A variant of the discrete Burgers equation derived from the correlated random walk and its ultradiscretization

TL;DR

This paper derives a new variant of the discrete Burgers equation from correlated random walks, applies ultradiscretization to obtain cellular automata, and interprets them as traffic flow models.

Contribution

It introduces a novel variant of the discrete Burgers equation linked to correlated random walks and develops its ultradiscrete form with cellular automata applications.

Findings

Derived a new discrete Burgers equation variant from correlated random walks.

Obtained an ultradiscrete Burgers equation leading to cellular automata.

Interpreted cellular automata as traffic flow models.

Abstract

In this paper, we show that a variant of the discrete Burgers equation can be obtained through the Cole--Hopf transformation to a generalized discrete diffusion equation corresponding to the correlated random walk, which is also known as a generalization of the well known random walk. By applying the technique called ultradiscretization, we obtain the generalized ultradiscrete diffusion equation, the ultradiscrete Cole--Hopf transformation and a variant of the ultradiscrete Burgers equation. Moreover, we show that the resulting ultradiscrete Burgers equation yields cellular automata which can be interpreted as a traffic flow model.