GROWTH RATES OF TOTAL VARIATIONS OF SNAPSHOTS OF THE 1D LINEAR WAVE EQUATION WITH COMPOSITE NONLINEAR BOUNDARY REFLECTION RELATIONS

2003 ◽  
Vol 13 (05) ◽  
pp. 1183-1195 ◽  
Author(s):  
YU HUANG
Keyword(s):  
Wave Equation ◽  
Nonlinear Boundary ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Distributed Parameter ◽  
Linear Wave ◽  
Linear Wave Equation ◽  
Energy Pumping ◽  
Long Time

The linear wave equation on an interval with a van der Pol nonlinear boundary condition at one end and an energy-pumping condition at the other end is a useful model for studying chaotic behavior in distributed parameter system. In this paper, we study the dynamics of the Riemann invariants (u, v) of the wave equation by means of the total variations of the snapshots on the spatial interval. Our main contributions here are the classification of the growth of total variations of the snapshots of u and v in long-time horizon, namely, there are three cases when a certain parameter enters a different regime: the growth (i) remains bounded; (ii) is unbounded (but nonexponential); (iii) is exponential, for a large class of initial conditions with finite total variations. In particular, case (iii) corresponds to the onset of chaos. The results here sharpen those in an earlier work [Chen et al., 2001].

RAPID FLUCTUATIONS OF SNAPSHOTS OF ONE-DIMENSIONAL LINEAR WAVE EQUATION WITH A VAN DER POL NONLINEAR BOUNDARY CONDITION

2005 ◽  
Vol 15 (02) ◽  
pp. 567-580 ◽  
Author(s):  
YU HUANG ◽  
JUN LUO ◽  
ZUOLING ZHOU
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Condition ◽  
Dynamical Behavior ◽  
Linear Wave ◽  
Van Der Pol ◽  
Linear Wave Equation ◽  
Energy Pumping ◽  
Long Run

In this paper, we consider a linear wave equation on an interval with a van der Pol nonlinear boundary condition at one end and an energy-pumping condition at the other end. We study the dynamical behavior of the Riemann invariants (u,v) of the wave equation in terms of the growth rates of the total variations of the snapshots on the spatial interval. Our main contributions here are the detection of rapid fluctuations of the snapshots of u and v in the long run. The results here sharpen those in the earlier works of [Chen et al., 2001] and [Huang, 2003].

A linear wave equation with a nonlinear boundary condition of viscoelastic type

2010 ◽  
Vol 72 (3-4) ◽  
pp. 1865-1885
Author(s):  
Le Thi Phuong Ngoc ◽  
Nguyen Anh Triet ◽  
Nguyen Thanh Long
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Condition ◽  
Linear Wave ◽  
Linear Wave Equation

A Solution for the Interior Sound Field With a Nonuniform Temperature Distribution in the Vehicle Passenger Room

2001 ◽  
Author(s):  
Jiuhui Wu ◽  
Hualing Chen
Keyword(s):  
Temperature Field ◽  
Wave Equation ◽  
Temperature Distribution ◽  
Theoretical Foundation ◽  
Initial Conditions ◽  
Sound Field ◽  
Linear Wave ◽  
Nonuniform Temperature ◽  
Linear Wave Equation ◽  
Nonuniform Temperature Distribution

Abstract A non-linear wave equation in a nonuniform sound space is deduced to determine the coupling relation between sound field and temperature field. Adopting Poisson’s integral a formula for calculating the interior sound field with uneven temperature field and certain initial conditions is developed. This establishes the corresponding theoretical foundation for studying the heat and vibroacoustic comfortability in a vehicle passenger room.

Randomly perturbed vibrations

1995 ◽  
Vol 32 (02) ◽  
pp. 417-428 ◽  
Author(s):  
M. Elshamy
Keyword(s):  
Dirichlet Boundary ◽  
Space Variable ◽  
Initial Conditions ◽  
Linear Wave ◽  
Stochastic Perturbations ◽  
Uniform Topology ◽  
Linear Wave Equation ◽  
Random Forcing ◽  
Continuity Properties ◽  
Positive Time

Let u ε(t, x) be the position at time t of a point x on a string, where the time variable t varies in an interval I: = [0, T], T is a fixed positive time, and the space variable x varies in an interval J. The string is performing forced vibrations and also under the influence of small stochastic perturbations of intensity ε. We consider two kinds of random perturbations, one in the form of initial white noise, and the other is a nonlinear random forcing which involves the formal derivative of a Brownian sheet. When J has finite endpoints, a Dirichlet boundary condition is imposed for the solutions of the resulting non-linear wave equation. Assuming that the initial conditions are of sufficient regularity, we analyze the deviations u ε(t, x) from u 0(t, x), the unperturbed position function, as the intensity of perturbation ε ↓ 0 in the uniform topology. We also discuss some continuity properties of the realization of the solutions u ε(t, x).

scholarly journals Energy decay rate for a quasi-linear wave equation with localized strong dissipation

2011 ◽  
Vol 62 (1) ◽  
pp. 164-172 ◽  
Author(s):  
Daewook Kim ◽  
Yong Han Kang ◽  
Mi Jin Lee ◽  
Il Hyo Jung
Keyword(s):  
Wave Equation ◽  
Decay Rate ◽  
Energy Decay ◽  
Linear Wave ◽  
Linear Wave Equation ◽  
Energy Decay Rate

scholarly journals Parameter identification for the linear wave equation with Robin boundary condition

2019 ◽  
Vol 27 (1) ◽  
pp. 25-41
Author(s):  
Valeria Bacchelli ◽  
Dario Pierotti ◽  
Stefano Micheletti ◽  
Simona Perotto
Keyword(s):  
Wave Equation ◽  
Mixed Boundary ◽  
Stability Result ◽  
Interval Length ◽  
Left Endpoint ◽  
Linear Wave ◽  
Reconstruction Procedure ◽  
Element Discretization ◽  
Linear Wave Equation ◽  
Bounded Interval

Abstract We consider an initial-boundary value problem for the classical linear wave equation, where mixed boundary conditions of Dirichlet and Neumann/Robin type are enforced at the endpoints of a bounded interval. First, by a careful application of the method of characteristics, we derive a closed-form representation of the solution for an impulsive Dirichlet data at the left endpoint, and valid for either a Neumann or a Robin data at the right endpoint. Then we devise a reconstruction procedure for identifying both the interval length and the Robin parameter. We provide a corresponding stability result and verify numerically its performance moving from a finite element discretization.

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