Fractional calculus has emerged as an active area of research due to its ability to model complex dynamical systems with memory effects and anomalous diffusion. In particular, the Mittag–Leffler function plays a fundamental role in solving fractional differential equations. This study aims to derive the analytical solution of the Linear Fractionally Damped Oscillator using the Laplace transform and the Mittag–Leffler function, where the derivative is of Caputo type with order 0<α<1. We further extend the analysis to both homogeneous and nonhomogeneous models, the latter corresponding to the presence of an external forcing term. The results indicate that the oscillatory behavior exhibits algebraic decay and eventual convergence due to damping or dissipation effects. The decay rate is directly influenced by the asymptotic properties of the Mittag–Leffler function, which depend on the fractional order α. These findings provide a deeper understanding of fractional-order damped oscillatory systems and offer a more generalized framework for analyzing dissipative processes in engineering, physics, and control systems.
Putra et al. (Wed,) studied this question.