A Comparative Study of Exact and Fractional Derivative Model via Mittag–Leffler Function

This paper is a comparative analysis of classical and fractional derivatives models using Mittag-Leffler function. The Caputo fractional derivative is used to generalize the classical exponential decay model in order to include memory effects. Transform methods are used to obtain solutions of the forms of the Mittag-Leffler function. Numerical simulations are performed to analyze the behavior of the system for different fractional orders. The findings reveal that the fractional model generalizes the classical solution and it features slower decay because of memory effects. The analysis is further generalized to second order system, in which the system results in a damped oscillatory behavior due to the presence of a fractional dynamics. The results indicate the significance of the fractional calculus in the modeling of complicated physical systems.