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Some exact solutions to the Lighthill–Whitham–Richards–Payne traffic flow equations
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Rowlands, G. (George), Infeld, E. and Skorupski, A. A. (2013) Some exact solutions to the Lighthill–Whitham–Richards–Payne traffic flow equations. Journal of Physics A: Mathematical and Theoretical, Volume 46 (Number 36). Article number 365202. doi:10.1088/1751-8113/46/36/365202 ISSN 1751-8113.
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Official URL: http://dx.doi.org/10.1088/1751-8113/46/36/365202
Abstract
We find a class of exact solutions to the Lighthill–Whitham–Richards–Payne (LWRP) traffic flow equations. Using two consecutive Lagrangian transformations, a linearization is achieved. Next, depending on the initial density, we either apply (again two) Lambert functions and obtain exact formulae for the dependence of the car density and velocity on x, t, or else, failing that, the same result in a parametric representation. The calculation always involves two possible factorizations of a consistency condition. Both must be considered. In physical terms, the lineup usually separates into two offshoots at different velocities. Each velocity soon becomes uniform. This outcome in many ways resembles the two soliton solution to the Korteweg–de Vries equation. We check general conservation requirements. Although traffic flow research has developed tremendously since LWRP, this calculation, being exact, may open the door to solving similar problems, such as gas dynamics or water flow in rivers. With this possibility in mind, we outline the procedure in some detail at the end.
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Physics | ||||
Journal or Publication Title: | Journal of Physics A: Mathematical and Theoretical | ||||
Publisher: | IOP Publishing Ltd | ||||
ISSN: | 1751-8113 | ||||
Official Date: | 2013 | ||||
Dates: |
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Volume: | Volume 46 | ||||
Number: | Number 36 | ||||
Page Range: | Article number 365202 | ||||
DOI: | 10.1088/1751-8113/46/36/365202 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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