Generalized discrete and ultradiscrete Burgers equations derived through the correlated random walk. (English) Zbl 1510.39005
Summary: 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.
MSC:
| 39A14 | Partial difference equations |
| 39A60 | Applications of difference equations |
| 82C41 | Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics |
| 76A30 | Traffic and pedestrian flow models |
| 37B15 | Dynamical aspects of cellular automata |
Keywords:
discrete Burgers equation; correlated random walk; Cole-Hopf transformation; cellular automata; traffic flow modelReferences:
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