Using Fourier method find eigenvalues and eigenfunctions of Laplacian in the rectangle $\{0<x<a, <y<b\}$ with Dirichlet boundary conditions:

\begin{align}

&u_{xx}+u_{yy}=-\lambda u\qquad 0<x<a,\ 0<y<b,\label{3-1}\\[3pt]

&u_{x=0}=u_{x=a}=u_{y=0}=u_{y=b}=0.\label{3-2}

\end{align}