By fractional diffusive waves we mean the solutions of the so-called time-fractional diffusion-wave equation. This equation is obtained from the classical D'Alembert wave equation by replacing the second-order time derivative with a fractional derivative of order β∈(0,2) and is expected to govern evolution processes intermediate between diffusion and wave propagation when β∈(1,2). Here it is shown to govern the propagation of stress waves in viscoelastic media which, by exhibiting a power law creep, are of relevance in acoustics and seismology since their quality factor turns out to be independent of frequency. The fundamental solutions for the Cauchy and signaling problems are expressed in terms of entire functions (of Wright type) in the similarity variable. Their behaviors turn out to be intermediate between those found in the limiting cases of a perfectly viscous fluid and a perfectly elastic solid. Furthermore, their scaling properties and the relations with some stable probability distributions are outlined.

Fractional Diffusive Waves

The one-dimensional propagation of seismic waves with constant Q is shown to be governed by an evolution equation of fractional order in time, which interpolates the heat equation and the wave equation. The fundamental solutions for the Cauchy and Signalling problems are expressed in terms of entire functions (of Wright type) in the similarity variable and their behaviours turn out to be intermediate between those for the limiting cases of a perfectly viscous fluid and a perfectly elastic solid. In view of the small dissipation exhibited by the seismic pulses, the nearly elastic limit is considered. Furthermore, the fundamental solutions for the Cauchy and Signalling problems are shown to be related to stable probability distributions with an index of stability determined by the order of the fractional time derivative in the evolution equation.

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Seismic pulse propagation with constant Q and stable probability distributions

The paper deals with a rigorous analysis of the «wave hierarchie» related to the operator Ln quoted in the title. Whatever the number n=1, 2, 3 of space dimensions may be, the fundamental solutions En are constructed. These distributions are tempered positive Radon measures associated with positive value functions which have numerous basic properties. So the Cauchy problemP
n(n=1,2,3) with quite arbitrary data is explicitly solved. As another example, also the solution of the signaling problem ℋ is established. Then, various basic aspects of the wave behavior such as diffusion, asymptotic properties, maximum principles and the generalized Huyghens principle are evaluated. Moreover, singular perturbation problems as ɛ→ 0, with estimates of the remainder terms uniformly valid for all t⩾0, are discussed too.

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On a wave theory for the operator ɛ∂t(∂ t 2 −c 1 2 Δn)+∂ t 2 −c 0 2 Δ n

Fractional calculus, in allowing integrals and derivatives of any positive order (the term "fractional" kept only for historical reasons), can be considered a branch of mathematical physics which mainly deals with integro-differential equations, where integrals are of convolution form with weakly singular kernels of power law type. In recent decades fractional calculus has won more and more interest in applications in several fields of applied sciences. In this lecture we devote our attention to wave propagation problems in linear viscoelastic media. Our purpose is to outline the role of fractional calculus in providing simplest evolution processes which are intermediate between diffusion and wave propagation. The present treatment mainly reflects the research activity and style of the author in the related scientific areas during the last decades.

Fractional Calculus in Wave Propagation Problems

Dispersion due to internal frictions as wave propagates is a consequence of the second principle of thermodynamics. When the wavelengths are several times higher than the mesoscopic inhomogeneities, internal diffractions can be ignored and the propagation medium can then be considered as a continuum at the scale of these wavelengths. Here, we consider the dissipation mechanism due to viscosity only. By mean of Laplace transforms both on time and space, a causal analysis leads us to a closed-form solution, which we think is the simplest analytical form. This is illustrated by searching the viscoelastic Green’s function associated with the horizontal shear wave generated by a uniform impulsive line source in an infinite homogeneous medium, whose example is almost mathematically equivalent to the study of the scalar wave generated in viscous fluid. The described method is thus restricted to one-wave propagation problems and is probably not generally applicable when the source generates several waves. In the ...

F. Mainardi

Papers

Wave front expansions for transient viscoelastic waves