Fractional driven-damped oscillator and its general closed form exact solution

New questions in fundamental physics and in other fields, which cannot be formulated adequately using traditional integral and differential calculus emerged recently. Fractional calculus was shown to describe phenomena where conventional approaches have been unsatisfactory. The driven damped fractional oscillator entails a rich set of important features, including loss of energy to the environment and resonances. In this paper, this oscillator with Caputo fractional derivatives is solved analytically in closed form. The exact solution is expressed in terms of generalized Mittag-Leffler functions. The standard driven-damped Harmonic Oscillator is recovered as a special case of non-fractional derivatives. In contradistinction to the standard oscillator, the solution of the fractional oscillator is shown to decay algebraically and to possess a finite number of zeros. Several decay patterns are uncovered and are a direct consequence of the asymptotic properties of the generalized Mittag-Leffler functions. Other interesting properties of the fractional oscillator like the momentum–position phase-plane diagrams and the time dependence of the energy terms are discussed as well.