Novel analytical solutions of the fractional Drude model

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.