Solution of Volterra Integro-Differential Equations with Generalized Mittag-Leffler Function in the Kernels

The present paper is intended for the investigation of the integro-differential equation of the formwith complex α, ρ, µ, γ and ω (Re (α), Re (ρ), Re (µ) > 0) in the space of summable functions L(a, b) on a finite interval [a, b] of the real axis.Here D α a+ is the operator of the Riemann-Liouville fractional derivative of complex order α (Re (α) > 0) and E γ ρ,µ (z) is the function defined bywhere, when γ = 1, E 1 ρ,µ (z) coincides with the classical Mittag-Leffler function E ρ,µ (z), and in particular E 1,1 (z) = e z .Thus, when f (x) ≡ 0, a = 0, α = 1, µ = 1, γ = 0, ρ = 1, λ = -iπg, ω = iν, g and ν are real numbers, the equation ( * ) describes the unsaturated behavior of the free electron laser.The Cauchy-type problem for the above integro-differential equation is considered.It is proved that such a problem is equivalent to the Volterra integral equation of the second kind, and its solution in closed form is established.Special cases are investigated. Introduction.It is well known that solutions of integrodifferential equations of Volterra type can be obtained as solutions of