Reduction procedure and generalized simple waves for systems written in Riemann variables
type: Item
instance of: scholarly article
title: Reduction procedure and generalized simple waves for systems written in Riemann variables
zbMATH Open document ID: 1015.35067
DOI: 10.1023/A:1020341610639
author: N. Manganaro
author: Sergey V. Meleshko
published in: Nonlinear Dynamics
publication date: 2003-04-03T00:00:00Z
review text: The authors of this interesting paper study quasilinear hyperbolic systems of conservation laws, which have the form, \(\partial u/\partial t+Q\partial u/\partial x =f\), where \(Q=Q(x,t,u)\) is an \(m\times m \) matrix, \(f=f(x,t,u)\) is a vector-function, \(x\in \mathbb{R}\) and \(t\in\mathbb{R}_+\) are real independents, and \(u\) is some unknown solution. There are first-order differential constraints \(\Phi_k(x,t,u,u_x)=0\) \((k=1,2,\dots ,q)\) which satisfy the natural requirement \(\text{rank}(\partial \Phi_k/\partial u_x)=q\). A solution of the system under consideration, which satisfies the above stated constraints, is called a solution characterized by differential constraints. The system with constraints is an overdetermined system and the known method of solving requires some compatibility. A class of solutions called generalized simple waves is investigated. The main feature of these solutions is that they are constructed by integrating a system of ordinary differential equations. The solutions of a system of two equations reduced in Riemann variables can be obtained by integrating some simple equations. The mathematical models of traffic flow and rate-type materials are analyzed as examples by which the above method is illustrated. Exact solutions called generalized simple waves are obtained.
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35L67
Mathematics Subject Classification ID: 90B20
Mathematics Subject Classification ID: 35L45
Mathematics Subject Classification ID: 35N10
Mathematics Subject Classification ID: 74D10
zbMATH DE Number: 1892292
zbMATH Keywords: method of differential constraints
zbMATH Keywords: compatibility
reviewed by: Dimitar A. Kolev
MaRDI profile type: Publication
full work available at URL: a:1020341610639
OpenAlex ID: W144074361
description: scientific article; zbMATH DE number 1892292