivative and the respective ones of its Caputo-type modification are presented together.

Caputo-Type Modification of the Erdélyi-Kober Fractional Derivative

In this paper, we establish the controllability for a class of abstract impulsive mixed-type functional integro-differential equations with finite delay in a Banach space. Some sufficient conditions for controllability are obtained by using the Monch fixed point theorem via measures of noncompactness and semigroup theory. Particularly, we do not assume the compactness of the evolution system. An example is given to illustrate the effectiveness of our results.

/pdf/controllability-results-for-impulsive-mixed-type-functional-na92a08574.pdf

Controllability results for impulsive mixed-type functional integro-differential evolution equations with nonlocal conditions

The objective of this paper is to show an approach to the fractional version of the Sturm-Liouville problem, by using different fractional operators that return to the ordinary operator for integer order. For each fractional operator we study some of the basic properties of the Sturm-Liouville theory. We analyze a particular example that evidences the applicability of the fractional Sturm-Liouville theory.

A fractional approach to the Sturm-Liouville problem

Cytotoxicity, acute toxicity, and general pharmacological effects in mice of an arylnaphthalene lignan isolated from Jisticia hyssopifolia were studied. Elenoside was cytotoxic to human cancer cell lines in a range of concentrations from 10(-5) to 10(-4)M. It has an LD(50), ip in mice, of 305 mg/kg and central depressive properties at doses of 25, 50, and 100 mg/kg. Thus, elenoside appears to be a sedative with broad spectrum cytotoxicity.

General behavior, toxicity, and cytotoxic activity of elenoside, a lignan from Justicia hyssopifolia.

This paper is devoted to the study of four integral operators, which are generalizations and modifications of integrals of Hadamard, in the space X c p of Lebesgue measurable functions f on R + = (0, ∞) such that for c ∈ R = (−∞, ∞) [Formula: See Text] Representations for the operators are given in the form of integral transforms involving the Meijer G-function in the kernels. The mapping properties such as the boundedness, representation and range are established.

Juan J. Trujillo

Papers

Caputo-Type Modification of the Erdélyi-Kober Fractional Derivative

Controllability results for impulsive mixed-type functional integro-differential evolution equations with nonlocal conditions

A fractional approach to the Sturm-Liouville problem

General behavior, toxicity, and cytotoxic activity of elenoside, a lignan from Justicia hyssopifolia.

Hadamard-type integrals as G-transforms