Fractional Calculus and Waves in Linear Viscoelasticity

Mainardi, Francesco

doi:10.1142/9781848163300

Cited by 1,092 publications

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“…Mainardi [15, Section 6.3] considers a version of (1.2) that employs the Caputo fractional derivative…”

Section: Fractional Wave Equationsmentioning

confidence: 99%

“…Mainardi [15, Section 6.4] derives the fractional wave equation (1.2) from a viscoelastic model with a power law stress-strain relationship. He notes that the Green’s function solution to the fractional wave equation (1.2) can be also expressed in terms of stable densities.…”

Section: Fractional Wave Equationsmentioning

confidence: 99%

“…Mainardi [15, Section 6.3] shows that the solution to the fractional wave equation (1.2) with initial conditions p ( x , 0) = ϕ ( x ) and

1 \frac{\partial}{\partial t} p false(x, 0 false) = 0

is given by the convolution formula…”

Section: General Wave Equationsmentioning

confidence: 99%

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Stochastic solutions for fractional wave equations

2014

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“…Mainardi [15, Section 6.3] considers a version of (1.2) that employs the Caputo fractional derivative…”

Section: Fractional Wave Equationsmentioning

confidence: 99%

Section: Fractional Wave Equationsmentioning

confidence: 99%

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Stochastic solutions for fractional wave equations

2014

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“…Fractional derivatives were invented by Leibnitz soon after the more familiar integer order derivatives [39, 54], but have only recently become popular in applications. They are now used to model a wide variety of problems in physics [25, 36, 39, 52, 53, 57, 68], finance [23, 28, 44, 38, 63, 64], biology [4, 2, 22, 27, 37], and hydrology [1, 7, 8, 15, 17, 65].…”

Section: Introductionmentioning

confidence: 99%

Tempered fractional calculus

Sabzikar

Meerschaert

Chen

2015

Journal of Computational Physics

287

168

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Fractional derivatives and integrals are convolutions with a power law. Multiplying by an exponential factor leads to tempered fractional derivatives and integrals. Tempered fractional diffusion equations, where the usual second derivative in space is replaced by a tempered fractional derivative, govern the limits of random walk models with an exponentially tempered power law jump distribution. The limiting tempered stable probability densities exhibit semi-heavy tails, which are commonly observed in finance. Tempered power law waiting times lead to tempered fractional time derivatives, which have proven useful in geophysics. The tempered fractional derivative or integral of a Brownian motion, called a tempered fractional Brownian motion, can exhibit semi-long range dependence. The increments of this process, called tempered fractional Gaussian noise, provide a useful new stochastic model for wind speed data. A tempered difference forms the basis for numerical methods to solve tempered fractional diffusion equations, and it also provides a useful new correlation model in time series.

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“…Typically, one introduces a parametric representation of G ( t ) that fits given experimental measurements. In this work we have considered a stress relaxation function derived from a fractional-order Standard Linear Solid model [20]:

G (t) = E {(\frac{τ_{∊}}{τ_{σ}})}^{α} + E true({[\frac{τ_{∊}}{τ_{σ}}]}^{α} - 1 true) E_{α, 1} true(- {[\frac{t}{τ_{σ}}]}^{α} true),

parametrized by the Young modulus E , the fractional order , and the viscoelastic relaxation times { τε , τδ }, while Eα,β ; ( t ) is a two parameter Mittag-Leffer function [16]. Arterial wall constitutive laws based on fractional calculus have been shown to provide a good fit to in-vivo stress relaxation data [8, 7], accounting for a continuum relaxation spectrum and exhibiting lower sensitivity on input parameters compared to integer-order models [8, 20].…”

Section: Methodsmentioning

confidence: 99%

An Effective Fractal-Tree Closure Model for Simulating Blood Flow in Large Arterial Networks

Perdikaris

Grinberg²,

Karniadakis

2014

Ann Biomed Eng

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The aim of the present work is to address the closure problem for hemodynamic simulations by developing a exible and effective model that accurately distributes flow in the downstream vasculature and can stably provide a physiological pressure out flow boundary condition. To achieve this goal, we model blood flow in the sub-pixel vasculature by using a non-linear 1D model in self-similar networks of compliant arteries that mimic the structure and hierarchy of vessels in the meso-vascular regime (radii 500 μm – 10 μm). We introduce a variable vessel length-to-radius ratio for small arteries and arterioles, while also addressing non-Newtonian blood rheology and arterial wall viscoelasticity effects in small arteries and arterioles. This methodology aims to overcome substantial cut-off radius sensitivities, typically arising in structured tree and linearized impedance models. The proposed model is not sensitive to out flow boundary conditions applied at the end points of the fractal network, and thus does not require calibration of resistance/capacitance parameters typically required for out flow conditions. The proposed model convergences to a periodic state in two cardiac cycles even when started from zero-flow initial conditions. The resulting fractal-trees typically consist of thousands to millions of arteries, posing the need for efficient parallel algorithms. To this end, we have scaled up a Discontinuous Galerkin solver that utilizes the MPI/OpenMP hybrid programming paradigm to thousands of computer cores, and can simulate blood flow in networks of millions of arterial segments at the rate of one cycle per 5 minutes. The proposed model has been extensively tested on a large and complex cranial network with 50 parent, patient-specific arteries and 21 outlets to which fractal trees where attached, resulting to a network of up to 4,392,484 vessels in total, and a detailed network of the arm with 276 parent arteries and 103 outlets (a total of 702,188 vessels after attaching the fractal trees), returning physiological flow and pressure wave predictions without requiring any parameter estimation or calibration procedures. We present a novel methodology to overcome substantial cut-off radius sensitivities

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Fractional Calculus and Waves in Linear Viscoelasticity

Cited by 1,092 publications

References 0 publications

Stochastic solutions for fractional wave equations

Stochastic solutions for fractional wave equations

Tempered fractional calculus

An Effective Fractal-Tree Closure Model for Simulating Blood Flow in Large Arterial Networks

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