Abstract:We report study exploring the fractional Drude model in the time domain, using fractional derivatives without singular kernels, Caputo-Fabrizio (CF), and fractional derivatives with a stretched Mittag-Le er function. It is shown that the velocity and current density of electrons moving through a metal depend on both the time and the fractional order 0 < ≤ 1. Due to non-singular fractional kernels, it is possible to consider complete memory effects in the model, which appear neither in the ordinary model, nor in the fractional Drude model with Caputo fractional derivative. A comparison is also made between these two representations of the fractional derivatives, resulting a considered difference when < 0.8.

This paper presents new solutions for twodimensional projectile motion in a free and resistive medium, obtained within the newly established conformable derivative. For free motion, we obtain analytical solutions and show that the trajectory, height, flight time, optimal angle, and maximum range depend on the order of the conformable derivative, 0 < γ ≤ 1. Likewise, we analyse and simulate the projectile motion in a resistive medium by assuming several scenarios. The obtained trajectories never exceed the ordinary ones, given by γ = 1, unlike results reported in other studies.

This work studies a mechanical system composed of two‐spring coupled masses arrangement without damping using fractional derivatives under the Caputo sense. We determine and implement an explicit model based on the Caputo‐Fabrizio operator. To achieve this model, we detail a systematic methodology that avoids dimensional inconsistencies. Then, we use the proposed model to explain the system dynamic acquired from an experimental (quite rudimentary) setup. We also compare our results to those achieved from the ordinary model and the numerical implementation using the Caputo derivative definition. Results prove that the proposed model describes the dynamic of this mechanical system better than the ordinary and numerical models. We also show that the fractional order allows the model (obtained from an idealized undamped system) to describe dynamics from fully dissipative to wholly oscillatory. Moreover, we discuss other exciting characteristics of the obtained responses.

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