Explicit Scheme, solving the equations
Since all the discretized initial values
\begin{equation*}
u_{i,0} = g(x_i),
\end{equation*}
are known, then after one time-step the only unknown quantity is
u_{i,1} which is given by
\begin{equation*}
u_{i,1}= \alpha u_{i-1,0}+(1-2\alpha)u_{i,0}+\alpha u_{i+1,0}=
\alpha g(x_{i-1})+(1-2\alpha)g(x_{i})+\alpha g(x_{i+1}).
\end{equation*}
We can then obtain
u_{i,2} using the previously calculated values
u_{i,1}
and the boundary conditions
a(t) and
b(t) .
This algorithm results in a so-called explicit scheme, since the next functions
u_{i,j+1} are explicitely given by Eq.
(7).