Prabhakar-like fractional viscoelasticity

Abstract

Abstract. The aim of this paper is to present a linear viscoelastic model based on Prabhakar fractional operators. In particular, we propose a modification of the classical fractional Maxwell model, in which we replace the Caputo derivative with the Prabhakar one. Furthermore, we also discuss how to recover a formal equivalence between the new model and the known classical models of linear viscoelasticity by means of a suitable choice of the parameters in the Prabhakar derivative. Moreover, we also underline an interesting connection between the theory of Prabhakar fractional integrals and the recently introduced Caputo-Fabrizio differential operator.

Generalized Langevin Equation and the Prabhakar Derivative

“…[5]. These derivatives are applicable in the fractional Poisson process [12], for description of dielectric relaxation phenomena [15,16], in the fractional Maxwell model in the linear viscoelasticity [17], in mathematical modeling of fractional differential filtration dynamics [18], in fractional dynamical systems [19], generalized reaction-diffusion equations [20], in generalized model of particle deposition in porous media [21], etc.…”

An anomalous diffusion model based on a new general fractional operator with the Mittag-Leffler function of Wiman type

“…Atangana et al used the MLF to replace the exponent function in the integral kernel of the above definition and obtained its new form in [18]. Giusti et al suggested the Prabhakar-like fractional derivative in [31]. A family of general FDOs based on the extensions of the classical MLFs (Gösta MittagLeffler, Wiman and Prabhakar functions) were proposed in [25,28].…”

“…Similarly, the Prabhakar-like fractional derivative can return to the Caputo-Fabrizio operator and the new FDOs defined by Yang when the parameters take the particular values (see [25,28,31]). These FDOs involving the MLFs in the integral kernel have been applied to model many physical phenomena, such as the anomalous relaxation, heat-transfer problems, viscoelastic problems, Euler-Lagrange equation and the boundary value problem, extensively (see [18,[28][29][30][31][32][33][34][35][36]). Especially, the new general FDOs with the aid of Laplace transform (LT) of the MLFs may be provided to describe different anomalous physical phenomena (see [28]).…”

“…Based on the MLFs, many fractional differential operators (FDOs) have been suggested (see [18,[25][26][27][28][29][30][31]). For example, Caputo and Fabrizio [30] defined a new fractional derivative without singular kernel.…”

Analysis of Drude model using fractional derivatives without singular kernels

“…There are classical applications where FC has showed its great capabilities, such as the tautochrone problem [13], models based on memory mechanisms [14], fractional diffusion equations [15], new linear capacitor theory [16], the non-local description of quantum dynamics like Brownian motion and anomalous diffusion [17,18]. Other interesting applications are given in viscoelastic materials [19][20][21][22][23][24], anomalous non-Gaussian transport [25] and dielectric materials [26][27][28][29].…”