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*}