The Prabhakar or three parameter Mittag–Leffler function: Theory and application

Abstract

The Prabhakar function (namely, a three parameter Mittag-Leffler function) is investigated. This function plays a fundamental role in the description of the anomalous dielectric properties in disordered materials and heterogeneous systems manifesting simultaneous nonlocality and nonlinearity and, more generally, in models of Havriliak-Negami type. After reviewing some of the main properties of the function, the asymptotic expansion for large arguments is investigated in the whole complex plane and, with major emphasis, along the negative semi-axis. Fractional integral and derivative operators of Prabhakar type are hence considered and some nonlinear heat conduction equations with memory involving Prabhakar derivatives are studied.

Storage and Dissipation of Energy in Prabhakar Viscoelasticity

“…Furthermore, it is rather easy to prove that some very well-known constant-Q models are nothing but some specific realizations of the model defined in Equation (1). Indeed, for example, the renowned Kjartansson model [36,37] can be obtained from (18) by setting γ = 0, β ≡ 2 η with η ∈ (0, 1/2) and a = 0, which is nothing but the Scott-Blair model [36]. In view of the last few comments, we believe that the Maxwell-Prabhakar model of viscoelasicity can potentially provide some stimulating new insights into the mathematical modelling of seismic processes and therefore is worthy of further studies.…”

“…It is important to remark that the Prabhakar fractional calculus has been attracting much attention in the mathematical community [15,17,18,[23][24][25][26], particularly because of its connection with the theoretical description of the Havriliak-Negami model [18,24,27,28]. Moreover, this growing interest in Prabhakar's calculus is also reflected by the increasing literature on the recently proposed Maxwell-Prabhakar model, which was also kindly referred to as the Giusti-Colombaro model in [29].…”

“…denotes the Prabhakar fractional integral [17] and E ρ µ,ν (t) represents the Prabhakar function [18][19][20][21]. Furthermore, it is important to stress that the function t ν−1 E ρ µ,ν (−t µ ), for t > 0, is locally integrable and completely monotone provided that 0 < µ ≤ 1 and 0 < µ ρ ≤ ν ≤ 1; see, for example, [22,23].…”

Stability and chaos control of regularized Prabhakar fractional dynamical systems without and with delay

“…38,48,55 The Prabhakar operators are not only applicable in mathematics, but also in the fractional Poisson process, 38 the fractional Maxwell model in linear viscoelasticity, 52 the generalized reaction-diffusion equations, 56 the generalized model of particle deposition in porous media 57 and the description of relaxation and response in anomalous dielectrics of the Havriliak-Negami type. [58][59][60] Garrappa 61 showed that the input-output equation of the Havriliak-Negami model in time domain, can be easily cast in terms of an integral equation involving the Prabhakar fractional integral. Further, the time-evolution of polarization processes in the Havriliak-Negami models can be suitably described by the Prabhakar derivative and integral.…”

A new Rabotnov fractional‐exponential function‐based fractional derivative for diffusion equation under external force

“…Prabhakar function: The Prabhaker function or generalized Mittag-Leffler function is defined as[54][55][56][57][58] …”