System Of Hyperbolic Differential Equations Research Articles

Calcul d'un ecoulement supersonique bidimensionnel d'un fluide diphasique en presence de condensation

Thanks to the replacement of the condensation kinetics equation of a vapor in a gas stationary flow in a two-dimensional configuration by four differential equations describing the particle size distribution in the condensed phase, the system of hyperbolic differential equations for the evolutions of the two phases may be solved numerically by a Lax-Wendroff finite difference scheme. The theoretical results show how the evolutions of the physical flow quantities along the two-phase fluid streamlines, the properties of the nucleation and growth zones of the condensed nuclei, and the shape of the line of onset of condensation, are influenced by the two-dimensional effect due to the nozzle geometry.

Formation of singularities in one‐dimensional nonlinear wave propagation

The waves considered here are solutions of a first-order strictly hyperbolic system of differential equations, written in the form $${u_t} + a(u){u_x} = 0$$ (1) , where u = u(x, t) is a vector with n components u 1,· ·· , u n depending on two scalar independent variables x, t, and a = a(u) is an n-th order square matrix. The question to be discussed is the formation of singularities of a solution u of (1) corresponding to initial data (2) u(x, 0) = f(x). It will be shown that if the system (1) is “genuinely nonlinear” in a sense to be defined below, and if the initial data are “sufficiently small” (but not identically 0), the first derivatives of u will become infinite for certain (x, t) with t > 0. The result is well known for n = 1 and n = 2 (see the papers by Lax and by Glimm and Lax [1], [2], [10]). In many cases such a singular behavior can be identified physically with the formation of a shock, and considerable interest attaches to the study of the subsequent behavior of the solution. In the present paper we shall be content just to reach the onset of singular behavior, without attempting to define and to follow a solution for all times. For that we would need a physical interpretation of our system to guide us in formulating proper shock conditions.

Estimate of the time of occurrence of discontinuities in the solution of a boundary value problem for a second order quasilinear hyperbolic system: PMM vol. 36, n≗3, 1972, pp. 528–532

An estimate is obtained for the time of occurrence of a discontinuity in the solutions of a hyperbolic system of quasilinear differential equations with zero initial and continuous boundary conditions. Examples are presented for one-dimensional gasdynamics and geometrically nonlinear elasticity theory problems. Problems of the analysis of nonlinear transient waves in gasdynamics and elasticity theory result in the integration of quasilinear hyperbolic systems of differential equations. Problems of the occurrence of discontinuities in the solutions of such systems under continuous initial conditions (the Cauchy problem), as well as under continuous boundary conditions (the boundary value problem) have been examined in [1– 6], The most general result has been obtained in [4] for the Cauchy problem, for which upper and lower bounds for the time of occurrence of discontinuities in the solution of a second order system have been established by using a congruence theorem. An analogous result is obtained herein for the boundary value problem by using the Jeffrey method [4].

A Finite Difference Scheme and an Existence Theorem for a Nonlinear Hyperbolic System of Differential Equations

We consider hyperbolic nonlinear systems of the form \[ (1)\qquad \frac{{\partial R}}{{\partial t}} = f(R,S)\frac{{\partial R}}{{\partial x}},\qquad \frac{{\partial S}}{{\partial t}} = - g(R,S)\frac{{\partial S}}{{\partial x}},\] where f and g are positive, differentiable functions of R and S, such that \[ \frac{{\partial f}}{{\partial R}} \geqq \alpha \frac{{\partial f}}{{\partial S}} \geqq 0,\qquad \frac{{\partial g}}{{\partial S}} \geqq \frac{1}{\alpha }\frac{{\partial g}}{{\partial R}} \geqq 0\] for some positive $\alpha $. We study a finite difference scheme to solve the Cauchy problem for (1). Under appropriate assumptions on the initial data, we establish stability estimates for the finite difference approximations, and by means of suitable compactness theorems, we prove the convergence of these approximations to a certain weak solution of our problem. This solution is such that only one of the two functions R and S may become discontinuous.

Estimates of solutions of hyperbolic systems of differential equations in two independent variables

Estimates of solutions of hyperbolic systems of differential equations in two independent variables