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.

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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.

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