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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 $$({\cal D}^\alpha_{a+} y)(x)= \lambda \int^x_a(x-t)^{\mu-1} E^\gamma_{\rho,\mu} \bigl[\omega(x-t)^\rho\bigr] y(t)dt +f(x),\quad a<x\le b, \tag*$$ 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 ${\cal D}^\alpha_{a+}$ 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.

##### MSC:
 45J05 Integro-ordinary differential equations 26A33 Fractional derivatives and integrals (real functions) 33E12 Mittag-Leffler functions and generalizations 45D05 Volterra integral equations
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