terminology - What's the difference between "boundary value problems" and "initial value problems"?

Mathematically speaking, is there any essential difference between initial value problems and boundary value problems?

The specification of the values of a function $f$ and the "velocities" $\frac{\partial f}{\partial t}$ at an initial time $t=0$ can also be seen, I think, as the specification of boundary values, since the boundaries of the variable $t$ are, usually, at $t=0$ and $t<\infty$.

Answer

In many cases, there really is no difference. Think of the specification of initial values as boundary values on a "time slice." (Incidentally, I addressed a question tangentially related to this the other day: Differentiating Propagator, Greens function, Correlation function, etc) However, sometimes the specificity of calling something an initial value question might indicate something useful about the boundary, e.g. that it is a Cauchy surface and all of the rest of space lies in its causal future/past if the problem is relativistic.