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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.

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