In recent works it has been demonstrated that using an appropriate rescaling, linear Boltzmann-type equations give rise to a scalar fractional diffusion equation in the limit of a small mean free path. The equilibrium distributions are typically heavy-tailed distributions, but also classical Gaussian equilibrium distributions allow for this phenomena if combined with a degenerate collision freq...

Mass transport such as movement of phosphorus in soils and solutes in rivers is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or super diffusion and is well described using a fractional adv...

The generalized Fick-Jacobs equation is widely used to study diffusion of Brownian particles in three-dimensional tubes and quasi-two-dimensional channels of varying constraint geometry. We show how this equation can be applied to study the slowdown of unconstrained diffusion in the presence of obstacles. Specifically, we study diffusion of a point Brownian particle in the presence of identical...

We show that the increments of generalized Wiener process, useful to describe nonGaussian white noise sources, have the properties of infinitely divisible random processes. Using functional approach and the new correlation formula for non-Gaussian white noise we derive directly from Langevin equation, with such a random source, the Kolmogorov’s equation for Markovian non-Gaussian process. From ...

In this study, we present a numerical scheme to solve the drift-diffusion traffic flow model under the steady state. The drift-diffusion traffic flow model consists of a continuity equation and a nonlinear Poisson equation. The continuity equation describes the propagation of density along the road, and the Poisson equation describes the interaction among vehicles. The system equations cannot b...

A lattice Boltzmann model for convection-diffusion equation with nonlinear convection and isotropic-diffusion terms is proposed through selecting equilibrium distribution function properly. The model can be applied to the common real and complex-valued nonlinear evolutionary equations, such as the nonlinear Schrödinger equation, complex Ginzburg-Landau equation, Burgers-Fisher equation, nonline...

The problem of the one dimensional electro-diffusion of ions in a strong binary electrolyte is considered. The mathematical description, known as the Poisson-Nerst-Planck (PNP) system, consists of a diffusion equation for each species augmented by transport due to a self consistent electrostatic field determined by the Poisson equation. This description is also relevant to other important probl...

where 3 > q > 1. According to the different exponents of m, p, we have the following classical terminologies about the equation (1). (i) The case p = 2,m = 1, is the ordinary semilinear diffusion equation. (ii) The case p = 2,m ̸= 1, is the porous media equation, it is degenerate at u = 0 for m > 1 and singular at u = 0 for 0 < m < 1. (iii) The case p ̸= 2,m = 1, is the p-diffusion equation, it i...

We obtain a limit when mass tends to zero of the relativistic diffusion of Schay and Dudley. The diffusion process has the log-normal distribution. We discuss Langevin stochastic differential equations leading to an equilibrium distribution. We show that for the Jüttner equilibrium distribution the relativistic diffusion is a linear approximation to the Kompaneetz equation describing a photon d...

A diagonally implicit method is shown to be an effective method for integrating the multicomponent species conservation equations. The constitutive equation for multicomponent diffusion is recast into a form analogous to that for binary diffusion, except that the diffusion coefficient is replaced with a matrix of effective multicomponent diffusion coefficients. The resulting matrix has properti...