Series solutions of a fractional oscillator by means of the homotopy perturbation method

Abstract

In this paper, we present a framework to obtain analytical solutions to a fractional oscillator by the homotopy perturbation method. The equation of motion of a driven fractional oscillator is obtained from the corresponding equation of motion of a driven harmonic oscillator by replacing the second-order time derivative by a fractional derivative of order α with 0<α≤2. The fractional derivative is described in the Caputo sense. Some examples are tested, and the results reveal that the technique introduced here is very effective and convenient for solving a fractional oscillator. The response characteristics of the fractional oscillator are studied for several cases.

Keywords:

• homotopy perturbation method
• fractional oscillator
• Caputo derivative

2000 AMS Subject Classifications::

• 65L05
• 41A58

Acknowledgements

The authors are grateful to the referees for their invaluable suggestions and comments for the improvement of the paper.