The Havriliak–Negami susceptibility as a nonlinear and nonlocal process

A Numerical Scheme Based on the Chebyshev Functions to Find Approximate Solutions of the Coupled Nonlinear Sine-Gordon Equations with Fractional Variable Orders

“…Some applications of the Prabhakar function can be seen in mathematics and physics as a fractional Poisson process [16], Havriliak-Negami relaxation functions [18,19], irregular case of the dielectric relaxation responses [20], a model of anomalous relaxation in dielectrics of fractional order [21], fractional thermoelasticity [10], telegraph equations [22], thermodynamics [23], and fractal time random [24]. By placing α 1 ðp, qÞ = 2, α 2 ðp, qÞ = 2 in Equation 3, the coupled nonlinear sine-Gordon equations of fractional variable orders given in (3) change into the classical coupled nonlinear sine-Gordon equations which are defined by Equation (2), and the classical coupled nonlinear sine-Gordon equations have many applications in physics as nonlinear models [25,26], plasma [27], quantum [28], optics [29], and mathematics [13,30,31].…”

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

“…Most of the interest in the Prabhakar function is related to the description of relaxation and response in anomalous dielectrics of Havriliak-Negami type (e.g., see [24,69,56]), a model of complex susceptibility introduced to keep into account the simultaneous nonlocality and nonlinearity observed in the response of disordered materials and heterogeneous systems [49]. Further applications of the Prabhakar function are however encountered in probability theory [29,36,61], in the study of stochastic processes [14,62] and of systems with strong anisotropy [10,73], in fractional viscoelasticity [26], in the solution of some fractional boundary-value problems [5,6,16,18,46], in the description of dynamical models of spherical stellar systems [3] and in connection with other fractional or integral differential equations [4,45,39].…”

Asymptotic Stability of Distributed-Order Nonlinear Time-Varying Systems with the Prabhakar Fractional Derivatives

“…Our interest in choosing this type of derivative is related to the three-parameter Mittag-Leffler function. One useful application of the three-parameter Mittag-Leffler function in mathematics has been related to their importance in fractional calculus as a model of complex susceptibility in the response of disordered materials and heterogeneous systems [12], in the response in anomalous dielectrics of Havriliak-Negami type [13], in fractional viscoelasticity [14], in the discussion of stochastic processes [15], in probability theory [16], in the description of dynamical models of spherical stellar systems [17], in the polarization processes in Havriliak-Negami models [13,18], and in fractional or integral differential equations [19]. In this paper, we intend to survey the stability or asymptotic stability analysis of a distributed-order fractional differential/integral operator containing the Prabhakar fractional derivatives.…”