Dynamics of the fractional oscillator

The integral equation of motion of a simple harmonic oscillator is generalized by taking the integral to be of arbitrary order according to the methods of fractional calculus to yield the equation of motion of a fractional oscillator. The solution is obtained in terms of Mittag–Leffler functions using Laplace transforms. The expressions for the generalized momentum and the total energy of the fractional oscillator are also obtained. Numerical application and the phase plane representation of the dynamics are discussed.