Developments in the theory of the Prigogine-Herman kinetic equation of vehicular traffic

The nonlinear kinetic equation of Prigogine and Herman is examined in regards to existence and uniqueness of solutions. The solution exists and is unique in the Banach space of bounded continuous functions over a particular subspace Ω. The equilibrium solution of the kinetic equation of Prigogine and Herman is used to derive asymptotic type series expansions in the form of Hilbert or Chapman and Enskog for concentrations (c ) corresponding to the stable flow regime of traffic (0 < c < c crit ). As expected the conservation of mass equation, the Lighthill-Whitham-Richards model, can be obtained from these expansions. We use the Chapman-Enskog expansion to obtain hydrodynamic-like equations equivalent to the Euler, Navier-Stokes or Burnett equations of fluid flow, depending on the order of the series expansions we used. The zeroth and first order hydrodynamic-like partial differential equations are solved using appropriate conservative numerical schemes. Analogous continuum approximations up to order one are obtained from the Hilbert expansion. Last a zeroth-order (extended Lighthill-Whitham-Richards) model is obtained for unstable flow at sufficiently high concentrations