Fractional diffusion limits of non-classical transport equations

Frank, Martin; Sun, Weiran

doi:10.3934/krm.2018059

Cited by 17 publications

(19 citation statements)

References 18 publications

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“…The theory has been extended over the last few years [3][4][5][6][7] and has found applications in other areas, including neutron transport in certain types of nuclear reactors [8][9][10], computer graphics [11][12][13], and problems involving anomalous diffusion (cf. [14]). Moreover, a similar kinetic equation has been independently derived for the periodic Lorentz gas in a series of papers by Golse (cf.…”

Section: Introductionmentioning

confidence: 99%

“…The assumption that Σ t is independent of both Ω and s is generally valid when the spatial locations of the scattering centers in the system are uncorrelated (Poisson distributed). This leads to the Beer-Lambert law, with particle flux decreasing as an exponential function of s. However, a nonexponential attenuation law for the particle flux occurs in certain heterogeneous media in which the scattering centers in the system are spatially correlated [2,[8][9][10][11][13][14][15][16][17][18][19]. The theory of nonclassical particle transport [1,[3][4][5][6][7] was developed to address this class of problems.…”

Section: Introductionmentioning

confidence: 99%

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A spectral approach for solving the nonclassical transport equation

Vasques

Moraes

Barros

et al. 2020

Journal of Computational Physics

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This paper introduces a mathematical approach that allows one to numerically solve the nonclassical transport equation in a deterministic fashion using classical numerical procedures. The nonclassical transport equation describes particle transport for random statistically homogeneous systems in which the distribution function for free-paths between scattering centers is nonexponential. We use a spectral method to represent the nonclassical flux as a series of Laguerre polynomials in the free-path variable s, resulting in a nonclassical equation that has the form of a classical transport equation. We present numerical results that validate the spectral approach, considering transport in slab geometry for both classical and nonclassical problems in the discrete ordinates formulation.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A spectral approach for solving the nonclassical transport equation

Vasques

Moraes

Barros

et al. 2020

Journal of Computational Physics

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“…We note that working with a weighted density seems to be a general setting when deriving (fractional) diffusion limits of kinetic equations with extended variables. See for example in [14], where the macroscopic equations for a non-classical kinetic equation are derived for the weighted density function against the path length distribution. Compared with [14], the choice of the weight function χ 0 in this paper is much less obvious.…”

Section: Introductionmentioning

confidence: 99%

“…See for example in [14], where the macroscopic equations for a non-classical kinetic equation are derived for the weighted density function against the path length distribution. Compared with [14], the choice of the weight function χ 0 in this paper is much less obvious. Second, the derivation of the fractional diffusion equations in [1,9,19] relies on the method of auxiliary functions or a related Hilbert expansion.…”

Section: Introductionmentioning

confidence: 99%

The fractional diffusion limit of a kinetic model with biochemical pathway

Perthame

Sun

Tang

2018

Z. Angew. Math. Phys.

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Kinetic-transport equations that take into account the intra-cellular pathways are now considered as the correct description of bacterial chemotaxis by run and tumble. Recent mathematical studies have shown their interest and their relations to more standard models. Macroscopic equations of Keller-Segel type have been derived using parabolic scaling. Due to the randomness of receptor methylation or intra-cellular chemical reactions, noise occurs in the signaling pathways and affects the tumbling rate. Then, comes the question to understand the role of an internal noise on the behavior of the full population. In this paper we consider a kinetic model for chemotaxis which includes biochemical pathway with noises. We show that under proper scaling and conditions on the tumbling frequency as well as the form of noise, fractional diffusion can arise in the macroscopic limits of the kinetic equation. This gives a new mathematical theory about how long jumps can be due to the internal noise of the bacteria.

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“…Here we assume that φ is bounded from above and below. The equilibrium (1.12) arises in numerous areas of applications such as granular plasmas with dissipative collision [5,8,7,13,23], astrophysical plasmas [22], economy [12] and transport in atmospheric clouds [14]. See also [23] for a review of granular materials in which power law like distribution appears as a typical equilibrium in inelastic kinetic theory.…”

mentioning

confidence: 99%

An Asymptotic-Preserving Scheme for the Kinetic Equation with Anisotropic Scattering: Heavy Tail Equilibrium and Degenerate Collision Frequency

Wang¹,

Yan²

2019

SIAM J. Sci. Comput.

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We design an asymptotic preserving scheme for the linear kinetic equation with anisotropic scattering that leads to a fractional diffusion limit. This limit may be attributed to two reasons: a heavy tail equilibrium or a degenerate collision frequency, both of which are considered in this paper. Our scheme builds on the ideas developed in [24] but with two major variations. One is a new splitting of the system that accounts for the anisotropy in the scattering cross section by introducing two extra terms. We then showed, via detailed calculation, that the scheme enjoys a relaxed AP property as opposed to the one step AP for the isotropic scattering. Another contribution is for the degenerate collision frequency case, which brings in additional stiffness. We propose to integrate a 'body' term, which appears to be the main component in the diffusion limit. This term is precomputed once with a prescribed accuracy, via a change of variable that alleviates the stiffness. Numerical examples are presented to validate its efficiency in both kinetic and fractional diffusion regimes.

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Fractional diffusion limits of non-classical transport equations

Cited by 17 publications

References 18 publications

A spectral approach for solving the nonclassical transport equation

A spectral approach for solving the nonclassical transport equation

The fractional diffusion limit of a kinetic model with biochemical pathway

An Asymptotic-Preserving Scheme for the Kinetic Equation with Anisotropic Scattering: Heavy Tail Equilibrium and Degenerate Collision Frequency

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