Fractional diffusion limits of non-classical transport equations

Martin Frank; Weiran Sun

doi:10.3934/krm.2018059

Article Contents

2018, Volume 11, Issue 6: 1503-1526. Doi: 10.3934/krm.2018059

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Fractional diffusion limits of non-classical transport equations

Martin Frank^1, , and
Weiran Sun^2,

1.
Karlsruhe Institute of Technology, Steinbuch Center for Computing, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany
2.
Dept. of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC V5A 1S6, Canada

* Corresponding author: Martin Frank
* Corresponding author: Martin Frank

Received: June 30, 2017

Revised: September 30, 2017

Published: June 2018

The first author is supported by the German research foundation DFG under grant FR2841/6-1.

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Abstract

We establish asymptotic diffusion limits of the non-classical transport equation derived in [12]. By introducing appropriate scaling parameters, the limits will be either regular or fractional diffusion equations depending on the tail behaviour of the path-length distribution. Our analysis is based on a combination of the Fourier transform and a moment method. We put special focus on dealing with anisotropic scattering, which compared to the isotropic case makes the analysis significantly more involved.

Keywords:

Fractional diffusion limit,
non-classical transport.

Mathematics Subject Classification: Primary: 82B40, 80M35; Secondary: 35R11.

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