Controlling the dynamics of Burgers equation with a high-order nonlinearity
type: Item
instance of: scholarly article
title: Controlling the dynamics of Burgers equation with a high-order nonlinearity
zbMATH Open document ID: 1070.35072
DOI: 10.1155/S0161171204404116
author: Nejib Smaoui
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2005-05-25T00:00:00Z
full work available at URL: https://eudml.org/doc/51596
review text: Summary: We investigate analytically as well as numerically Burgers equation with a high-order nonlinearity (i.e., \(u_{t}=\nu u_{xx}-u^{n}u_{x}+mu+h(x)\)). We show existence of an absorbing ball in \(L^{2}[0,1]\) and uniqueness of steady state solutions for all integer \(n \geq 1\). 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 the Chebychev collocation method with the 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.
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 35B37
Mathematics Subject Classification ID: 93D21
zbMATH DE Number: 2171782
zbMATH Keywords: absorbing ball
zbMATH Keywords: adaptive nonlinear boundary controller
zbMATH Keywords: asymptotic stability
zbMATH Keywords: Chebyshev collocation
MaRDI profile type: Publication
OpenAlex ID: W2145434844
description: scientific article; zbMATH DE number 2171782