Consider the problem
$\begin{align} u''''=f&& \text{on} \; (0,a) \\ u'=u=0&& \text{on} \; \{0,a\} \end{align} $
for which I have an orthogonal system $\phi_k$
How can I "construct" an orthogonal system for the two dimensional problem
$\begin{align} \Delta^2 u=f&& \text{on} \; (0,a)\times (0,b) \\ u_n=u=0&& \text{on} \; \partial (0,a)\times (0,b) \end{align} $
where $u_n$ is the derivative in normal direction
Would appreciate any help