An operational method for solving fractional differential equations with the Caputo derivatives. (English) Zbl 0931.44003

For the Caputo fractional differential operator \(D^\mu_*\) defined by \(D^\mu_*f= J^{m-\mu} f^{(m)}\) with \(m-1< \mu\leq m\in \mathbb{N}\), where \(J\) is the integral operator, an operational calculus is developed. The calculus is used in order to solve fractional differential equations with constant coefficients and given initial values. The explicit solutions are expressed by means of Mittag-Leffler type functions \(E_{(\cdot), \beta}(x)\) in form of a series or an integral representation. Two examples are given.


44A40 Calculus of Mikusiński and other operational calculi
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
45J05 Integro-ordinary differential equations
26A33 Fractional derivatives and integrals
33E30 Other functions coming from differential, difference and integral equations