Differential Constraints and Exact Solution to Riemann Problems for a Traffic Flow Model

The constant astigmatism equation. New exact solution

“…In each particular case, the method of differential constraints utilizes specific features of a corresponding nonlinear system (see Refs. [13], [15], [12], [4], [5], [1], [2], [3]). First, for further convenience we change the sign u → −u in the Constant Astigmatism equation (1)…”

The Constant Astigmatism Equation. New Exact Solution

“…In each particular case, the method of differential constraints utilizes specific features of a corresponding nonlinear system (see Refs. [13], [15], [12], [4], [5], [1], [2], [3]). First, for further convenience we change the sign u → −u in the Constant Astigmatism equation ( 1)…”

“…Without loss of generality, we consider the first such a particular solution determined by ( 19), (20), i.e. u = (w + t) 2 , where w(x, t) is a solution of the algebraic equation…”

Solutions via double wave ansatz to the 1-D non-homogeneous gas-dynamics equations

“…Therefore, the main aim of the present paper is to develop, along the lines of the analysis carried on in [22], a reduction procedure for determing classes of exact double wave solutions to (1)-( 3) and, consequently, some nonlinear wave problems of relevant interest such as Riemann problems ( [30]- [32]) and nonlinear wave interactions ( [33]- [35]) are analysed. In particular, after reducing the full set of governing equations to a suitable 2 × 2 hyperbolic auxiliary system, following the idea developed in [17,18], a Riemann problem will be solved as well as an exact analitical description of nonlinear waves interaction admitted by the governing system under interest will be given. Furthermore, possible functional forms of the pressure p(ρ, s) as well as of the production term f (ρ, u) allowing the reduction procedure under interest to hold are characterized.…”