After learning some distribution theory, I find that in my book, all PDEs given as examples are in free space (without any boundary conditions). I wonder if distribution theory can be used to tackle PDEs with boundary conditions.
To be more specific, let's consider this problem. Let there be a string of length $\pi$ with both ends fixed. Transverse waves can be produced on the string, satisfying the classical wave equation $$ \partial^2_t u(x,t)=c^2\partial^2_x u(x,t). $$ The boundary conditions are $u(0)=u(\pi)=0$.
Now, let impose a wired initial condition: let's pluck the string in the middle, so initially, the string is at rest, in the position $$ u(x,0)=A(\pi/2-|x-\pi/2|), A\in \mathbb R. $$ As one can see, the initial condition is not everywhere differentiable. However, $u$ can be seen as an element of $\mathcal D'(\mathbb R)$ or $\mathcal S'(\mathbb R)$, the space of (tempered) distributions. The differential equation therefore make sense in the sense of distributions.
Using Fourier transform and convolution, we can manage to get a solution, IF there are no boundary conditions. However, in this situation, I do not know how to state the boundary condition in term of distributions.
So, my question now is: can we make sense out of this problem, possibly in the sense of distributions, and solve the equation?
Edit: we can use Fourier series expansion to solve this, but then I don't feel it really a way of "understanding" how it really works - after all, the original equation ceases to make sense when it is not differentiable. I want to somehow have some formalism in making sense of the derivative of a function which is not differentiable. Possibly weak derivative?
Edit: Fourier transform over a bounded interval doesn't seem to be obvious to define; it appears that Fourier series are really easier.