Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency

Chen, W.; Holm, Sverre

doi:10.1121/1.1646399

Cited by 282 publications

(184 citation statements)

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“…Nowadays it is well recognized that evolution equations can be interpreted as differential equations of fractional order in time when some hereditary mechanisms of power-law type are present in diffusion or wave phenomena. This has been shown for example in [Chen and Holm(2003), Chen and Holm(2004), Mainardi and Tomirotti (1997)] and more recently in [Mainardi (2010)] and [Näsholm and Holm (2013], where propagation of pulses in linear lossy media governed by constitutive equations of fractional order has been revisited.…”

Section: Introductionmentioning

confidence: 99%

Propagation speed of the maximum of the fundamental solution to the fractional diffusion–wave equation

Luchko

Mainardi

Povstenko

2013

Computers & Mathematics with Applications

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In this paper, the one-dimensional time-fractional diffusion-wave equation with the fractional derivative of order α, 1 < α < 2 is revisited. This equation interpolates between the diffusion and the wave equations that behave quite differently regarding their response to a localized disturbance: whereas the diffusion equation describes a process, where a disturbance spreads infinitely fast, the propagation speed of the disturbance is a constant for the wave equation. For the timefractional diffusion-wave equation, the propagation speed of a disturbance is infinite, but its fundamental solution possesses a maximum that disperses with a finite speed. In this paper, the fundamental solution of the Cauchy problem for the time-fractional diffusion-wave equation, its maximum location, maximum value, and other important characteristics are investigated in detail. To illustrate analytical formulas, results of numerical calculations and plots are presented. Numerical algorithms and programs used to produce plots are discussed.

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Section: Introductionmentioning

confidence: 99%

Propagation speed of the maximum of the fundamental solution to the fractional diffusion–wave equation

Luchko

Mainardi

Povstenko

2013

Computers & Mathematics with Applications

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“…where s is the physical quantity of interest (e.g., temperature in anomalous heat conduction), γ the corresponding physical coefficient, ( ) β 2 ∇ − represents the symmetric fractional Laplacian 14 , and α and β can be real numbers. The fundamental solution of equation (2) is the time-dependent Lévy probability density distribution (fat tailed distribution α=1, β<1), in which 2β is the stability index of Lévy distribution [1][2][3] .…”

Section: Fractal Time-space Transformsmentioning

confidence: 99%

Time–space fabric underlying anomalous diffusion

Chen

2006

Chaos, Solitons & Fractals

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This study unveils the time-space transforms underlying anomalous diffusion process. PACS numbers: 61.43. Hv, 47.27.Qb, 65.40.Fb, 05.40.Ca, 43.20.Bi Anomalous diffusion is one of the most important concepts in modern physics [1][2][3][4] and is present in extremely diverse engineering fields such as charges transport in amorphous semiconductor, 5 vibration and acoustic dissipation in soft matter, 6 magnetic plasma, 7 polymer dynamics, 8 turbulence 9 and quantum processes 10 among many other problems. 4 However, it is noted that the anomaly is mostly introduced in a descriptive level of statistical representation or phenomenological modeling. The purpose of this

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“…[3, 4,5,10,12,13,17,20,29,31,33,34,35,38,40]. In all these analytical models the fractional Laplacian has been introduced in heuristic manner.…”

Section: Introductionmentioning

confidence: 99%

The fractional Laplacian as a limiting case of a self-similar spring model and applications to n-dimensional anomalous diffusion

Michelitsch

Maugin

Nowakowski

et al. 2013

fcaa

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We analyze the "fractional continuum limit" and its generalization to n dimensions of a self-similar discrete spring model which we introduced recently [21]. Application of Hamilton's (variational) principle determines in rigorous manner a self-similar and as consequence non-local Laplacian operator. In the fractional continuum limit the discrete self-similar Laplacian takes the form of the fractional Laplacian −(−Δ) α 2 with 0 < α < 2. We analyze the fundamental link of fractal vibrational features of the discrete self-similar spring model and the smooth regular ones of the corresponding fractional continuum limit model in n dimensions: We find a characteristic scaling law for the density of normal modes ∼ ω 828of Lévi flights in n-dimensions. In the limit of "large scaled times" ∼ t/r α >> 1 we show that all distributions exhibit the same asymptotically algebraic decay ∼ t −n/α → 0 independent from the initial distribution and spatial position. This universal scaling depends only on the ratio n/α of the dimension n of the physical space and the Lévi parameter α.MSC 2010: 28A80, 35Q84, 35R11, 60E07, 26A33, 82C31 Keywords and phrases: fractional Laplacian, self-similarity, power laws, scale invariance, fractals, Weierstrass-Mandelbrot function, continuum limit, fractional calculus, Fokker-Planck equation, anomalous diffusion, Lévi flights, Lévi (stable) distributions, non-locality IntroductionSelf-similarity (scaling-invariance) can be found in many problems of physics. There is a need for understanding the dynamics that leads to fractal and self-similarity properties in domains of the physics as varied as flow turbulence and complex materials. In "traditional modelling" the continuum is assumed to have a characteristic length scale which determines the wavelengths where wave fields interact with the microstructure. However, there are numerous materials in nature which are constituted by a scale hierarchy of recurrent microstructure which can be conceived in a good approximation as self-similar. This is true for solids and porous media but also in fluid mechanics, or multicomponent and multiphase flows. As a consequence, there are many areas of modelling that could benefit from a better understanding of the mechanisms underlying the dynamics of objects exhibiting self-similarity properties.In fluid mechanics, synthetic turbulence models are one of many examples of tempering with the input spectrum in order to understand better the physics underlying Lagrangian diffusion [26] and such spectra can be traced to the fractal distributions of velocity accumulation points in the synthetic flow. Fractal approaches are developing rapidly in fluid mechanics. Such approaches consist in either experimentally or numerically interfering with the flow, forcing it through self-similar objects (or through numerical forcing), [36,11,27].On the other hand, there is a large area of research devoted to the so called Fractional Laplacian and a huge number of references exists, employing this operator in various contexts of ...

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Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency

Cited by 282 publications

References 19 publications

Propagation speed of the maximum of the fundamental solution to the fractional diffusion–wave equation

Propagation speed of the maximum of the fundamental solution to the fractional diffusion–wave equation

Time–space fabric underlying anomalous diffusion

The fractional Laplacian as a limiting case of a self-similar spring model and applications to n-dimensional anomalous diffusion

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