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.