# 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 $t<\infty$.

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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.

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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.

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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.

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