# Natural Boundary Condition

## Main Question or Discussion Point

Hi, I was reading Lemon's Perfect Form and it talked about "natural boundary conditions". But I don't understand exactly how one determines them. It seems to me that one imposes some random condition then deduce stuff from it...?!

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Born2bwire
Gold Member
That's probably it though I do not have access to Lemon's text to confirm how he is using it but just a quick glance at the book's description I think you are probably correct. A lot of times if we approach a problem using a set of differential equations we find that we can deduce the appropriate boundary conditions just from inspection. For example, let's say we have a second order ODE that describes a wave in an inhomogeneous medium. One logical restriction would be that the elements of the ODE should be finite and this imposes restrictions on the derivatives that exist in the equation. From this we can find a set of boundary conditions that must arise to satisfy this restriction.

Likewise in say a problem of the minimum of an action we may introduce variables via taking the variation that start without any explicit boundary conditions imposed upon them. But by virtue of the need of minimization we may find that there must be a set of boundary conditions that have to be satisfied. Take for example the following expression:

$$\int w f(x,y) dxdy = 0$$

Now if we are taking a variational we could probably have said that the function or value w is arbitrary. If that is so, then the only way for the above integral to always be zero regardless of w would be that f(x,y) is identically zero over the domain of the integration. In this way we can infer a boundary condition on the function f(x,y) by virtue of how the mathematics falls out from the original problem.

thanks! this is very helpful :)