We can solve some Laplace equations on the lectangular area by using Fourier transform. However many textbooks only introduce the cases when the area is given as a half plane or a strip.
; $y>0, -\infty<x<\infty $ or $ 0<y<\pi, -\infty<x<\infty$
I want to know if there is a similar mehod on a square area.
For the general case like $$u_{xx}+u_{yy}=0 \ \ \ \forall \ (x,y) \in (0,2\pi)^2$$ $$u(0,y)=f_1(y),\ u(2\pi,y)=f_2(y) \ \forall y \in [0,2\pi]$$ $$u(x,0)=g_1(x),\ u(x,2\pi)=g_2(x) \ \forall x \in [0,2\pi]$$
Or for the mixed case like
$$u_{xx}+u_{yy}=0 \ \ \ \forall \ (x,y) \in (0,2\pi)^2$$ $$u(0,y)=u_x(0,y),\ u(2\pi,y)=f_2(y)\ \forall y \in [0,2\pi]$$ $$u(x,0)=g_1(x),\ u(x,2\pi)=g_2(x)\ \forall x \in [0,2\pi]$$
Any help will be appreicated.