Fractional models of anomalous relaxation based on the Kilbas and Saigo function

We revisit the Kilbas and Saigo functions of the Mittag-Leffler type of a real variable (t) , with two independent real order-parameters. These functions, subjected to the requirement to be completely monotone for (t>0) , can provide suitable models for the responses and for the corresponding spectral distributions in anomalous (non–Debye) relaxation processes, found e.g. in dielectrics. Our analysis includes as particular cases the classical models referred to as Cole–Cole (the one-parameter Mittag-Leffler function) and to as Kohlrausch (the stretched exponential function). After some remarks on the Kilbas and Saigo functions, we discuss a class of fractional differential equations of order (alpha in (0,1]) with a characteristic coefficient varying in time according to a power law of exponent (beta ) , whose solutions will be presented in terms of these functions. We show 2D plots of the solutions and, for a few of them, the corresponding spectral distributions, keeping fixed one of the two order-parameters. The numerical results confirm the complete monotonicity of the solutions via the non-negativity of the spectral distributions, provided that the parameters satisfy the additional condition (0 , assumed by us.