SOLUTION OF VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITH GENERALIZED MITTAG-LEFFLER FUNCTION IN THE KERNELS

• Published in: The Journal of integral equations and applications 2002-12, Vol.14 (4), p.377-396
• Main Authors: KILBAS, ANATOLY A., SAIGO, MEGUMI, SAXENA, R.K.
• Format: Article
• Language: English
• Subjects: 26A33; 33E12; 45J05; Approximation; Differential equations; Free electron lasers; generalized Mittag-Leffler function; Integro-differential equation of Volterra type; Mathematical functions; Mathematical integrals; Mathematics; method of successive approximations; RiemannLiouville fractional integrals and derivatives; Series convergence; Special functions; Volterra equations
• ISSN: 0897-3962; 1938-2626
• DOI: 10.1216/jiea/1181074929

The present paper is intended for the investigation of the integro-differential equation of the form $\begin{array}{c}(*) \quad \left({\mathcal{D}}_{\mathrm{a}+}^{\mathrm{a}}\mathrm{y}\right)\left(\mathrm{x}\right)=\mathrm{\lambda }{\int }_{\mathrm{a}}^{\mathrm{x}}(\mathrm{x}-\mathrm{t}{)}^{\mathrm{\mu }-1}{\mathrm{E}}_{\mathrm{\rho },\mathrm{\mu }}^{\mathrm{\gamma }}\left[\mathrm{\omega }\right(\mathrm{x}-\mathrm{t}{)}^{\mathrm{\rho }}\left]\mathrm{y}\right(\mathrm{t})\mathrm{d}\mathrm{t}+\mathrm{f}(\mathrm{x}),\\ \mathrm{a} 0) in the space of summable functions L(a, b) on a finite interval [a, b] of the real axis. Here ${\mathrm{D}}_{\mathrm{\alpha }+}^{\mathrm{\alpha }}$ is the operator of the Riemann-Liouville fractional derivative of complex order α (Re (α) > 0) and ${\mathrm{E}}_{\mathrm{\rho },\mathrm{\mu }}^{\mathrm{\gamma }}\left(\mathrm{z}\right)$ is the function defined by ${\mathrm{E}}_{\mathrm{\rho },\mathrm{\mu }}^{\mathrm{\gamma }}\left(\mathrm{z}\right)=\sum _{\mathrm{k}=0}^{\mathrm{\infty }}\frac{(\mathrm{\gamma }{)}_{\mathrm{k}}}{\mathrm{\Gamma }(\mathrm{\rho }\mathrm{k}+\mathrm{\mu })}\frac{{\mathrm{z}}^{\mathrm{k}}}{\mathrm{k}!}$, where, when ${\mathrm{E}}_{\mathrm{\rho },\mathrm{\mu }}^{1}\left(\mathrm{z}\right)$ coincides with the classical Mittag-Leffler function Eρ,μ(z), and in particular E1,1(z) = ez. 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.