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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*}