The generalized Burgers equation with and without a time delay

Nejib Smaoui and Mona Mekkaoui

International Journal of Stochastic Analysis, 2004, vol. 2004, 1-24

Abstract:

We consider the generalized Burgers equation with and without a time delay when the boundary conditions are periodic with period 2 π . For the generalized Burgers equation without a time delay, that is, u t = v u x x − u u x + u + h ( x ) , 0 < x < 2 π , t > 0 , u ( 0 , t ) = u ( 2 π , t ) , u ( x , 0 ) = u 0 ( 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, u t ( x , t ) = v u x x ( x , t ) − u ( x , t − τ ) u x ( x , t ) + u ( x , t ) , 0 < x < 2 π , t > 0 , u ( 0 , t ) = u ( 2 π , t ) , t > 0 , u ( x , s ) = u 0 ( x , s ) , 0 < x < 2 π , − 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.

Date: 2004
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnijsa:175294

DOI: 10.1155/S1048953304210012

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