The aim of this tutorial survey is to revisit the basic theory of relaxation processes governed by linear differential equations of fractional order. The fractional derivatives are intended both in the Rieamann-Liouville sense and in the Caputo sense. After giving a necessary outline of the classical theory of linear viscoelasticity, we contrast these two types of fractional derivatives in their ability to take into account initial conditions in the constitutive equations of fractional order. We also provide historical notes on the origins of the Caputo derivative and on the use of fractional calculus in viscoelasticity.
In the present review we survey the properties of a transcendental function of the Wright type, nowadays known as M-Wright function, entering as a probability density in a relevant class of self-similar stochastic processes that we generally refer to as time-fractional diffusion processes. Indeed, the master equations governing these processes generalize the standard diffusion equation by means of time-integral operators interpreted as derivatives of fractional order. When these generalized diffusion processes are properly characterized with stationary increments, the M-Wright function is show…
In this study we obtained analytically relaxation function in terms of rotational correlation functions based on Brownian motion for complex disordered systems in a stochastic framework. We found out that rotational relaxation function has a fractional form for complex disordered systems, which indicates relaxation has non-exponential character obeys to Kohlrausch-William-Watts law, following the Mittag-Leffler decay.
We introduce a new relaxation function depending on an arbitrary parameter as solution of a kinetic equation in the same way as the relaxation function introduced empirically by Debye, Cole-Cole, Davidson-Cole and Havriliak-Negami, anomalous relaxation in dielectrics, which are recovered as particular cases. We propose a differential equation introducing a fractional operator written in terms of the Hilfer fractional derivative of order ξ, with 0<ξ<1 and type η, with 0<η<1. To discuss the solution of the fractional differential equation, the methodology of Laplace transform is required. As a b…
Stress relaxation and oscillation damping of complex viscoelastic media often manifest history- and path-dependent physical behaviors and cannot accurately be described by the classical models. Recent research found that fractional derivative models can characterize such complex relaxation and damping. However, to our best knowledge, easy-to-use numerical software is not available for fractional relaxation-oscillation (FRO) equations. This paper is to introduce an open source free Matlab toolbox which we developed in recent years for numerical solution of the FRO equations. This FRO toolbox us…
This work concerns causal acoustical wave equations which imply frequency power-law attenuation. A connection between the five-parameter fractional Zener wave equation, which is derived from a fractional stress-strain relation plus conservations of mass and momentum, and the physically well established multiple relaxation framework is developed. It is shown that for a certain continuous distribution of relaxation mechanisms, the two descriptions are equal.
We study waves in a rod of finite length with a viscoelastic constitutive equation of fractional distributed-order type for the special choice of weight functions. Prescribing boundary conditions on displacement, we obtain case corresponding to stress relaxation. In solving system of differential and integro-differential equations we use the Laplace transformation in the time domain.
The relaxation functions introduced empirically by Debye, Cole-Cole, Cole-Davidson and Havriliak-Negami are, each of them, solutions to their respective kinetic equations. In this work, we propose a generalization of such equations by introducing a fractional differential operator written in terms of the Riemann-Liouville fractional derivative of order $γ$, $0 < γ\leq 1$. In order to solve the generalized equations, the Laplace transform methodology is introduced and the corresponding solutions are then presented, in terms of Mittag-Leffler functions. In the case in which the derivative's orde…
The purpose of this paper is twofold: from one side we provide a general survey to the viscoelastic models constructed via fractional calculus and from the other side we intend to analyze the basic fractional models as far as their creep, relaxation and viscosity properties are considered. The basic models are those that generalize via derivatives of fractional order the classical mechanical models characterized by two, three and four parameters, that we refer to as Kelvin-Voigt, Maxwell, Zener, anti-Zener and Burgers. For each fractional model we provide plots of the creep compliance, relaxat…
The introduction of a fractional differential operator defined in terms of the Riemann-Liouville derivative makes it possible to generalize the kinetic equations used to model relaxation in dielectrics. In this context such fractional equations are called fractional kinetic relaxation equations and their solutions, called fractional kinetic relaxation functions, are given in terms of Mittag-Leffler functions. These fractional kinetic relaxation functions generalize the kinetic relaxation functions associated with the Debye, Cole-Cole, Cole-Davidson and Havriliak-Negami models, as the latter fu…
The first-order differential equation of exponential relaxation can be generalized by using either the fractional derivative in the Riemann-Liouville (R-L) sense and in the Caputo (C) sense, both of a single order less than 1. The two forms turn out to be equivalent. When, however we use fractional derivatives of distributed order (between zero and 1), the equivalence is lost, in particular on the asymptotic behaviour of the fundamental solution at small and large times. We give an outline of the theory providing the general form of the solution in terms of an integral of Laplace type over a p…
Inspired by the article "Anomalous relaxation model based on the fractional derivative with a Prabhakar-like kernel" (Z. Angew. Math. Phys. (2019) 70:42) which authors D. Zhao and HG. Sun studied the integro-differential equation with the kernel given by the Prabhakar function $e^{-γ}_{α, β}(t, λ)$ we provide the solution to this equation which is complementary to that obtained up to now. Our solution is valid for effective relaxation times which admissible range extends the limits given in \cite[Theorem 3.1]{DZhao2019} to all positive values. For special choices of parameters entering the equ…
Physically natural assumption says that the any relaxation process taking place in the time interval $[t_{0},t_{2}]$, $t_{2} > t_{0}\ge 0$ may be represented as a composition of processes taking place during time intervals $[t_{0}, t_{1}]$ and $[t_{1},t_{2}]$ where $t_{1}$ is an arbitrary instant of time such that $t_{0} \leq t_{1} \leq t_{2}$. For the Debye relaxation such a composition is realized by usual multiplication which claim is not valid any longer for more advanced models of relaxation processes. We investigate the composition law required to be satisfied by the Cole-Cole relaxation…
We develop a kind of fractional calculus and theory of relaxation and diffusion equations associated with operators in the time variable, of the form $(Du)(t)=\frac{d}{dt}\int\limits_0^tk(t-τ)u(τ)\,dτ-k(t)u(0)$ where $k$ is a nonnegative locally integrable function. Our results are based on the theory of complete Bernstein functions. The solution of the Cauchy problem for the relaxation equation $Du=-λu$, $λ>0$, proved to be (under some conditions upon $k$) continuous on $[(0,\infty)$ and completely monotone, appears in the description by Meerschaert, Nane, and Vellaisamy of the process $N(E(t…
We prove that the electromagnetic fields in dielectric media whose susceptibility follows a fractional power-law dependence in a wide frequency range can be described by differential equations with time derivatives of noninteger order. We obtain fractional integro-differential equations for electromagnetic waves in a dielectric. The electromagnetic fields in dielectrics demonstrate a fractional power-law relaxation. The fractional integro-differential equations for electromagnetic waves are common to a wide class of dielectric media regardless of the type of physical structure, the chemical co…
We consider equations of the form $(D_{(ρ)}u)(t)=-λu(t)$, $t>0$, where $λ>0$, $D_{(ρ)}$ is a distributed order derivative, that is the Caputo-Dzhrbashyan fractional derivative of order $α$, integrated in $α\in (0,1)$ with respect to a positive measure $ρ$. Such equations are used for modeling anomalous, non-exponential relaxation processes. In this work we study asymptotic behavior of solutions of the above equation, depending on properties of the measure $ρ$.
We present a new approach based on linear integro-differential operators with logarithmic kernel related to the Hadamard fractional calculus in order to generalize, by a parameter $ν\in (0,1]$, the logarithmic creep law known in rheology as Lomnitz law (obtained for $ν=1$). We derive the constitutive stress-strain relation of this generalized model in a form that couples memory effects and time-varying viscosity. Then, based on the hereditary theory of linear viscoelasticity, we also derive the corresponding relaxation function by solving numerically a Volterra integral equation of the second …
We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating these operators in a way accessible to applied scientists, avoiding unproductive generalities and excessive mathematical rigor. By applying this technique we shall derive the analytical solutions of the most simple linear integral and differential equations of fractional order. We show the fundamental role of the Mittag-Leffler function, whose properties are r…
Multi-system interaction is an important and difficult problem in physics. Motivated by the experimental result of an electronic circuit element "Fractor", we introduce the concept of dynamic-order fractional dynamic system, in which the differential-order of a fractional dynamic system is determined by the output signal of another dynamic system. The new concept offers a comprehensive explanation of physical mechanism of multi-system interaction. The properties and potential applications of dynamic-order fractional dynamic systems are further explored with the analysis of anomalous relaxation…
Spatial non-locality of space-fractional viscoelastic equations of motion is studied. Relaxation effects are accounted for by replacing second-order time derivatives by lower-order fractional derivatives and their generalizations. It is shown that space-fractional equations of motion of an order strictly less than 2 allow for a new kind anisotropy, associated with angular dependence of non-local interactions between stress and strain at different material points. Constitutive equations of such viscoelastic media are determined. Explicit fundamental solutions of the Cauchy problem are construct…
We analyze time-discrete and continuous `fractional' random walks on undirected regular networks with special focus on cubic periodic lattices in $n=1,2,3,..$ dimensions. The fractional random walk dynamics is governed by a master equation involving {\it fractional} powers of Laplacian matrices $L^{\fracα{2}}$}where $α=2$ recovers the normal walk. First we demonstrate that the interval $0<α\leq 2$ is admissible for the fractional random walk. We derive analytical expressions for fractional transition matrix and closely related the average return probabilities. We further obtain the fundamental…
Frequency-dependent acoustical loss due to a multitude of physical mechanisms is commonly modeled by multiple relaxations. For discrete relaxation distributions, such models correspond with causal wave equations of integer-order temporal derivatives. It has also been shown that certain continuous distributions may give causal wave equations with fractional-order temporal derivatives. This paper demonstrates analytically that if the wave-frequency ω satisfies Ω_L << ω << Ω_H, a continuous relaxation distribution populating only Ω belongs to [Ω_L,Ω_H] gives the same effective wave equation as fo…
We have provided a fractional generalization of the Poisson renewal processes by replacing the first time derivative in the relaxation equation of the survival probability by a fractional derivative of order $α~(0 < α\leq 1)$. A generalized Laplacian model associated with the Mittag-Leffler distribution is examined. We also discuss some properties of this new model and its relevance to time series. Distribution of gliding sums, regression behaviors and sample path properties are studied. Finally we introduce the $q$-Mittag-Leffler process associated with the $q$-Mittag-Leffler distribution.
The application of quantum Langevin equations for the study of non-equilibrium relaxations is illustrated in the exactly solved quantum spherical model. Tutorial sections on the physical background of non-markovian quantum noise, the spherical model quantum phase transition, the long-time limit of the quantum Langevin equation of the spherical model and physical ageing are followed by a brief review of the solution of the non-markovian time-dependent spherical constraint and about the consequences for quantum ageing at zero temperature, after a quantum quench.
The relaxation properties of dielectric materials are described, in the frequency domain, according to one of the several models proposed over the years: Kohlrausch-Williams-Watts, Cole-Cole, Cole-Davidson, Havriliak-Negami (with its modified version) and Excess wing model are among the most famous. Their description in the time domain involves some mathematical functions whose knowledge is of fundamental importance for a full understanding of the models. In this work, we survey the main dielectric models and we illustrate the corresponding time-domain functions. In particular, we stress the a…
