| course |
course_year |
question_number |
tags |
title |
year |
Numerical Analysis |
II |
105 |
II |
2019 |
Numerical Analysis |
|
Paper 4, Section II, C |
2019 |
For a 2-periodic analytic function $f$, its Fourier expansion is given by the formula
$$f(x)=\sum_{n=-\infty}^{\infty} \widehat{f}{n} e^{i \pi n x}, \quad \widehat{f}{n}=\frac{1}{2} \int_{-1}^{1} f(t) e^{-i \pi n t} d t$$
(a) Consider the two-point boundary value problem
$$-\frac{1}{\pi^{2}}(1+2 \cos \pi x) u^{\prime \prime}+u=1+\sum_{n=1}^{\infty} \frac{2}{n^{2}+1} \cos \pi n x, \quad-1 \leqslant x \leqslant 1,$$
with periodic boundary conditions $u(-1)=u(1)$. Construct explicitly the infinite dimensional linear algebraic system that arises from the application of the Fourier spectral method to the above equation, and explain how to truncate the system to a finitedimensional one.
(b) A rectangle rule is applied to computing the integral of a 2-periodic analytic function $h$ :
$$\int_{-1}^{1} h(t) d t \approx \frac{2}{N} \sum_{k=-N / 2+1}^{N / 2} h\left(\frac{2 k}{N}\right)$$
Find an expression for the error $e_{N}(h):=\mathrm{LHS}-\mathrm{RHS}$ of $(*)$, in terms of $\widehat{h}{n}$, and show that $e{N}(h)$ has a spectral rate of decay as $N \rightarrow \infty$. [In the last part, you may quote a relevant theorem about $\widehat{h}_{n}$.]