International Journal of Stochastic Analysis

International Journal of Stochastic Analysis / 2004 / Article

Open Access

Volume 2004 |Article ID 175294 | https://doi.org/10.1155/S1048953304210012

Nejib Smaoui, Mona Mekkaoui, "The generalized Burgers equation with and without a time delay", International Journal of Stochastic Analysis, vol. 2004, Article ID 175294, 24 pages, 2004. https://doi.org/10.1155/S1048953304210012

The generalized Burgers equation with and without a time delay

Received03 Oct 2002
Revised27 Apr 2003

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=vuxxuux+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,tτ)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, τs0, 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.

Copyright © 2004 Hindawi Publishing Corporation. 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.


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