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 $h = 0.5$.
The first step was to draw a grid but now the question is:
Formulate the system of equations for a finite difference discretisation of the problem.
I'm not entirely sure what is meant by this.
Suppose you sample $u(x,y)$ on a grid of points, and call them $x_n,y_m$. They are given by $x_m = mh$ and $y_n=nh$, i.e. you let them be integer multiples of the step size $h$. Then the sampled values of $u$ can be denoted as $u_m^n$, with $$ u_m^n = u(x_m,y_n) $$ These are your unknowns, and your job is to derive a system of equations for these unknowns, which represent the values of the solution on the grid points. To do this, you use the finite difference approximations of the derivatives of $u$, $$ \frac{\partial^2u}{\partial x^2} \approx \frac{u_{m-1}^n-2u_{m}^n+u_{m+1}^n}{h^2} $$ And do the same for the $y$ derivatives. Then plug into the original equation, and enforce the equations at the interior points of the grid. In your case, the grid is given by $$ x_0 = 0,\; x_1=1/2,\; x_2 = 1,\; x_3 = 3/2,\; x_4=2,\; x_5 = 5/2,\; x_6 = 3 $$ and $$ y_0 = 0,\; y_1=1/2,\; y_2 = 1 $$
So you only have five interior points. The question is asking you to write down five equations, using the original equation, approximated by finite differences as above, evaluated at the five interior grid points. Your five unknowns will be $u_1^1,u_1^2,u_1^3,u_1^4,u_1^5$, and the rest will be known from the given boundary conditions.