The linearly damped free fractional mechanical oscillator equation is solved by Laplace Transform method and series solution technique. In both methods, the solution is expressed in terms of the Mittag-Leffler function defined by The Rieman-Liouville and Caputo’s formulations of the fractional differentiation are both considered. The parameters carry over their meanings from discrete calculus as the damping coefficient and circular frequency respectively, is the order of the fractional derivative. The damping coefficient is a measure of resistive force present in the medium through which the oscillator vibrates while the resonant frequency is its natural frequency in the absence of external excitations.
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