Generalized discrete and ultradiscrete Burgers equations derived through the correlated random walk. Issue 1 (2nd January 2023)

• Record Type: Journal Article
• Authors: Fukuda, Akiko; Segawa, Etsuo; Watanabe, Sennosuke
• Is Part Of: Journal of difference equations and applications. Volume 29:Issue 1(2023)
• Page Start: 84
• Page End: 101
• Publication Date: 2023-01-02
• Subjects: Discrete Burgers equation -- correlated random walk -- Cole–Hopf transformation -- cellular automata -- traffic flow model; 39A60 -- 82C41 -- 37J70 -- 37B15; Difference equations -- Periodicals; 515.625
• DOI: 10.1080/10236198.2023.2172969
• Languages: English
• ISSNs: 1023-6198

Abstract

Abstract : The correlated random walk is known as a generalization of the well-known random walk. In this study, we present that a generalized discrete Burgers equation corresponding to the correlated random walk can be obtained through a Cole–Hopf transformation to a generalized discrete diffusion equation. By applying a technique called ultradiscretization, the generalized ultradiscrete diffusion equation, the ultradiscrete Cole–Hopf transformation, and a variant of the ultradiscrete Burgers equation are obtained. Additionally, this study shows that the resulting ultradiscrete Burgers equation yields cellular automata that can be interpreted as a traffic flow model with controllability of vehicle flow.