Let's say I have a very simple steady-state BVP with Dirichlet BCs:
$$u_{xx} = 2, \ x\in [0,1]; u(x_{i=0}) = 0, u(x_{i=N-1}) = 1$$
If I take the Fourier Transform of the 2nd order finite difference equation, I obtain:
$$\mathcal{F}\big(\frac{u(x_{i+1})+u(x_{i-1})-2u(x_i)}{h^2} = 2\big) \rightarrow \frac{(1-e^{-jk})(e^{jk}-1)}{h^2}\hat{u} = 2 \rightarrow^{divide} \boxed{\hat{u} = 2\frac{h^2}{(1-e^{-jk})(e^{jk}-1)}}$$
I know I can't just inverse Fourier Transform to get the answer because I'd get division by 0 at the boundaries. So, what are the steps for applying the BCs?