How can I deal with an NDSolve::ivone message regarding boundary values?

Question

I have a question regarding the error message:

NDSolve::ivone: Boundary values may only be specified for one independent variable.
Initial values may only be specified at one value of the other independent variable. >>

I'm curious as to when this error shows up. If I run the heat equation example from the NDSolve documentation, I get a valid solution. However, consider an edit to the p.d.e to

NDSolve[{D[u[t, x], {t, 2}] == D[u[t, x], {x, 2}], u[0, x] == 0, 
  u[t, 0] == 0, u[t, 5] == 0}, u, {t, 0, 10}, {x, 0, 5}]

That is, I changed $u_t = u_{xx}$ to $u_{tt} = u_{xx}$ and also changed an initial condition. (A solution of course is $u(x, t) = 0$.) Now, if I run this, I will get the error quoted above. I believe NDSolve doesn't even try to run.

How do I need to edit the boundary conditions in order to get a run attempt? Does NDSolve only accept certain types of boundary conditions dependent on the order of the p.d.e? How do I find out what boundary conditions are acceptable?

I'm asking about this because I'm trying to solve a very complicated third-order, nonlinear, p.d.e resulting from an optic equation and every set of initial conditions that I have given NDSolve for this 3rd-order equation has resulted in the above error.

Answer 1

For a wave equation (2nd order in time) you need an initial condition and a derivative of an initial condition, like so:

NDSolve[{D[u[t, x], {t, 2}] == D[u[t, x], {x, 2}], u[0, x] == 0, 
  Derivative[1, 0][u][0, x] == 0, u[t, 0] == 0, u[t, 5] == 0}, u, {t, 
  0, 10}, {x, 0, 5}]

As a side note sometimes (not in this case though) it may be helpful to NDSolve to specify the temoral variable. This can be done like this:

NDSolve[{D[u[t, x], {t, 2}] == D[u[t, x], {x, 2}], u[0, x] == 0, 
  Derivative[1, 0][u][0, x] == 0, u[t, 0] == 0, u[t, 5] == 0}, u, {t, 
  0, 10}, {x, 0, 5}, Method -> {MethodOfLines, TemporalVariable -> t}]