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Expand the unknown in Fourier modes:By Termwise differentiation of a Fourier series,Also,Therefore the coefficient of $e^{i\pi mx}$ on the left-hand side isThe right-hand side has coefficientsbecause the coefficient of each of $e^{\pm i\pi nx}$ is $1/(n^2+1)$ and the constant coefficient is one. Hence the Fourier spectral system for a cosine-modulated second derivative isIts coefficient array is an infinite tridiagonal matrix.

For a spectral truncation, retain the modes $-M\le n\le M$, setand impose the displayed coefficient equation for every $-M\le m\le M$. This gives an explicit $(2M+1)$-dimensional tridiagonal system. The resulting finite Fourier series automatically satisfies the periodic boundary conditions.

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 isThe normalization and compatibility conditions areThus, on ${\mathbb{Z}}^2\setminus \{(0,0)\}$, the infinite Fourier spectral method system isFor a square spectral truncation, chooseretain 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.