Chapter 10

Consider the solution of the two-point boundary value problem $$ (2-\cos \pi x) u^{\prime \prime}+u=1, \quad-1 \leq x \leq 1, \quad u(-1)=u(1), $$ using the spectral method. a Plugging the Fourier expansion of \(u\) into this differential equation, show that the \(\hat{u}_{n}\) obey a three-term recurrence relation. b Computing \(\hat{u}_{0}\) separately and using the fact that \(\bar{u}_{-n}=\hat{u}_{n}\) (why?), prove that the computation of \(\hat{u}_{n}\) for \(-N / 2+1 \leq n \leq N / 2\) (assuming that \(\hat{u}_{n}=0\) outside this range of \(n\) ) reduces to the solution of an \((N / 2) \times(N / 2)\) tridiagonal system of algebraic equations.

The two-point \(\mathrm{ODE} u^{\prime \prime}+u=1, u(-1)=u(1)=0\), is solved by a Chebyshev method. a Show that the odd coefficients are zero and that \(u(x)=\sum_{n=0}^{\infty}{\tilde{u}}_{2 n} T_{2 n}(x)\). Express the boundary conditions as a linear condition of the coefficients \(\vec{u}_{2 n}\). b Express the differential equation as an infinite set of linear algebraic equations in the coefficients \(\vec{u}_{2 \mathrm{n}}\). c Discuss how to truncate the linear system and implement it as a proper, well-defined numerical method. d Since \(u(-1)=u(1)\), the solution is periodic. Yet we cannot expect a standard spectral method to converge at spectral speed. Why?