Specification of the boundary condition in high order PDE

[jollage]

Hi all,

I'm asking a question about the number of the boundary conditions in high-order PDE. Say, we are solving the nonlinear Burger's equation
$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\nu \frac{\partial^2 u}{\partial x^2}$ subject to the initial condition $u(x,0)=g(x)$ with addition boundary condition.
I saw in some references that the boundary condition is specified as $u(0,t)=f(t)$. So there is only one boundary condition at the origin, while for completeness, I think there should be another boundary condition for the viscous term. What I suppose is that the implicit unsaid is $u(\infty,t)=0$. But I don't know, it's a guess.

Could you who are experienced in PDE clarify what's the physical meaning of only specifying a boundary condition at the origin even for a high-order PDE? And to what extent is this formulation mathematically sound? Thank.

Jo

 

[pasmith]

Hi all,

I'm asking a question about the number of the boundary conditions in high-order PDE. Say, we are solving the nonlinear Burger's equation
$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\nu \frac{\partial^2 u}{\partial x^2}$ subject to the initial condition $u(x,0)=g(x)$ with addition boundary condition.
I saw in some references that the boundary condition is specified as $u(0,t)=f(t)$. So there is only one boundary condition at the origin, while for completeness, I think there should be another boundary condition for the viscous term. What I suppose is that the implicit unsaid is $u(\infty,t)=0$. But I don't know, it's a guess.

Could you who are experienced in PDE clarify what's the physical meaning of only specifying a boundary condition at the origin even for a high-order PDE? And to what extent is this formulation mathematically sound? Thank.

Jo

The usual assumption is $u(\infty,t) = 0$.