On a fractional generalization of the free electron laser equation

The main object of this paper is to investigate the solution of the following free electron laser equations associated with generalized MittagLeffler function and a confluent hypergeometric function in two variables: Dα τ h(τ) = ω ∫ τ 0 t β−1 h(τ − t)E σ,β(iνt σ)dt+ δf(τ), where γ, δ, ω ∈ C; ν ∈ R,α > 0, σ > 0, β > 0; E σ,β(z) is the generalized Mittag-Leffler function and Dα τ h(τ) = λ Γ(γ) ∫ τ 0 t γ−1 h(τ − t)φ2 (β, β′; γ;ωt, σt) dt+ μf(τ) under prescribed conditions. Uniqueness of the derived solutions is obtained. In the first case a closed form solution is obtained in terms of a generalized Mittag-Leffler function by the application of Riemann-Liouville fractional integration operators. The results obtained are general in character and provide extension of the results given earlier by Boyadjiev et al. and Al-Shammery et al. In the second case the results are derived in terms of confluent hypergeometric function of two variables. Use of FEL equation has been given by several authors and in particular by Ciocci et al. [1] and Dattoli et al. [2-3]. Our aim here is show role played by Mittag-Leffler function and operators of fractional integration in such problems. We are treating the mathematical aspect of FEL equations with operators of fractional integration and special functions.