Riemann problems and exact solutions to a traffic flow model

Abstract

Within the theoretical framework of differential constraints method a nonhomogeneous model describing traffic flows is considered. Classes of exact solutions to the governing equations under interest are determined. Furthermore, Riemann problems and generalized Riemann problems which model situations of interest for traffic flows are solved.

Nonlinear wave interaction problems in the three-dimensional case

“…In (13), (14) and in what follows the prime denotes the derivative with respect to the indicated argument. Next, we make use of the general expressions (13), (14) in order to describe nonlinear wave interactions prescribed by the quasilinear system (4).…”

“…and, from (13), (14), the arbitrary functions A k (α k ) as well as dA k dR k (α k ) specialize as follows…”

“…Within such a theoretical framework, the Riemann method was extended to the nonhomogeneous case in [4,5,6] and a large new classes of solutions to quasilinear systems of PDEs have been obtained in [7,8]. Furthermore a combined use of the hodograph method and of the differential constraints technique was considered in [9] and quite recently was used in order to study nonlinear wave interactions [10,11] as well as discontinuous initial value problems [12,13,14] for homogeneous and nonhomogeneous 2 × 2 systems.…”

The limiting behavior of the Riemann solution to the isentropic Euler system for logarithmic equation of state with a source term

“…There are various methods to solve the problem related to delta shock wave in the literature, such as the weak asymptotic method, 21,22 the vanishing viscosity method, 23,24 the shadow wave method, 9 the distributional product method, 25,26 and the split delta function method. [27][28][29] Curro and Manganaro 30 studied Riemann problems and exact solutions to a traffic flow model. Recently, Zeidan and Bira studied weak shock waves and its interaction with characteristic shock in polyatomic gas.…”

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)…”