Implicit Scheme
It means that we can rewrite the problem as
$$
\begin{equation*}
V_{j} = \mathbf{A}^{-1}V_{j-1}=\mathbf{A}^{-1}\left(\mathbf{A}^{-1}V_{j-2}\right)=\dots = \mathbf{A}^{-j}V_0.
\end{equation*}
$$
This is an implicit scheme since it relies on determining the vector
\( u_{i,j-1} \) instead of \( u_{i,j+1} \).
If \( \alpha \) does not depend on time \( t \), we need
to invert a matrix only once. Alternatively we can solve this system of equations using our methods
from linear algebra.
These are however very cumbersome ways of solving since they involve \( \sim O(N^3) \) operations
for a \( N\times N \) matrix.
It is much faster to solve these linear equations using methods for tridiagonal matrices,
since these involve only \( \sim O(N) \) operations.