International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 62, Pages 3321-3332
doi:10.1155/S0161171204404116
Controlling the dynamics of Burgers equation with a high-order nonlinearity
Department of Mathematics & Computer Science, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait
Received 7 April 2004
Copyright © 2004 Nejib Smaoui. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
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 in L2[0,1]
and uniqueness of steady state solutions for all integer n≥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.