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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.

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