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Prabhakar-like fractional viscoelasticity
Andrea Giusti, Ivano Colombaro
TL;DR
The paper investigates a Prabhakar-based linear viscoelastic model and whether its many parameters can recover known linear-viscoelasticity results. It develops a Prabhakar-Maxwell model, analyzes its connections with classical fractional models, and relates Prabhakar integrals to the Caputo-Fabrizio operator.
Problem
The paper asks whether a Prabhakar-based viscoelastic model can recover known linear-viscoelasticity results through suitable parameter choices.
Method
The paper replaces the Caputo derivative in the fractional Maxwell model with a Prabhakar derivative and analyzes parameter choices connecting the resulting model with classical models.
Results
The paper develops a Prabhakar-Maxwell viscoelastic model and analyzes its connections with the classical fractional Maxwell, Voigt, and Zener models, alongside a connection to the Caputo-Fabrizio operator.
Takeaways & Limitations
Prabhakar operators provide a framework for modeling linear viscoelasticity while connecting Prabhakar fractional integrals with the Caputo-Fabrizio differential operator.
Takeaways & Limitations
A complete characterization of physically acceptable Prabhakar-like viscoelastic models is beyond the paper’s scope and is left for future development.
Abstract
from arXiv · showhide
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.
1. Introduction
The paper introduces Prabhakar fractional calculus as a three-parameter extension of fractional operators and applies it to linear viscoelasticity. It also connects Prabhakar integrals with the Caputo–Fabrizio differential operator.
- Prabhakar fractional operators: The paper reviews Mittag-Leffler functions, Prabhakar calculus, and the regularized Prabhakar derivative before developing viscoelastic applications.The regularized derivative is defined for functions with sufficiently regular derivatives and is emphasized for physically reasonable initial-value problems.
- Mittag-Leffler functions and Prabhakar calculus: Prabhakar’s three-parameter Mittag-Leffler function extends fractional integro-differential calculus beyond the classical Riemann–Liouville–Caputo framework.The generalized function is linked to fractional differential equations and relaxation processes through exponential and power-law asymptotic behaviors.
- Prabhakar fractional operators: The Prabhakar integral recovers the Riemann–Liouville fractional integral when γ goes to zero or when γ tends to one and ω vanishes.This establishes a formal reduction from the generalized operator to a classical fractional integral under specified parameter choices.
- Caputo–Fabrizio operator: The paper relates the Caputo–Fabrizio operator’s non-singular-kernel formulation to Prabhakar fractional integrals through a parameter choice ω(α) = −α/(1 − α).This provides a Prabhakar-based perspective on the analysis of the Caputo–Fabrizio operator.
- Physical meaning: Prabhakar operators are connected to physically relevant dielectric-relaxation models, including the Havriliak–Negami response and its time-domain integral equation.The Havriliak–Negami response is represented by a specific Prabhakar-kernel choice, with β = αγ and ω = −λ.
2. Fractional Maxwell model with Prabhakar derivatives
The paper formulates a fractional Maxwell viscoelastic model by replacing the classical Caputo derivative with a regularized Prabhakar derivative. It then identifies parameter choices under which this model is formally equivalent to classical fractional Maxwell, Voigt, and Zener models.
- 2. Fractional Maxwell model with Prabhakar derivatives: The proposed Maxwell-Prabhakar model replaces the Caputo or Riemann-Liouville derivative in the classical fractional Maxwell constitutive equation with a regularized Prabhakar derivative.The model is studied for causal stress and strain functions under stated parameter and regularity assumptions.
- 2.1. Connection with the classical models of linear viscoelasticity: A complete characterization of physically acceptable Prabhakar-like viscoelastic models is left for future work because the broad parameter freedom complicates the analysis.The paper notes that positive a and b are not mandatory for physically meaningful models.
- 2. Fractional Maxwell model with Prabhakar derivatives: The model’s Laplace-domain constitutive equation follows from the transform of the regularized Prabhakar derivative, subject to the initial-data condition aσ(0+) = bε(0+).This condition is described as a reasonable constraint for the initial data.
- 2. Fractional Maxwell model with Prabhakar derivatives: The creep compliance is obtained by inspecting the Laplace-domain constitutive relation and inverting the resulting expression, whereas deriving the relaxation modulus is less straightforward.The relaxation modulus requires an absolutely convergent power-series expansion and subsequent convergence analysis.
- 2.1. Connection with the classical models of linear viscoelasticity: The Maxwell-Prabhakar model formally recovers the fractional Maxwell model through two parameter configurations, including γ = 0, a = A, b = B, and β = ν.The second configuration sets ω = 0 while allowing γ to vary over the reals.
- 2.1. Connection with the classical models of linear viscoelasticity: Formal equivalence with the fractional Voigt model occurs for γ = 1, a = 0, b = −B, α = β = ν, and ω = −M/B.The paper also gives two parameter configurations that recover the fractional Zener model.
3. Conclusions
The paper develops a Prabhakar-based fractional viscoelastic model and relates it to classical fractional models. It also identifies a formulation of the Caputo-Fabrizio derivative through a particular Prabhakar fractional integral.
- 3. Conclusions: The paper reviews Prabhakar fractional operators as a three-parameter generalization of Riemann-Liouville-Caputo fractional calculus.This review follows an introduction to Mittag-Leffler functions and their generalizations.
- 3. Conclusions: It develops a fractional Maxwell model with Prabhakar derivatives and analyzes its connections with the classical Maxwell, Voigt, and Zener models.The connections are obtained through suitable parameter choices in the Prabhakar-Maxwell model.
- 3. Conclusions: The paper formulates the Caputo-Fabrizio derivative in terms of a particular Prabhakar fractional integral.This connection is highlighted as an additional formulation in the introductory section.