भौतिक गणित

Relaxation Equations: Fractional Models

Abstract

Capelas de Oliveira E* and Ester CFA Rosa

The relaxation functions introduced empirically by Debye, Cole-Cole, Cole-Davidson and Havriliak-Negami are, each of them, solutions to their respective kinetic equations. In this work, we propose a generalization of such equations by introducing a fractional differential operator written in terms of the Riemann-Liouville fractional derivative of order γ, 0<γ≤1. In order to solve the generalized equations, the Laplace transform methodology is introduced and the corresponding solutions are then presented, in terms of Mittag-Leffler functions. In the case in which the derivative’s order is γ = 1, the traditional relaxation functions are recovered. Finally, we presented some 2D graphs of these function.