Explicit Scheme, simplifications
These equations can be further simplified as
$$
\begin{equation*}
u_t\approx \frac{u_{i,j+1}-u_{i,j}}{\Delta t},
\end{equation*}
$$
and
$$
\begin{equation*}
u_{xx}\approx \frac{u_{i+1,j}-2u_{i,j}+u_{i-1,j}}{\Delta x^2}.
\end{equation*}
$$
The one-dimensional diffusion equation can then be rewritten in its
discretized version as
$$
\begin{equation*}
\frac{u_{i,j+1}-u_{i,j}}{\Delta t}=\frac{u_{i+1,j}-2u_{i,j}+u_{i-1,j}}{\Delta x^2}.
\end{equation*}
$$
Defining \( \alpha = \Delta t/\Delta x^2 \) results in the explicit scheme
$$
\begin{equation}
\tag{7}
u_{i,j+1}= \alpha u_{i-1,j}+(1-2\alpha)u_{i,j}+\alpha u_{i+1,j}.
\end{equation}
$$