Step 2: Fulfilling the equation at the mesh points

Require the PDE (1) to be fulfilled at an arbitrary interior mesh point $(x_i,t_n)$ leads to $$\begin{equation} \frac{\partial}{\partial t} u(x_i, t_n) = \dfc\frac{\partial^2}{\partial x^2} u(x_i, t_n) \tag{5} \end{equation}$$

Applies to all interior mesh points: $i=1,\ldots,N_x-1$ and $n=1,\ldots,N_t-1$

For $n=0$ we have the initial conditions $u=I(x)$ and $u_t=0$

At the boundaries $i=0,N_x$ we have the boundary condition $u=0$.