$$\frac{∂^2 u}{∂x^2}+\frac{∂^2 u}{∂y^2}=S(x)$$
Consider solution $u(x,y)$ in finite Cartesian domain $(0,L_x)\times(0,L_y)$. The domain is subjected to the source of $S(x)$, which is only dependent on $x$ and well defined. For now, consider the boundary conditions to be fully zero Dirichlet.
I had tried the series expansion but it did not work well because getting Fourier coefficients demand integration for both $x$ and $y$ yet as source is only $x$-dependent, it vanishes to zero when I am integrating for $y$ coordinate. Any clever trick to solve this?
Was also thinking about Green's function but the calculation on convolution is very tedious and therefore, am looking something else.