On Rosenau-type approximations to fractional diffusion equations

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.