# mathematical statement of “open” boundary condition

For your information, the original equation comes from here. Note: You DON'T have to read the paper. I will make the question as self-contained as possible.

The central equation to solve is equation (16), which is of the form (my question extends to other system of PDE with similar form):

$\dfrac{\partial f_n(x;\tau)}{\partial \tau} =\left[ [-(n-6)+\dot{\bar{A}}x]\dfrac{\partial}{\partial x}+2\dot{\bar{A}}-(c_++c_-)n \right]f_n(x;\tau) +c_+(n-1)f_{n-1}(x;\tau)+c_-(n+1)f_{n+1}(x;\tau)$

$x$ is continuous and positive, $\tau$ is basically time, $n$ is positive integer number and $f_n(x;t)$ goes to zero very quickly as $x\rightarrow\infty$ so that we can have a cutoff $x_{max}$. The boundary conditions used by the paper is called "used open boundary conditions, i.e., probability ($f_n(x;\tau)$ is proportional to probability density) could flow out of the system across the cutoffs, but no probability could flow into the system, since $f_n(x)$vanished beyond the cutoffs."

additional clarifications for the symbols: $\dot{\bar{A}}=\sum_{k=0}^5(6-k)f_k(0;\tau)$, $c_+=(1/6)f_5(0;\tau)$ and $c_-=(1/6)\sum_{k=0}^5(6-k)f_k(0;\tau)$.

My understanding of the above description is that the BC looks like some kind of absorbing boundary conditions. The "mass" of $f_n$ can flow outside the interval $[0,x_{max}]$ but nothing outside can flow into the interval, such that the integral of $f_n(x;\tau)$ over the interval will decrease with time.

If we discretize $x$, it's not very difficult to write a program for explicit scheme according to the above description, but the implicit scheme would be difficult (there are two index $x$ and $n$).

So my question is: how to write down the mathematical statement (for continuous $x$) for such boundary conditions? The motivation for writing down a mathematical statement instead of directly writing down the BC for discretized $x$ is that I can let software such as Mathematica. Then why don't I ask the question in Mathematica exchange? Yes, I did. But the folks there seemingly thinks that I should figure out the mathematical statement first.

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I only see first-order derivative w.r.t $x$. Do you have a BC at $x=0$? If so, maybe you do not need a BC at $x=+\infty$. –  Hui Zhang Feb 23 '13 at 8:10
@HuiZhang,My question is exactly about how to mathematically express the BC at x=0. –  wdg Feb 24 '13 at 6:01
So your original (before cut-off) problem is really defined in $x\in (-\infty,\infty)$. Then, the BCs for truncated domain at $x=0$ and $x=x_{\max}$ should come from the time initial condition. I thought about the simpler constant-coefficient linear system of first-order PDEs $\mathbf{u}_t+B\mathbf{u}_x+C\mathbf{u}=0$. If $B$ and $C$ are simultaneously diagonalizable, we need only to consider the scalar equation $v_t+bv_x+cv=0$, for which a characteristic line gives on it a ODE $dv/dt+cv=0$. The time initial condition propagates to the truncation boundaries along characteristic lines. –  Hui Zhang Feb 24 '13 at 10:17

I would sum this up as a coupled system of advection-reaction equations. You should only need a boundary condition in $x$ at the boundary where characteristics flow into the domain, which depends on the sign of the coefficient of the $x$-derivative term.
I'm not sure if I understand your answer. I do need BC in $x$ for $x=0$ and $x=x_{max}$. Could you write the BC down as mathematical expressions? –  wdg Feb 24 '13 at 9:32
@wdg: You do not need a BC at the outflow part of boundaries. Think about the scalar equation $v_t+bv_x+cv=0$, for which the characteristic line is $dx/dt=b$ or $x=bt$. If $b>0$, the direction of propagation $dv/dt+cv=0$ along the characteristic is up-right. So only $x=0$ is inflow boundary but not $x=x_{\max}>0$, relatively to your domain $(0,x_{\max})$. –  Hui Zhang Feb 24 '13 at 10:29
@wdg What is the sign of the coefficient of the $x$-derivative term at each of the boundaries? –  David Ketcheson Feb 26 '13 at 13:16