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Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives. (English) Zbl 1182.26011
Summary: The operational calculus is an algorithmic approach for the solution of initial-value problems for differential, integral, and integro-differential equations. In this paper, an operational calculus of the Mikusiński type for a generalized Riemann-Liouville fractional differential operator with types introduced by one of the authors is developed. The traditional Riemann-Liouville and Liouville-Caputo fractional derivatives correspond to particular types of the general one-parameter family of fractional derivatives with the same order. The operational calculus constructed in this paper is used to solve the corresponding initial value problem for the general $$n$$-term linear equation with these generalized fractional derivatives of arbitrary orders and types with constant coefficients. Special cases of the obtained solutions are presented.

##### MSC:
 26A33 Fractional derivatives and integrals 33E12 Mittag-Leffler functions and generalizations 44A40 Calculus of Mikusiński and other operational calculi 45J05 Integro-ordinary differential equations