Solving the wave equation, with boundary conditions, in the sense of distributions (Generalized functions)

Question

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.

Answer 1

It is possible to use Fourier series here, by making use of the periodicity and the continuity of the initial condition. Indeed, the problem can be extended to $x$ in $\Bbb R$ by periodization (successive identical strings with fixed ends of length $\pi$). Consider the spatially $\pi$-periodic function $$u(x,t) = \sum_n c_n(t)\, \text{e}^{2\text{i}nx} \, ,$$ which is written as a spatial Fourier series. Its restriction to $x\in [0,\pi]$ may solve the problem if the boundary condition $\sum_n c_n(t) = 0$ is satisfied. Also, at time $t=0$, the Fourier coefficients $c_n(0)$ must be the Fourier coefficients of the initial condition $u(\cdot, 0)$. Similarly, their derivatives $c'_n(0)$ are the Fourier coefficients of the null function, i.e. $c'_n(0) = 0$. Injecting this function in the PDE gives $$ c_n''(t) + 4n^2c^2 c_n(t) = 0 $$ for all $n$, by uniqueness of Fourier series. Therefore, we have $c_n(t) = c_n(0)\cos(2nct)$. The same approach can be followed for other triangular signals, Gaussian signals, rectangular signals, etc.

Instead of the initial Fourier series, we could have written the Fourier transform representation $$u(x,t) = \frac1{2\pi} \int_{\Bbb R} \hat u(k,t)\, \text{e}^{\text{i}kx}\,\text d k \, ,$$ where $\hat u$ is the spatial Fourier transform of $u$.