Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 39C (Numerical Analysis)

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.

Insert the Fourier seriesinto the rectangle rule. The finite geometric sum obeysConsequently the quadrature value iswhereasThus the exact Periodic N-point rectangle-rule error is

By the Fourier coefficient decay of an analytic periodic function, there are constants $C>0$ and $0<\rho <1$ such thatIt follows thatThereforewhich is exponential, and hence spectral, convergence.