Subject: Partial Differential Equations.
Here are the details of the question:
$$ \frac{\partial ^2u}{\partial x^2} + \frac{\partial ^2u}{\partial y^2} = 0 $$ for $0 < x < 3, 0 < y < 1$
$$u(0, y) = 0,\text{ for }0 ≤ y ≤ 1 $$ $$u(3, y) = 0,\text{ for }0 ≤ y ≤ 1, $$ $$u(x, 0) = 0,\text{ for }0 < x < 3, $$ $$u(x, 1) = x(x − 3),\text{ for }0 < x < 3.$$
Let the step size in both the $x$ and the $y$ direction be $h = 0.5$.
Question:
Make a diagram of the domain and the grid (with step size h) that clearly shows all the information available. Label the grid points and indicate which ones are interior points and which ones are boundary points.
I have come up with the following but I'm not convinced it's correct:
http://i.imgur.com/MetclnU.png
Do I need to incorporate $ x(x − 3) $ into the top line of the graph?