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Mathematics 2017, 5(4), 73; https://doi.org/10.3390/math5040073

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

1
ENEA—Bologna Research Center, Via Martiri di Monte Sole, 4, 40129 Bologna, Italy
2
ENEA—Frascati Research Center, Via Enrico Fermi 45, 00044 Frascati, Rome, Italy
3
Department of Mathematics and Computer Science, University of Catania, Viale A. Doria 6, 95125 Catania, Italy
*
Author to whom correspondence should be addressed.
Received: 17 October 2017 / Revised: 20 November 2017 / Accepted: 24 November 2017 / Published: 4 December 2017
(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)
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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. View Full-Text
Keywords: free electron laser (FEL); Volterra equations; iterative solutions; Hermite polynomials; Legendre polynomials free electron laser (FEL); Volterra equations; iterative solutions; Hermite polynomials; Legendre polynomials
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Artioli, M.; Dattoli, G.; Licciardi, S.; Pagnutti, S. Fractional Derivatives, Memory Kernels and Solution of a Free Electron Laser Volterra Type Equation. Mathematics 2017, 5, 73.

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