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 \( \Delta t/\Delta x^2 \le 1/2 \).