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

This paper considers the Riemann-Liouville fractional operator as a tool to reduce linear ordinary equations with variable coefficients to simpler problems, avoiding the singularities of the original equation. The main result is that this technique allow us to obtain an extension of the classical integral representation of the special functions related with the original differential equations. In particular, we will use as examples the cases of the well-known Generalized, Gauss and Confluent Hypergeometric equations, Laguerre equation, Hermite equation, Legendre equation and Airy equation.

Fractional operators and some special functions

In this paper a method based on Sinc approximation is developed for the numerical solution of a nonlinear fractional pantograph equation. In order to use Sinc approximation, the problem needs to have an analytic solution. So we investigated the existence and uniqueness of analytic solutions in the proposed domain. Single and double exponential transformations are used to approximate the solution. Some test problems are given to demonstrate the efficiency and applicability of the methods. The results are compared with some other existing numerical methods to show the performance and good accuracy of the methods. The numerical orders of convergence show that the method has exponential rate of convergence.

Existence and numerical simulation of solutions for nonlinear fractional pantograph equations

This paper deals with some existence and Ulam stability results for a class of partial integral equations via Hadamard’s fractional integral, by applying Schauder’s fixed-point theorem.

/pdf/ulam-stabilities-for-partial-hadamard-fractional-integral-5dgyijirlz.pdf

Ulam stabilities for partial Hadamard fractional integral equations

The purpose of this paper is to present a new approach to generalizations of Stirling numbers of the first kind by the application of differential and integration operators of fractional order and generalized, infinite differences. Such an approach allows us to extend the classical Stirling numbers of the first kind, s(n, k), to functions s(α, β), where both parameters n, k have been extended to complex α, β. Under such a construction the s(α, β) turn out to have the series representation—a major result of this paper for , with for any when β = 0. Various properties of the new Stirling functions are established, most generalize those known for the numbers s(n, k); some are new, i.e. a multiple sum formula for s(α, k), and an interesting connection between the s(α, β) and the Riemann zeta function for complex β with . Several connections between the s(α, β) and the Stirling functions of second kind, s(α, β), studied earlier by the authors, are deduced. Thus the s( − n, β) coincide with the Stirling functio...

Juan J. Trujillo

Papers

Fractional operators and some special functions

Existence and numerical simulation of solutions for nonlinear fractional pantograph equations

Ulam stabilities for partial Hadamard fractional integral equations

Stirling functions of first kind in the setting of fractional calculus and generalized differences

An application of Szegö quadratures to the computation of the Fourier transform