Fractional Derivatives, Memory Kernels and Solution of a Free Electron Laser Volterra Type Equation

Marcello Artioli,Simonetta Pagnutti,Giuseppe Dattoli,Silvia Licciardi

doi:10.3390/math5040073

Abstract

The high gain free electron laser (FEL) equation is a Volterra type integro-differential equation amenable for analytical solutions in a limited number of cases. In this note, a novel technique, based on an expansion employing a family of two variable Hermite polynomials, is shown to provide straightforward analytical solutions for cases hardly solvable with conventional means. The possibility of extending the method by the use of expansion using different polynomials (two variable Legendre like) expansion is also discussed.

Highlights

A novel technique, based on an expansion employing a family of two variable Hermite polynomials, is shown to provide straightforward analytical solutions for cases hardly solvable with conventional means
The constituting elements of the device are a beam of high energy electrons, injected into an alternating magnetic field where they emit laser like radiation whose growth is ruled by the following integro differential equation [1,2,3]
The relevant kernel can be viewed as a memory operator, it is associated with the nature of the physics it describes, namely the amplification of a laser field

Summary

Introduction

The free electron laser (FEL) is a device capable of transforming the kinetic energy of a releativistic electron beam into electromagnetic radiation. The constituting elements of the device are a beam of high energy electrons, injected into an alternating magnetic field where they emit laser like radiation whose growth is ruled by the following integro differential equation [1,2,3]. The FEL equation reported in (1) belongs to the Volterra integro-differential family. It has playd an important role in the design of modern FEL devices, it is the most elementary version of the FEL linear intensity growth equation. Following such a point of view the authors of references [7,8] have proposed a further definition of fractional derivative without a singular kernel, defined as. Even though the method is efficient and fast converging the integrals cannot be done analytically but requires a numerical treatment, which increases the computation times when a larger number of terms is involved

Using the Hermite Expansion

Implementation and Comparison

Conclusions