Explicit Scheme, boundary conditions
The boundary conditions are
$$
\begin{equation*}
u(0,t)= a(t) \hspace{0.5cm} t \ge 0,
\end{equation*}
$$
and
$$
\begin{equation*}
u(L,t)= b(t) \hspace{0.5cm} t \ge 0,
\end{equation*}
$$
where \( a(t) \) and \( b(t) \) are two functions which depend on time only, while
\( g(x) \) depends only on the position \( x \).
Our next step is to find a numerical algorithm for solving this equation. Here we recur
to our familiar equal-step methods
and introduce different step lengths for the space-variable \( x \) and time \( t \) through
the step length for \( x \)
$$
\begin{equation*}
\Delta x=\frac{1}{n+1}
\end{equation*}
$$
and the time step length \( \Delta t \). The position after \( i \) steps and
time at time-step \( j \) are now given by
$$
\begin{equation*}
\begin{array}{cc} t_j=j\Delta t & j \ge 0 \\
x_i=i\Delta x & 0 \le i \le n+1\end{array}
\end{equation*}
$$