**Preprint No.**
A-98-08

#
Yuri Luchko, Rudolf Gorenflo

## The initial value problem for some fractional differential equations with the Caputo derivatives

**Abstract:** In the present paper, we first develop the operational calculus of Mikusinski's
type for the Caputo fractional differential operator. This calculus is used to optain
exact solutions of an initial value problem for linear fractional differential
equations with constant coefficients and fractional derivatives in Caputo's sense.
The initial conditions are given in terms of the field variable and its derivatives of
integer order. The optained solutions are expressed through the Mittag-Leffler type functions.
Special cases and integral representations of solutions are presented.
**Keywords: **fractional differential equations, operational calculus, Caputo fractional derivative, Mittag-Leffler type functions

**Mathematics Subject Classification (MSC91):** 45J05 Integro-ordinary differential equations
, 44A40 Calculus of Mikusinski and other operational calculi

**Language:** ENG
**Available:** Pr-A-98-08.ps

**Contact**: Luchko, Yuri; Gorenflo, Rudolf ; Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2-6, D-14195 Berlin, Germany (luchko@math.fu-berlin.de)

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