When can I use Fourier Series to solve the heat/wave equation on $[0,L]$?

I don't have a complete answer, but just some preliminary thoughts: the idea behind using Fourier analysis to solve constant coefficient linear PDEs is to transform a partial differential equation into an ordinary differential equation. In symbols, suppose $psi:Itimesmathbb{R}^nto mathbb{C}$ solves the PDE

$$ sum_{0leq i leq N} P_i(nabla) partial_t^i psi = 0 $$

where $P_i$ are constant coefficient polynomials, the Fourier transform "gives an equation"

$$ sum_{0 leq i leq N} P_i(xi) partial_t^i hat{psi} = 0~. $$

The problem is: how do you interpret this equation? To treat it as an ODE, you need to treat $hat{psi}$ as a map $I to X$ where $X$ is, say, the Hilbert space of $L^2$ functions over $mathbb{R}^n$ or some such.

Now, in your case of prescribing boundary conditions for the interval $[0,1]$, if your boundary conditions were time independent, and if the boundary conditions plays well with the Fourier transform, then you can again recover the ODE formulation. (The solvability of the ODE, as you noted, depends on which Hilbert space you use and the properties of the polynomials $P_i$ on the Hilbert space.)

But if your boundary conditions are time dependent, then an immediate problem is that the Hilbert space in which $hat{psi}$ lives will be time-dependent. So the naive application of "Fourier" methods won't make sense. Geometrically the case where $X$ is a fixed Hilbert space is the analogue of solving an ODE on the trivial vector bundle $V$ over $I$ with trivial connection. The case where $X$ also varies with time can be thought of as having some sort of an attempt at writing down an ODE on an arbitrary vector bundle $V$ over $I$. Without specifying the connection, even the notion of an ODE is not well-defined.

To put it differently, since a connection over a curve is just an identification of the fibres (roughly speaking), what you need to use an analogue of the Fourier method is a collection of 1-parameter families of functions $phi_i(t;x)$, such that

• For each fixed $t$, the functions $phi_i(t;x)$ forms an ON basis of some appropriate Hilbert space


• Each $phi_i(t;x)$ solves the PDE you are looking at

Just directly assuming the trace of $phi_i$ on constant $t$ slices are the trigonometric functions is probably not the right way to go in general.

Not having thought about this problem in detail before, I don't have much more to say. But I suspect that the suitability of individual boundary conditions need to be examined on a case by case basis.