Damped oscillations in view of the fractional oscillator equation

Abstract

This paper discusses the fractional oscillator equation involving fractional time derivatives of the Riemann-Liouville type. The exact solution of the fractional oscillator equation was obtained. On this basis the correspondence between the fractional time derivative and the dissipative properties was established. The relationship between the order of the fractional time derivative and the dissipative constant was derived. In addition to the exact solution, the perturbation approach was developed as well. It was shown that the perturbation method and exact solution lead to similar results. On the basis of the exact solution, the concept of fractional time evolution was illustrated.