Complete Monotonicity of Fractional Kinetic Functions

Abstract

The introduction of a fractional differential operator defined in terms of the Riemann-Liouville derivative makes it possible to generalize the kinetic equations used to model relaxation in dielectrics. In this context such fractional equations are called fractional kinetic relaxation equations and their solutions, called fractional kinetic relaxation functions, are given in terms of Mittag-Leffler functions. These fractional kinetic relaxation functions generalize the kinetic relaxation functions associated with the Debye, Cole-Cole, Cole-Davidson and Havriliak-Negami models, as the latter functions become particular cases of the fractional solutions, obtained for specific values of the parameter specifying the order of the derivative. The aim of this work is to analyse the behavior of these fractional functions in the time variable. As theoretical tools we use the theorem by Bernstein on the complete monotonicity of functions together with Titchmarsh's inversion formula. The last part of the paper contains the graphics of some of those functions, obtained by varying the value of the parameter in the fractional differential operator and in the corresponding Mittag-Leffler functions. The graphics were made with Mathematica 10.4.