Explicit Scheme, simple case
We specialize to the case
\( a(t)=b(t)=0 \) which results in \( u_{0,j}=u_{n+1,j}=0 \).
We can then reformulate our partial differential equation through the vector
\( V_j \) at the time \( t_j=j\Delta t \)
$$
\begin{equation*}
V_j=\begin{bmatrix}u_{1,j}\\ u_{2,j} \\ \dots \\ u_{n,j}\end{bmatrix}.
\end{equation*}
$$