Fractional Zener Wave Equation Research Articles

Micro-local and qualitative analysis of the fractional Zener wave equation

This paper concerns the micro-local and qualitative analysis of the fractional Zener wave equation. The classical and Gevrey-type wave front sets of the fundamental solution are determined, and questions on dispersion, dissipation, wave propagation speed and wave packet shape are addressed.

Exponential sum approximation for Mittag-Leffler function and its application to fractional Zener wave equation

The Mittag-Leffler function (MLF) is fundamental to solving fractional calculus problems. An exponential sum approximation for the single-parameter MLF with negative input is proposed. Analysis shows that the approximation, which is based on the Gauss-Legendre quadrature, converges uniformly for all non-positive input. The application to modelling of wave propagation in viscoelastic material with the finite element method is also presented. The propagation speed of the solution to the approximated wave equation is proved to have the same upper bound as the original. The discretized scheme, based on the generalised alpha method, involves only a single matrix inverse whose size is the degree of freedom of the geometric model per time step. It is proved that the scheme is unconditionally stable. Furthermore, the solution converges to the true solution at O(T2), where T is the interval each time step, provided that the approximation error of the MLF is sufficiently small.

Well-Posedness of the Fractional Zener Wave Equation for Heterogeneous Viscoelastic Materials

AbstractZener’s model for viscoelastic solids replaces Hooke’s law σ = 2με(u) + λ tr(ε(u)) I, relating the stress tensor σ to the strain tensor ε(u), where u is the displacement vector, μ > 0 is the shear modulus, and λ ≥ 0 is the first Lamé coefficient, with the constitutive law (1 + τDt) σ = (1 + ρDt)[2με(u) + λ tr(ε(u)) I], where τ > 0 is the characteristic relaxation time and ρ ≥ τ is the characteristic retardation time. It is the simplest model that predicts creep/recovery and stress relaxation phenomena. We explore the well-posedness of the fractional version of the model, where the first-order time-derivative Dt in the constitutive law is replaced by the Caputo time-derivative $\begin{array}{} D_t^\alpha \end{array} $ with α ∈ (0, 1), μ, λ belong to L∞(Ω), μ is bounded below by a positive constant and λ is nonnegative. We show that, when coupled with the equation of motion ϱü = Div σ + f, considered in a bounded open Lipschitz domain Ω in ℝ3 and over a time interval (0, T], where ϱ ∈ L∞(Ω) is the density of the material, assumed to be bounded below by a positive constant, and f is a specified load vector, the resulting model is well-posed in the sense that the associated initial-boundary-value problem, with initial conditions u(0, x) = g(x), u̇(0, x) = h(x), σ(0, x) = S(x), for x ∈ Ω, and a homogeneous Dirichlet boundary condition, possesses a unique weak solution for any choice of g ∈ [$\begin{array}{} \mathrm{H}^1_0 \end{array} $(Ω)]3, h ∈ [L2(Ω)]3, and S = ST ∈ [L2(Ω)]3×3, and any load vector f ∈ L2(0, T; [L2(Ω)]3), and that this unique weak solution depends continuously on the initial data and the load vector.

Microlocal analysis of fractional wave equations

We determine the wave front sets of solutions to two special cases of the Cauchy problem for the space‐time fractional Zener wave equation, one being fractional in space, the other being fractional in time. For the case of the space fractional wave equation, we show that no spatial propagation of singularities occurs. For the time fractional Zener wave equation, we show an analogue of non‐characteristic regularity.

Space-time fractional Zener wave equation.

The space-time fractional Zener wave equation, describing viscoelastic materials obeying the time-fractional Zener model and the space-fractional strain measure, is derived and analysed. This model includes waves with finite speed, as well as non-propagating disturbances. The existence and the uniqueness of the solution to the generalized Cauchy problem are proved. Special cases are investigated and numerical examples are presented.

Comparison of Fractional Wave Equations for Power Law Attenuation in Ultrasound and Elastography

A set of wave equations with fractional loss operators in time and space are analyzed. The fractional Szabo equation, the power law wave equation and the causal fractional Laplacian wave equation are all found to be low-frequency approximations of the fractional Kelvin-Voigt wave equation and the more general fractional Zener wave equation. The latter two equations are based on fractional constitutive equations, whereas the former wave equations have been derived from the desire to model power law attenuation in applications like medical ultrasound. This has consequences for use in modeling and simulation, especially for applications that do not satisfy the low-frequency approximation, such as shear wave elastography. In such applications, the wave equations based on constitutive equations are the viable ones.

Numerical model of a causal and fractional all-frequency wave equation for lossy media

A numerical algorithm, to simulate a lossy acoustic wave equation with fractional time derivative terms, is presented. The inclusion of fractional derivatives yields a causal acoustic wave equation which can model and predict power law attenuation for which the level of absorption is proportional to frequency raised to a non-integer power. The fractional Zener wave equation is derived from the fractional Zener stress-strain constitutive relation and contains two absorption terms. One term which includes the Laplacian, similar to the traditional wave equation in viscous fluids, has a coefficient, which is proportional to a relaxation time. The other term is a fractional time derivative, higher than second order, and is proportional to the creep or retardation time. Both relaxation and retardation parameters will affect the frequency behavior of the attenuation. The inclusion of two terms enables the modeling of power-law attenuation in all frequency ranges. It results in more stable numerical simulations as both the overall mass and stiffness matrices of the finite element algorithm are changed according to the level of absorption. Results show the animation of diffraction patterns of focused and planar acoustic waves by inclusions with different values for the two different parameters, retardation time and relaxation time.

Deriving fractional acoustic wave equations from mechanical and thermal constitutive equations

It is argued that fractional acoustic wave equations come in two kinds. The first kind is constructed ad hoc to have loss operators that fit power law measurements. The second kind is more fundamental as they in addition are based on underlying physical equations. Here that means constitutive equations. These equations are the fractional Kelvin–Voigt and the more general fractional Zener stress–strain relationships as well as a fractional version of the Fourier heat law. The properties of the wave equations are given in terms of attenuation, and phase/group velocities for low-, intermediate- and high-frequency regions. In the most general case, the attenuation exhibits power law behavior in all frequency ranges while the phase and group velocities increase sharply in the intermediate frequency range and converge to a constant, finite value for high frequencies. It is also shown that the fractional Zener wave equation is equivalent to the multiple relaxation model for attenuation.

Model-based discrete relaxation process representation of band-limited power-law attenuation

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 ΩL≪ω ≪ΩH, a continuous relaxation distribution populating only Ω∈[ΩL,ΩH] gives the same effective wave equation as for a fully populated distribution. This insight sparks the main contribution: the elaboration of a method to determine discrete relaxation parameters intended for mimicking a desired attenuation behavior for band-limited waves. In particular, power-law attenuation is discussed as motivated by its prevalence in complex media, e.g., biological tissue. A Mittag-Leffler function related distribution of relaxation mechanisms has previously been shown to be related to the fractional Zener wave equation of three power-law attenuation regimes. Because these regimes correspond to power-law regimes in the relaxation distribution, the idea is to sample the distribution's compressibility contributions evenly in logarithmic frequency while appropriately taking the stepsize into account. This work thence claims to provide a model-based approach to determination of discrete relaxation parameters intended to adequately model attenuation power-laws.

On a fractional Zener elastic wave equation

Abstract This survey concerns a causal elastic wave equation which implies frequency power-law attenuation. The wave equation can be derived from a fractional Zener stress-strain relation plus linearized conservation of mass and momentum. A connection between this four-parameter fractional wave equation and a physically well established multiple relaxation acoustical wave equation is reviewed. The fractional Zener wave equation implies three distinct attenuation power-law regimes and a continuous distribution of compressibility contributions which also has power-law regimes. Furthermore it is underlined that these wave equation considerations are tightly connected to the representation of the fractional Zener stress-strain relation, which includes the spring-pot viscoelastic element, and by a Maxwell-Wiechert model of conventional springs and dashpots. A purpose of the paper is to make available recently published results on fractional calculus modeling in the field of acoustics and elastography, with special focus on medical applications.