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# Communications in Mathematical Sciences

## Volume 13 (2015)

### Number 5

### On Rosenau-type approximations to fractional diffusion equations

Pages: 1163 – 1191

DOI: http://dx.doi.org/10.4310/CMS.2015.v13.n5.a5

#### Authors

#### Abstract

Owing to the Rosenau argument [P. Rosenau, *Physical Review A,* 46, 12–15, 1992], originally proposed to obtain a regularized version of the Chapman-Enskog expansion of hydrodynamics, we introduce a non-local linear kinetic equation which approximates a fractional diffusion equation. We then show that the solution to this approximation, apart of a rapidly vanishing in time perturbation, approaches the fundamental solution of the fractional diffusion (a Lévy stable law) at large times.

#### Keywords

fractional diffusion equations, non-local models, Fourier metrics, Rosenau approximation, Lévy-type distributions

#### 2010 Mathematics Subject Classification

35B40, 35K55, 35K60, 35K65