Solutions of the Nonlinear Wave Equation in the case of a Finite Tube
by
Victoria Iordan
University of the West TIMISOARA, ROMANIA
Coauthors: Agneta M. Balint(University of the West Timisoara), Stefan Balint(University of the West Timisoara)

In this paper we consider the nonlinear wave equation which governs the motion of a compressible fluid in a finite tube. We semidiscretise (in spatial coordinates) this equation and we study the system of nonlinear differential equations which results.// We obtain results concerning the existence of some particular solutions for the system and the existence of all the solutions, on R^1. For the discretisation step tending to zero, we obtain weak solutions for the nonlinear wave equation. These weak solutions present discontinuites, which are shocks.// The used technique was proposed by John von Neumann [3,4] and it was developed by P.Lax in [2] and J.M.Greenberg in [1].// {References} [1] Greenberg,J.M. - The Shock Generation Problem for a Discrete Gas with Short-Range Repulsive Forces,Comm. Pure Appl. Math., Vol.XLV, 1992, pp. 1125-1139.// [2] Hou,T.Y., Lax,P.D.- Dispersive approximations in fluid dynamics, Comm.Pure Appl.

Math. 44, 1991, pp. 1-40.// [3] Von Neumann,J.- Proposal and analysis of a numerical method for the treatment of hydrodynamical shock problems, pp.361-379 in Collected Works Vol. VI,Oxford, 1963.// [4] Von Neumann,J., Richtmyer, R.D..- A Method for the Numerical Calculation of Hydrodynamic Shocks, Journal Appl. Physics, Vol.21, nr.3, pp.380-385,1950

Date received: January 14, 2000