Role of bi-order Atangana-Aguilar fractional differentiation on Drude model: an analytic study for distinct sources

The Drude model has captured minds of many researchers in the past decades because this model is able to explain the transport properties of electrons in materials more precisely metals. This model is an application of kinetic theory that accepts the microscopic behavior of electrons in a solid may be treated characteristically and looks much like a pinball machine. This model is applied for constantly jittering electrons bouncing and re-bouncing off heavier, relatively immobile positive ions. In this manuscript, we consider the mathematical model using the concept of differential operator with two fractional orders. We employ the Laplace transform, inverse Laplace transform and the convolution theorem with analytical method to derive the exact solution of the new equations of the Drude model. Additionally, the three types of sources namely periodic, exponential and unit step are invoked on Drude model for knowing the hidden phenomena of velocity of electrons.