Novel analytical solutions of the fractional Drude model

Abstract

Alternative solutions for the equation of motion of the electrons in a metal in the presence of an external electric field are presented. Novel fractional derivatives of type Atangana–Koca–Caputo with constant and variable-order and fractional conformable derivative in the Liouville–Caputo sense were considered. The variable-order fractional derivative can be set as a smooth function, bounded on (0;1], while, the constant-order fractional derivative can be set as a fractional equation, bounded on (0;1]. We presented the exact solution using the properties of Laplace transform operator with its inversions. Numerical simulations are presented for evaluating the difference between these operators. Based on the generalized Mittag–Leffler function and power-law function with the conformable derivative, new behaviors for the optical properties in metals were obtained.