On a fractional generalization of the free electron laser equation

Abstract

The following fractional generalization of the free electron laser equation is investigated: D τ αh(τ)=λ∫ 0 τt δh(τ−t)Φ(b;δ+1; iνt) dt+μτ γΦ(β,γ+1; iντ), 0⩽τ⩽1, where β, γ, λ∈ C; ν, b, β∈ R, α>0, γ>−1 and δ>−1. A closed form solution is derived in terms of Kummer’s function Φ( α, β; z) by the application of Riemann–Liouville fractional integration operators. Tau method approximation is used in the evaluation of the results in a suitable form for numerical computation. The results derived are of general character and provide extension of the work reported by Boyadjiev et al. and Al-Shammery et al. [Math. Comput. Model. 25 (1997) 1; Integral Transform. Spec. Funct. 9 (2000) 81; J. Fract. Cal. Appl. Anal. 2 (1999) 501].