So for a 1-d spatial, and 1-d time PDE,
I represent the solution as $U_N= \sum_{n=0}^N c_n(t) \phi_n(x)$,
where $\phi_n(x) = \frac{1}{\sqrt{\pi}}\cos(nx)$ and $x\in [-\pi,\pi]$
I was curious if anyone had suggestions on how to generalize this to 2-d spatial dimension. e.g $x\in [-\pi,\pi]^2$.
I hypothosized that it would look perhaps like this:
$U_N= \sum_{n,m} c_n(t) \phi_{n,m}(x)$
where $\phi_{n,m}=\frac{1}{\pi}cos(nx_1)cos(mx_2)$, but I could not find a reference that confirmed my hypothesis.
Does anyone have suggestions, or can confirm that I am correct?