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Solution of Volterra integrodifferential equations with generalized Mittag-Leffler function in the Kernels. (English) Zbl 1041.45011

Summary: The present paper is intended for the investigation of the integro-differential equation of the form

(𝒟 a+ α y)(x)=λ a x (x-t) μ-1 E ρ,μ γ ω (x-t) ρ y(t)dt+f(x),a<xb,(*)

with 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 a+ α is the operator of the Riemann-Liouville fractional derivative of complex order α(Re(α)>0) and E ρ,μ γ (z) is the function defined by

E ρ,μ γ (z)= k=0 (γ) k Γ(ρk+μ)z k k!,

where, when γ=1, E ρ,μ 1 (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.

45J05Integro-ordinary differential equations
26A33Fractional derivatives and integrals (real functions)
33E12Mittag-Leffler functions and generalizations
45D05Volterra integral equations