Let $\frac{du}{dt} - \frac{du}{dx}=0$.
$u(x,0) = e^x, u(1,t) = e^{t+1}$
I've been asked to write up a finite difference scheme using the forward Euler method for the step $\tau$ for t and the backward Euler for the step h for x.
I've gotten: [$\frac{u_{n+1}-u_n}{\tau}-G(t_n,u_n)$]-[$\frac{u_{n+1}-u_n}{h}-G(t_{n+1},u_{n+1})$]=0.
Now I need to compute (by hand) the approximate solution given h=1/4, $\tau =1/2$, and a few values for t. How would I do this?