Ary O. Chiacchio

Differentiation to fractional orders and the fractional telegraph equation

Using methods of differential and integral calculus, we present and discuss the calculation of a fractional Green function associated with the one-dimensional case of the so-called general fractional telegraph equation with one space variable. This is a fractional partial differential equation with constant coefficients. The equation is solved by means of juxtaposition of transforms, i.e., we introduce the Laplace transform in the time variable and the Fourier transform in the space variable. Several particular cases are discussed in terms of the parameters. Some known results are recovered. As a by-product of our main result, we obtain two new relations involving the two-parameter Mittag–Leffler function.

Solution of the fractional Langevin equation and the Mittag–Leffler functions

We introduce the fractional generalized Langevin equation in the absence of a deterministic field, with two deterministic conditions for a particle with unitary mass, i.e., an initial condition and an initial velocity are considered. For a particular correlation function, that characterizes the physical process, and using the methodology of the Laplace transform, we obtain the solution in terms of the three-parameter Mittag–Leffler function. As particular cases, some recent results are also presented.

One-sided and two-sided Green’s functions

We discuss the one-sided Green’s function, associated with an initial value problem and the two-sided Green’s function related to a boundary value problem. We present a specific calculation associated with a differential equation with constant coefficients. For both problems, we also present the Laplace integral transform as another methodology to calculate these Green’s functions and conclude which is the most convenient one. An incursion in the so-called fractional Green’s function is also presented. As an example, we discuss the isotropic harmonic oscillator.

A real integral by means of complex integration

This note presents and discusses a class of real integrals involving a hyperbolic function by means of complex integration. An integral representation is obtained which appears in several fields of physics, including statistical mechanics, condensed matter and quantum optics and more in the so called dispersion relations.

On an integral with two branch points

The paper considers a class of real integrals performed by using a convenient integral in the complex plane. A complex integral containing a multi-valued function with two branch points is transformed into another integral containing a pole and a unique branch point. As a by-product we obtain a new class of integrals which can be calculated in a very simple way.

An integral for the scalar propagator

Using a convenient contour in the complex plane, an integral representation is obtained for the hypergeometric function, in the case when this function does not reduce to the polynomial case. As an application, a contour integral associated with the massive scalar propagator used in Feynman diagrams is discussed.

A Note on Generalizations of a Result of Browder

Abstract In this article we extend a result of Browder to more general situations. The main tool used is a lemma due to Ky Fan.

Chebyshev centers in spaces of continuous functions

Definition et caracterisation des centres de Tchebychev dans les espaces de Banach de fonctions continues