Distributed-order fractional constitutive stress–strain relation in wave propagation modeling

Abstract

Distributed order fractional model of viscoelastic body is used in order to describe wave propagation in infinite media. Existence and uniqueness of fundamental solution to the generalized Cauchy problem, corresponding to fractional wave equation, is studied. The explicit form of fundamental solution is calculated, and wave propagation speed, arising from solution's support, is found to be connected with the material properties at initial time instant. Existence and uniqueness of the fundamental solutions to the fractional wave equations corresponding to four thermodynamically acceptable classes of linear fractional constitutive models, as well as to power type distributed order model, are established and explicit forms of the corresponding fundamental solutions are obtained.Keywords: wave equation; distributed order model of viscoelastic body; linear fractional model; power type distributed order model;

Fractional Burgers wave equation

“…Since σ (g) sr = a3 b , see [28,Eq. (57)], the wave propagation speed (44) is exactly the wave propagation speed (8) that is obtained in [22] for the constitutive models having fractional differentiation orders not exceeding one.…”

“…obtained and analyzed in [2] for thermodynamical consistency and used in [22] as constitutive equations in wave propagation modeling. Namely, the results of [20,21], where the wave propagation speed is found via the conic solution support, i.e., |x| < ct, in the case of the fractional Zener model and its generalization, respectively given by…”

“…22) yields 0 < β−(µ − α) < β+(µ − α) < 1, so that by setting ζ = β−(µ − α) and ξ = β + (µ − α) in function g given by (49), using (50) one hassin ((α + β − µ) ϕ) sin ((µ − α + β) ϕ) < sin (µ−α−β)πTherefore, again by(22), one has that b − a3 a1 |sin((µ−α−β)ϕ)| sin((µ−α+β)ϕ) > 0, which, along with the positivity of all other terms in (52), implies that f ρ (ϕ) > 0 if ϕ ∈ 0, π 2 .…”

An Inverse Problem for a Generalized Fractional Derivative with an Application in Reconstruction of Time- and Space-Dependent Sources in Fractional Diffusion and Wave Equations

“…Often a memory is not of power-type. A direct generalization of (k1) leads to multiterm and distributed order fractional derivatives [15][16][17]. These derivatives have the following kernels:…”

A Unified Petrov–Galerkin Spectral Method and Fast Solver for Distributed-Order Partial Differential Equations

“…A faithful description of such anomalous transport requires exploiting distributed-order derivatives, in which the derivative order has a distribution over a range of values. The reader is referred to [14,15,19,28,36,37,42,53,58] and the references given therein for more details on the distributed-order fractional equations.…”