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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.

45J05Integro-ordinary differential equations
26A33Fractional derivatives and integrals (real functions)
33E12Mittag-Leffler functions and generalizations
45D05Volterra integral equations
Full Text: DOI
[1] A.H. Al-Shammery, S.L. Kalla and H.G. Khajah, A fractional generalization of the free electron laser equation , Frac. Calc. Appl. Anal. 2 (1999), 501-508. · Zbl 1033.65118
[2] --------, On a generalized fractional integro-differential equation of Volterra-type , Integral Transform. Spec. Funct. 9 (2000), 81-90. · Zbl 0962.45002 · doi:10.1080/10652460008819246
[3] L. Boyadjiev and H.-J. Dobner, On the solution of fractional integro-differential equation of Volterra type , Frac. Calc. Appl. Anal. 1 (1998), 385-400. · Zbl 1033.45005
[4] --------, Analytical and numerical treatment of a multidimensional fractional FEL equation , Appl. Anal. 70 (1998), 1-18. · Zbl 1020.65101 · doi:10.1080/00036819808840671
[5] L. Boyadjiev, H.-J. Dobner and S.L. Kalla, A fractional integro-differential equation of Volterra type , Math. Comput. Modelling 28 (1998), 103-113. · Zbl 0993.65153 · doi:10.1016/S0895-7177(98)00158-7
[6] L. Boyadjiev and S.L. Kalla, Series representations of analytic functions and applications , Frac. Calc. Appl. Anal. 1 (2001), 379-408. · Zbl 1033.30002
[7] L. Boyadjiev, S.L. Kalla and H.G. Khajah, Analytical and numerical treatment of a fractional integro-differential equation of Volterra type , Math. Comput. Modelling 25 (1997), 1-9. · Zbl 0932.45012 · doi:10.1016/S0895-7177(97)00090-3
[8] G. Dattoli, L. Gianessi, L. Mezi, D. Tocci and R. Colai, FEL time-evolution operator , Nucl. Instr. Methods A 304 (1991), 541-544.
[9] G. Dattoli, S. Lorezutta, G. Maino, D. Tocci and A. Torre, Results for an integro-differential equation arising in a radiation evolution problem , in Proc. Internat. Workshop on Dynamics of Transport in Plasmas and Charged Beams (Torino, July 1994), World Scientific, Singapore, 1995, pp. 202-221.
[10] G. Dattoli, S. Lorezutta, G. Maino and A. Torre, Analytic treatment of the high-gain free electron laser equation , Radiat. Phys. Chem. 48 (1996), 29-40.
[11] M.M. Dzherbashyan, Integral transforms and representations of functions in complex domain , Nauka, Moscow, 1966. (Russian)
[12] --------, Harmonic analysis and boundary value problems in the complex domain , Birkhauser Verlag, Basel, 1993. · Zbl 0798.43001
[13] A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher transcendental functions , Vol. I, McGraw-Hill, New York, 1953. · Zbl 0052.29502
[14] --------, Higher transcendental functions , Vol. III, McGraw-Hill, New York, 1954. · Zbl 0056.28104
[15] G. Gripenberg, S.-O. Londen and O. Staffans, Volterra integral and functional equations , Cambridge Univ. Press, Cambridge, 1990. · Zbl 0695.45002
[16] A.A. Kilbas, B. Bonilla and J.J. Trujillo, Existence and uniqueness theorems for nonlinear fractional differential equations , Demonstratio Math. 33 (2000), 583-602. · Zbl 0964.34004
[17] A.A. Kilbas, M. Saigo and R.K. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus operators , Integral Transform. Spec. Funct., · Zbl 1047.33011 · doi:10.1080/10652460310001600717
[18] M.L. Krasnov, A.I. Kiselev and G.I. Makarenko, Integral equations , Nauka, Moscow, 1976. (Russian) · Zbl 0217.15601
[19] G.L. Ortiz, The tau method , SIAM J. Numer. Anal. 6 (1969), 480-492. JSTOR: · Zbl 0195.45701 · doi:10.1137/0706044 · http://links.jstor.org/sici?sici=0036-1429%28196909%296%3A3%3C480%3ATTM%3E2.0.CO%3B2-X&origin=euclid
[20] T.R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel , Yokohama Math. J. 19 (1971), 7-15. · Zbl 0221.45003
[21] L.B. Rall, Computational solution of nonlinear operator equations , John Wiley and Sons, New York, 1969. · Zbl 0175.15804
[22] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional integrals and derivatives , Theory and applications , Gordon and Breach, Yverdon and New York, 1993. · Zbl 0818.26003