I have generated an approximate solution to $u''(x)=f(x)=x$ on the unit interval with $u(0)=u(1)=0$ and a step size of 0.25:
This process was fairly straightforward, so I'm relatively confident in the answer to this point.
However, I am now tasked with generating the five Green's functions that sum to this solution. I am given by LeVeque that the relevant form of the Green's function is
\begin{equation}
G(x,\bar{x})=
\begin{cases}
(\bar{x}-1)x&0\leq x\leq\bar{x}\\
\bar{x}(x-1)&\bar{x}\leq x\leq 1
\end{cases}\,.
\end{equation}
But when I plot these and their sum, it is not even close:
Am I making a computational error, or a conceptual one, or both?
The Green's function solves $\frac{\partial^2 G}{\partial x^2}=\delta(x-\overline{x})$. You have
$$\int_0^1 \overline{x} \delta(x-\overline{x}) d \overline{x}=x$$
for each $x \in (0,1)$. Therefore the solution to your problem is
$$\int_0^1 \overline{x} G(x,\overline{x}) d\overline{x}.$$
In effect you sum Green's functions with various values of $\overline{x}$, which are "weighted" according to the right hand side of the original equation.
Discretizing this integral in an appropriate way should recover your approximate solution.