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 $$ \begin{equation} \tag{7} u_{i,j+1}= \alpha u_{i-1,j}+(1-2\alpha)u_{i,j}+\alpha u_{i+1,j}. \end{equation} $$