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 $t<\infty$.
When there is only one spatial variable then mathematically the two are indistinguishable. But often boundary value problems are solved over a higher dimensional domain. For example, a common problem in physics is to solve Laplace's equation over a spatial region of three dimensions, with a two dimensional surface providing the boundary conditions. If the boundary condition specifies the value of the solution on the surface, then it is called a Dirichlet boundary condition. However, sometimes the boundary condition specifies the normal derivative of the solution at the surface, and then it is called a Neumann boundary condition. Boundary value problems over multi-dimensional domains are necessarily tied to partial differential equations rather than ordinary differential equations, and so they are more complicated than ordinary differential equations with a single initial value specified.
Seems to me the difference is semantic:
It is implicit that one is seeking a specific solution to a problem in time and space given the initial values.
The boundary conditions bound the solutions but do not pick up a specific solution, unless the initial values are used.
Initial values pick up a specific solution from the family of solutions allowed/defined by the boundary conditions.
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.
