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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 \[ ({\mathcal D}^\alpha_{a+} y)(x)= \lambda \int^x_a(x-t)^{\mu-1} E^\gamma_{\rho,\mu} \bigl[\omega(x-t)^\rho\bigr] y(t)dt +f(x),\quad a<x\leq b, \tag{*} \] with complex \(\alpha, \rho,\mu, \gamma\) and \(\omega\) \((\text{Re}(\alpha), \text{Re} (\rho), \text{Re} (\mu)>0)\) in the space of summable functions \(L(a,b)\) on a finite interval \([a,b]\) of the real axis. Here \({\mathcal D}^\alpha_{a+}\) is the operator of the Riemann-Liouville fractional derivative of complex order \(\alpha(\text{Re} (\alpha)>0)\) and \(E^\gamma_{\rho,\mu} (z)\) is the function defined by \[ E^\gamma_{\rho,\mu} (z)= \sum^\infty_{k=0} \frac{( \gamma)_k} {\Gamma(\rho k+\mu)} \frac {z^k}{k!}, \] where, when \(\gamma=1\), \(E^1_{ \rho, \mu}(z)\) coincides with the classical Mittag-Leffler function \(E_{\rho, \mu}(z)\), and in particular \(E_{1,1}(z) =e^z\). Thus, when \(f(x)\equiv 0\), \(a=0\), \(\alpha=1\), \(\mu=1\), \(\gamma=0\), \(\rho=1\), \(\lambda= -i\pi g\), \(\omega =i\nu\), \(g\) and \(\nu\) 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.

45J05 Integro-ordinary differential equations
26A33 Fractional derivatives and integrals
33E12 Mittag-Leffler functions and generalizations
45D05 Volterra integral equations
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