The Generalized Burgers Equation with and without a Time Delay
نویسنده
چکیده
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 = νuxx − 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) = νuxx(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.
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