Rigorous solution of wave equation for string with fixed ends Consider the PDE $\dfrac{\partial^2 u}{\partial t^2} = c^2 \dfrac{\partial^2 u}{\partial x^2}$ with $c>0$ and boundary conditions u(0,t) = u(L,t) = 0 corresponding to the problem of a vibrating string fixed at two ends in physics.
By initial considerations, $u$ must be a twice differentiable real valued function, so we'd expect to look for solutions in $C^2(\mathbb{R})$.
Solving the PDE mechanically using common methods results in solutions
$$u(x,t) = \sum_{k=1}^{n} A_k \sin\left(\dfrac{\pi k x}{L}\right) \cos\left(\dfrac{\pi c k t}{L}\right) + B_k \sin\left(\dfrac{\pi k x}{L}\right) \sin\left(\dfrac{\pi c k t}{L}\right)$$
for arbitrary coefficients $A_k, B_k \in \mathbb{C}$. When convergence is guaranteed, infinite sums of this form qualify as solutions as well.
During the solution steps, as a trick, we allow $u$ to be complex valued along the way. Furthermore, the convergence of Fourier series is non-trivial and may further require extending $u$ to be in $L_2$. At this point, we're quite far from $C^2(\mathbb{R})$.


*

*What are appropriate function spaces $X,Y$ for the domain and range of the operator $\left(\dfrac{\partial^2}{\partial t^2} - c^2 \dfrac{\partial^2}{\partial x^2}\right) : X \to Y$ when we are interested in physically realizable, real valued $u$?

*If we set $X$ to be some space of real valued functions, how does the trick to extend $u$ to be complex valued work in full detail while solving the PDE? Similarly, in full detail, how are the obtained solutions restricted back to real valued functions?

*How do we rigorously show that the solutions obtained above are all the solutions in the appropriate sense?
 A: The functions
$$
        s_n(x)= \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right),\;\;\; n \ge 1,
$$
form an orthonormal basis of $L^2[0,L]$, as do the functions
$$
              c_o(x)=\sqrt{\frac{1}{L}},\;\;c_n(x)=\sqrt{\frac{2}{L}}\cos\left(\frac{n\pi x}{L}\right),\;\;\; n \ge 1.
$$
Your solution of the wave equation may be written as
$$
         u(x,t)=\sum_{n=1}^{\infty}\langle u(x,0),s_n\rangle s_n(x)c_n(t)+\sum_{n=0}^{\infty}\frac{L}{n\pi}\langle u_t(x,0),s_n\rangle s_n(x)s_n(t).
$$
This expression for $u$ is consistent with
$$
       u(x,0)=\sum_{n=1}^{\infty}\langle u(x,0),s_n\rangle s_n(x), \\
       u_t(x,0)=\sum_{n=1}^{\infty}\langle u_t(x,0),s_n\rangle s_n(x).
$$
If $u(x,0),u_t(x,0)$ are specified as $L^2[0,L]$ functions, then $u(x,t)$ makes sense, though it takes some work to deal with the function $u$ classically. Working with $t\mapsto u(\cdot,t) \in L^2[0,L]$ turns out to be a useful model instead of thinking in terms of $u(x,t)$. This falls under $C_0$ semigroup theory if you keep the initial data $u(x,0),u_t(x,0)$ in $L^2$. And $L^2$ fits with Physics, too.
