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).