Rosenau type approximations to the heat equation
My collaborator Giuseppe Toscani and myself submitted a new paper, entitled Large-time behavior of the solutions to Rosenau type approximations to the heat equation.
We study in this paper the validity of the approximation to the linear diffusion equation proposed by Rosenau as a regularized version of the Chapman-Enskog expansion of hydrodynamics. This approximation essentially is realized by substituting the heat equation with a linear kinetic equation of Boltzmann type, describing collisions of particles with a fixed background. This remark allows to consider the Rosenau approximation as a particular realization of a model Boltzmann equation, in which the background distribution is a general probability density with bounded variance. In addition to the Rosenau distribution, we considered also a point masses background, which furnishes the central difference scheme to solve numerically the linear diffusion equation.