Analysis of Drude model using fractional derivatives without singular kernels

Abstract

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.

A novel generalized fractional‐order memristor model with fully explicit memory description

“…where 0 ≤ α < 1, B(α) denotes a normalization function obeying B(0) = B(1) = 1, and E α ½ stands for the Mittag-Leffler function, 30 which has been applied for mathematically describing the dynamics of various phenomena in practice, e.g., oscillation of human liver, 2 heat transfer of Casson nanofluids, 14 electrical conductivity of metal, 27 flow of currents in electrical circuits, 31 and option pricing in financial business. 33 By applying the Mittag-Leffler function as the kernel of both fractional derivatives, the singularity at the end point of the integrating interval, i.e., τ ¼ t, can be avoided because lim…”

“…Note also that such Atangana-Baleanu fractional derivative has been applied, cited, and studied in many research areas ranged from electrical engineering to epidemiology. 27,[31][32][33][34][35][36][37][38][39][40][41][42][43][44] In particular, we choose the Atangana-Baleanu fractional derivative in Liouville-Caputo sense rather than the derivative in Riemann-Liouville sense. This is because the former takes the initial condition into its Laplace transformation, 30,33 and we solve the fractional-order memristor's state equation of by means of the Laplace transformation-based methodology.…”

“…It should be mentioned here that such disadvantage occurs because these fractional‐order memristor models are relied on the Riemann‐Liouville and Caputo fractional derivatives, which are based on the power law function kernels. Since these power law kernel‐based fractional derivatives employ singularity at the end of the integrating interval, 27 they can also be referred to as the singular kernel fractional derivatives. At the singularity, the value of such power law kernel function at singular point cannot be explicitly defined as it approaches infinity, thus so does that of the derivative.…”

“…For obtaining such fully explicit memory description, a nonsingular kernel fractional derivative must be adopted as the basis. This is because the kernel of such fractional derivative does not have any singularity 27 ; thus, its value and so does that of the derivative can always be explicitly defined. As a result, a fully explicit memory description of the fractional‐order memristor can be obtained.…”

“…Between two well‐known nonsingular kernel fractional derivatives, i.e., the Caputo‐Fabrizio fractional derivative 29 and the Atangana‐Baleanu fractional derivative 30 ; we choose the latter as it is a generalization of the former and also nonlocal. Note also that such Atangana‐Baleanu fractional derivative has been applied, cited, and studied in many research areas ranged from electrical engineering to epidemiology 27,31–44 . In particular, we choose the Atangana‐Baleanu fractional derivative in Liouville‐Caputo sense rather than the derivative in Riemann‐Liouville sense.…”