Fox <i>H</i>-Functions in Self-Consistent Description of a Free-Electron Laser

Abstract

A fractional calculus concept is considered in the framework of a Volterra type integro-differential equation, which is employed for the self-consistent description of the high-gain free-electron laser (FEL). It is shown that the Fox H-function is the Laplace image of the kernel of the integro-differential equation, which is also known as a fractional FEL equation with Caputo–Fabrizio type fractional derivative. Asymptotic solutions of the equation are analyzed as well.

Keywords

• Volterra type equation
• Caputo–Fabrizio fractional derivative
• Laplace transform
• free-electron laser
• Fox H-function