Journal of Applied Mathematics and Stochastic Analysis
Volume 2004 (2004), Issue 1, Pages 73-96
doi:10.1155/S1048953304210012

The generalized Burgers equation with and without a time delay

Nejib Smaoui and Mona Mekkaoui

Department of Mathematics and Computer Science, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait

Received 3 October 2002; Revised 27 April 2003

Copyright © 2004 Nejib Smaoui and Mona Mekkaoui. 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 consider the generalized Burgers equation with and without a time delay when the boundary conditions are periodic with period 2π. For the generalized Burgers equation without a time delay, that is, ut=vuxx−uux+u+h(x), 0<x<2π, t>0, u(0,t)=u(2π,t), u(x,0)=u0(x), a Lyapunov function method is used to show boundedness and uniqueness of a steady state solution and global stability of the equation. As for the generalized time-delayed Burgers equation, that is, ut(x,t)=vuxx(x,t)−u(x,t−τ)ux(x,t)+u(x,t), 0<x<2π, t>0, u(0,t)=u(2π,t), t>0, u(x,s)=u0(x,s), 0<x<2π, −τ≤s≤0, we show that the equation is exponentially stable under small delays. Using a pseudospectral method, we present some numerical results illustrating and reinforcing the analytical results.