Explicit Scheme, matrix-vector formulation
This results in a matrix-vector multiplication
$$
\begin{equation*}
V_{j+1} = \mathbf{A}V_{j}
\end{equation*}
$$
with the matrix \( \mathbf{A} \) given by
$$
\begin{equation*}
\mathbf{A}=\begin{bmatrix}1-2\alpha&\alpha&0& 0\dots\\
\alpha&1-2\alpha&\alpha & 0\dots \\
\dots & \dots & \dots & \dots \\
0\dots & 0\dots &\alpha& 1-2\alpha\end{bmatrix}
\end{equation*}
$$
which means we can rewrite the original partial differential equation as
a set of matrix-vector multiplications
$$
\begin{equation*}
V_{j+1} = \mathbf{A}V_{j}=\dots = \mathbf{A}^{j+1}V_0,
\end{equation*}
$$
where \( V_0 \) is the initial vector at time \( t=0 \) defined by the initial value
\( g(x) \).
In the numerical implementation
one should avoid to treat this problem as a matrix vector multiplication
since the matrix is triangular and at most three elements in each row are different from zero.