Dynamics of Systems with Memory: A Fractional Calculus Approach to the Harmonic Oscillator

This work presents a pedagogical yet mathematically rigorous approach to Fractional Differential Equations (FDEs), focusing on the Caputo fractional derivative. By extending the classical second-order differential equation of the harmonic oscillator to a non-integer order ($1 < \alpha \le 2$), the paper formally demonstrates how the non-local nature of the fractional operator introduces damping and intrinsic energy dissipation.

The main result lies in demonstrating that this damping occurs without the need to add explicit phenomenological friction terms proportional to velocity. Using the Laplace transform, we derive the exact analytical solution of the system, expressed in terms of Mittag-Leffler functions.