Consider the Poisson equation in the rectangle $Q=\{(x,y):0<x<1,0<y<1\}$,
$$u_{xx}+u_{yy}=F(x,y)$$ $$u(0,y)=0,\,u(1,y)=0$$ $$u(x,0)=\phi(x),\,\,u(x,1)=\psi(x)$$
My Question: Is my solution procedure correct?
Solution Procedure:
I apply Fourier method. The corresponding auxiliary problem is $$X''(x)+\lambda X(x)=0$$ $$X(0)=0, \, X(1)=0.$$ The eigenvalues and corresponding eigenfunctions are $$\lambda_n=n^2\pi^2,\,\,X_n(x)=\sin(n\pi x),\,n=1,2,\ldots.$$
Expansion of $F(x,y),\phi(x)$ and $\psi(x)$ in terms of eigenfunctions are $$F(x,y)=\sum\limits_{n=1}^{\infty}f_n(y)\sin(n\pi x),$$ $$\phi(x)=\sum\limits_{n=1}^{\infty}\phi_n\sin(n\pi x),$$ $$\psi(x)=\sum\limits_{n=1}^{\infty}\psi_n\sin(n\pi x),$$ where $$f_n(y)=2\int\limits_0^1F(x,y)\sin(n\pi x)dx$$ $$\phi_n=2\int\limits_0^1\phi(x)\sin(n\pi x)dx$$ and $$\psi_n=2\int\limits_0^1\psi(x)\sin(n\pi x)dx,\,n=1,2,\ldots.$$ I assume that the solution is in the following form $$u(x,y)=\sum\limits_{n=1}^{\infty}a_n(y)\sin(n\pi x),$$ where $a_n(y),n=1,2,...$ are solutions of $$a_n^{''}-n^2\pi^2a_n(y)=f_n(y)$$ $$a_n(0)=\phi_n,\,a_n(1)=\psi_n,$$ $\,n=1,2,\ldots.$