I understand that there is a method which can be used to solve PDEs wherein you assume a function has a Fourier series representation to get rid of spatial derivatives. For example, if I consider Poisson's equation on a rectangle I can write: \begin{equation} \nabla^2 \varphi = f(x, y) \\ \implies \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) \sum_{k_x, k_y} \hat{\varphi}_{k_x, k_y} e^{i(k_x x + k_y y)} = \sum_{k_x, k_y} \hat{f}_{k_x, k_y} e^{i(k_x x + k_y y)} \\ \implies \left( k_x^2 + k_y^2 \right)\hat{\varphi}_{k_x, k_y} = \hat{f}_{k_x, k_y} \\ \implies \varphi(x, y) = \sum_{k_x, k_y} \frac{\hat{f}_{k_x, k_y}}{k_x^2 + k_y^2} \end{equation} where $k_x = 2\pi n/L_x$ and $k_y = 2\pi m/L_y$ for domain side lengths $L_x$ and $L_y$ and integers $m$, $n$. Then we may solve for $\varphi(x, y)$ in a specific case by doing some Fourier integrals. I have been told that this works if the functions involved are periodic, which I have taken to mean $f(x, 0) = f(x, L_y)$ and $f(0, y) = f(L_x, y)$ for all $x$ and $y$.
Can this scheme work with Dirichlet (or Neumann) boundary conditions? If so, where do we take these into account? If not, are there any similar techniques that will work with these boundary conditions? I have seen something about these half-range Fourier series, but it seems like you need to know whether your function is even/odd (or neither) a priori.
I have tried the specific example of $\varphi$ being $0$ at the boundaries. In this case, if we take $x = 0$ we get \begin{equation} \sum_{k_x, k_y} \hat{\varphi}_{k_x, k_y} e^{ik_y y} = 0 \end{equation} Operating on both sides with $\int_0^{L_y} e^{-i k_y' y}dy$ yields: \begin{equation} L_y\sum_{k_x} \hat{\varphi}_{k_x, k_y'} = 0 \end{equation} for all $k_y'$. Similarly, we may calculate \begin{equation} L_x\sum_{k_y} \hat{\varphi}_{k_x', k_y} = 0 \end{equation} for all $k_x'$. It's not really clear to me how I can use this information to solve for $\hat{\varphi}_{k_x, k_y}$ or even whether these conditions are compatible with the Laplace equation solution for $\hat{\varphi}_{k_x, k_y}$.