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

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].

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].

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