Ok so I have a question with tutorial solutions but there are a few points I don't understand.
Q: Find the solution to Laplace's equation $$(∇^2)u=0$$ in the unit square $d=\{(x,y):0≤x≤1,0≤y≤1\}$ subject to the boundary conditions \begin{align*} u&=0&\text{ on }x=0,x=1, \\ u&=\sinπx&\text{ on } y=0, y=1. \end{align*}
If someone could explain in detail all the steps that would be great! Mainly I don't understand how you know whether to use:
- $λ=k^2>0$, $F=Ae^{(kx)} + Be^{(-kx)}$
- $λ=0$, $F=Ax+B$
- $λ=-k^2<0$, $F=A\cos(kx) + B\sin(kx)$
and then why $k=nπ$ making $λ=-n^2π^2$
then why we use form (1.) for G eqn
and then just generally solving for the solution