Explicit Scheme, stability condition
If we assume that
x can be expanded in a basis of
x=(\sin{(\theta)}, \sin{(2\theta)},\dots, \sin{(n\theta)})
with
\theta = l\pi/n+1 , where we have the endpoints given by
x_0 = 0 and
x_{n+1}=0 , we can rewrite the
last equation as
\begin{equation*}
2\sin{(i\theta)}-\sin{((i+1)\theta)}-\sin{((i-1)\theta)}=\mu_i\sin{(i\theta)},
\end{equation*}
or
\begin{equation*}
2\left(1-\cos{(\theta)}\right)\sin{(i\theta)}=\mu_i\sin{(i\theta)},
\end{equation*}
which is nothing but
\begin{equation*}
2\left(1-\cos{(\theta)}\right)x_i=\mu_ix_i,
\end{equation*}
with eigenvalues
\mu_i = 2-2\cos{(\theta)} .
Our requirement in
Eq. (8) results in
\begin{equation*}
-1 < 1-\alpha2\left(1-\cos{(\theta)}\right) < 1,
\end{equation*}
which is satisfied only if \alpha < \left(1-\cos{(\theta)}\right)^{-1} resulting in
\alpha \le 1/2 or \Delta t/\Delta x^2 \le 1/2 .