I am currently taking a course on partial differential equations. So far it has been pretty much on "techniques" used to solve, and not a lot of theory or proofs. So needless to say, I certainly do not fully understand how these methods work.
I understand we make the initial assumption that solutions take the form of products of independent functions, and then see the conditions on them that must follow (and in fact see if solutions of this form could actually exist.) What seems awfully convenient to me is where we end up with an infinite family of solutions to the homogeneous problem that "happen" to form a complete orthogonal family. If we neglect just one of the non-trivial solutions here, it is of course no longer complete given that there would exist a function not identically zero orthogonal to the entire family (i.e. the non-trivial solution.) Something more fundamental must be going on here. Why should these family of solutions have such properties?
Another more specific question I have regards convergence in the L2 respect. Take the 1D heat equation, for example. Keep it simple by having it and all boundary conditions homogeneous. Using the separation of variables method, one would find a general solution to be of the form:
where x = l
is the "right" boundary.
If we have an initial distribution, f(x), where:
then it should be that the Fourier series converges in the L2 sense. That said, it is not true in general that:
This can be easily seen graphically in some cases. The derivative will be not that of what seems to be the general curve but rather the ripples made by the periodic functions. It is more chaotic than practical. This being the case, I have to wonder then, what does the PDE really say about how the solution changes in t for an initial distribution f(x), when you consider the "general" curve of f(x)?
Could one look at the PDE that u satisfies and conclude that the curvature of f(x) is approximately proportional to d/dt(u) near t = 0, even though d2/dx2(u) may be oscillating rapidly (does not look like f(x) in the limit)? Mathematically, it is obvious that any finite sum of terms taken from the Fourier series is a solution to the PDE and so of course d/dt(u) = d2/dx2(u). But as far as the interpretation goes with the general curve of an initial condition, it is not so clear to me given on some scale the curve may actually be very chaotic.