Rewrite of CN scheme
Using our previous definition of \( \alpha=\Delta t/\Delta x^2 \) we can rewrite Eq.
(9) as
$$
\begin{equation*}
-\alpha u_{i-1,j}+\left(2+2\alpha\right)u_{i,j}-\alpha u_{i+1,j}=
\alpha u_{i-1,j-1}+\left(2-2\alpha\right)u_{i,j-1}+\alpha u_{i+1,j-1},
\end{equation*}
$$
or in matrix-vector form as
$$
\begin{equation*}
\left(2\hat{I}+\alpha\hat{B}\right)V_{j}=
\left(2\hat{I}-\alpha\hat{B}\right)V_{j-1},
\end{equation*}
$$
where the vector \( V_{j} \) is the same as defined in the implicit case while the matrix
\( \hat{B} \) is
$$
\begin{equation*}
\hat{B}=\begin{bmatrix}2&-1&0&0 & \dots\\
-1& 2& -1 & 0 &\dots \\
\dots & \dots & \dots & \dots & \dots \\
\dots & \dots & \dots & \dots &-1 \\
0& 0 & \dots &-1& 2\end{bmatrix}.
\end{equation*}
$$