= Solution
Expand $a,u,f$ in the two-dimensional <Fourier basis>. Applying the derivative and product rules from part (a), the $(m,n)$ coefficient of the equation is
$$
-\pi^2\sum_{p,q\in\mathbb Z}
(mp+nq)\widehat a_{m-p,n-q}\widehat u_{p,q}
=\widehat f_{m,n}.
$$
The normalization and compatibility conditions are
$$
\boxed{\widehat u_{0,0}=0,
\qquad \widehat f_{0,0}=0}.
$$
Thus, on $\mathbb Z^2\setminus\{(0,0)\}$, the infinite <Fourier spectral method> system is
$$
\boxed{\sum_{p,q}A_{(m,n),(p,q)}\widehat u_{p,q}=-\widehat f_{m,n}},
\qquad
\boxed{A_{(m,n),(p,q)}=\pi^2(mp+nq)\widehat a_{m-p,n-q}}.
$$
For a square <spectral truncation>, choose
$$
\Lambda_N=\{(m,n):|m|,|n|\leq N\}\setminus\{(0,0)\},
$$
retain equations and unknowns indexed by $\Lambda_N$, and set omitted coefficients to zero. This is the <Fourier–Galerkin method>: it projects the residual onto the retained <Fourier modes>. Other finite mode sets may be used in the same way.
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