Asymptotic stability of rarefaction waves for the generalized KdV–Burgers equation on the half-line

2009 ◽  
Vol 70 (1) ◽  
pp. 372-384 ◽  
Author(s):  
Lei Yao ◽  
Lizhi Ruan
Keyword(s):  
Asymptotic Stability ◽  
Burgers Equation ◽  
Half Line ◽  
Rarefaction Waves

scholarly journals Controlling the dynamics of Burgers equation with a high-order nonlinearity

2004 ◽  
Vol 2004 (62) ◽  
pp. 3321-3332 ◽  
Author(s):  
Nejib Smaoui
Keyword(s):  
Steady State ◽  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Nonlinear Boundary ◽  
Burgers Equation ◽  
High Order ◽  
Time Stepping ◽  
Backward Euler ◽  
High Order Nonlinearity ◽  
Steady State Solutions

We investigate analytically as well as numerically Burgers equation with a high-order nonlinearity (i.e.,ut=νuxx−unux+mu+h(x)). We show existence of an absorbing ball inL2[0,1]and uniqueness of steady state solutions for all integern≥1. Then, we use an adaptive nonlinear boundary controller to show that it guarantees global asymptotic stability in time and convergence of the solution to the trivial solution. Numerical results using Chebychev collocation method with backward Euler time stepping scheme are presented for both the controlled and the uncontrolled equations illustrating the performance of the controller and supporting the analytical results.

Behaviors of Solutions for the Burgers Equation with Boundary corresponding to Rarefaction Waves

1998 ◽  
Vol 29 (2) ◽  
pp. 293-308 ◽  
Author(s):  
Tai-Ping Liu ◽  
Akitaka Matsumura ◽  
Kenji Nishihara
Keyword(s):  
Burgers Equation ◽  
Rarefaction Waves

scholarly journals Stability of Traveling Waves to a Burgers Equation with 2nd-Order Nonlinear Diffusion

2022 ◽  
Vol 4 (1) ◽  
pp. 77-85
Author(s):  
Mohammad Ghani
Keyword(s):  
Asymptotic Stability ◽  
Small Perturbation ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Nonlinear Diffusion ◽  
Wave Amplitude ◽  
Energy Estimate ◽  
Burgers Equation ◽  
Original Equation ◽  
The Stability

We are interested in the study of asymptotic stability for Burgers equation with second-order nonlinear diffusion. We first transform the original equation by the ansatz transformation to establish the existence of traveling wave. We further employ the energy estimate under small perturbation and arbitrary wave amplitude. This energy estimate is then used to establish the stability.

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