Differential constraints and exact solutions to Riemann problems for a traffic flow model

In view of solving in a closed form initial and/or boundary value problems of interest in nonlinear hyperbolic and dissipative wave processes it is considered a reduction approach based upon appending differential constraints to quasilinear nonhomogeneous hyperbolic systems of first order PDEs. In this context a governing model of traffic flow is analyzed thoroughly and a classification of possible constraints along with sets of consistent response functions involved therein is worked out whereupon the classes of corresponding exact solutions are determined. To some extent these solutions generalize the classical simple wave solutions and may also incorporate dissipative effects. Furthermore, in order to solve a Riemann Problem, an exact rarefaction wave-like solution is built. Finally an application of the results to the so-called “green traffic light problem” is also illustrated.