MATH50003 Numerical Analysis: Problem Sheet 6¶
This problem sheet explores condition numbers, indefinite integration,
and Euler’s method.
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Questions marked with a ⋆ are meant to be completed without using a computer.
Problems are denoted A/B/C to indicate their difficulty.
using LinearAlgebra, Plots, Test
1. Two-point boundary value problems¶
Problem 1.1 (C) Construct a finite-difference approximation to the
forced Helmholtz equation
\begin{align*}
u(0) &= 0 \\
u(1) &= 0 \\
u” + k^2 u &= {\rm e}^x
\end{align*}
and find an $n$ such the error is less than $10^{-4}$ when compared
with the true solution for $k=10$:
u(x) = (-\cos(k x) + {\rm e}^x \cos(k x)^2 + \cot(k) \sin(k x) – {\rm e} \cos(k) \cot(k) \sin(k x) – {\rm e} \sin(k) \sin(k x) + {\rm e}^x \sin(k x)^2)/(1 + k^2)
function helm(k, n)
x = range(0, 1; length = n)
h = step(x)
# TODO: Create a SymTridiagonal discretisation
u = x -> (-cos(k*x) + exp(x)cos(k*x)^2 + cot(k)sin(k*x) – ℯ*cos(k)cot(k)sin(k*x) – ℯ*sin(k)sin(k*x) + exp(x)sin(k*x)^2)/(1 + k^2)
n = 10 # TODO: choose n to get convergence
x = range(0, 1; length=n)
@test norm(helm(k, n) – u.(x)) ≤ 1E-4
Problem 1.2 (B) Discretisations can also be used to solve eigenvalue equations.
Consider the Schrödinger equation with quadratic oscillator:
u(-L) = u(L) = 0, -u” + x^2 u = λ u
Approximate the eigenvalues using eigvals(A) (which returns the eigenvalues of a
matrix A) with $L = 10$.
Can you conjecture their exact value if $L = ∞$? (Hint: they are integers and the eigenvalues
closest to zero are most accurate.)
Problem 1.3⋆ (A) Consider Helmholtz with Neumann conditions:
u'(0) = c_0 \\
u'(1) = c_1 \\
u_{xx} + k^2 u = f(x)
Write down the finite difference approximation approximating
$u(x_k) ≈ u_k$ on
an evenly spaced grid $x_k = (k-1)/(n-1)$ for $k=1,…,n$
using the first order derivative approximation conditions:
\begin{align*}
u'(0) &≈ (u_2-u_1)/h = c_0 \\
u'(1) &≈ (u_n-u_{n-1})/h = c_1
\end{align*}
Use pivoting to reduce the equation to one involving a
symmetric tridiagonal matrix.
2. Convergence¶
Problem 2.1⋆ (B) For the equation
\begin{align*}
u(0) &= c_0 \\
u’ + au &= f(x)
\end{align*}
where $a ∈ ℝ$ and $0 ≤ x ≤ 1$,
prove convergence as $n → ∞$ for the method constructed in PS6 using the approximation
where we take the average of the two grid points:
{u'(x_{k+1}) + u'(x_k) \over 2} ≈ {u_{k+1} – u_k \over h}.
Problem 2.2⋆ (A) Consider the matrices
L = \begin{bmatrix} 1 \\
-a_1 & 1 \\
& -a_2 & 1\\
&& ⋱ & ⋱ \\
&&& -a_{n-1} & 1
\end{bmatrix}, \qquad T = \begin{bmatrix} 1 \\
& -a & 1\\
&& ⋱ & ⋱ \\
&&& -a & 1
\end{bmatrix}.
By writing down the inverse explicitly prove that if $|a_k| ≤ a$ then
\|L^{-1}\|_{1 → ∞} ≤ \|T^{-1}\|_{1 → ∞}.
Use this to prove convergence as $n → ∞$ of forward Euler for
\begin{align*}
u(0) &= c_0 \\
u'(x) – a(x)u(x) &= f(x)
\end{align*}
3. Fourier series¶
Problem 3.1⋆ (C) Give explicit formulae for $f̂_k$ and $f̂_k^n$ for the following functions:
\cos θ, \cos 4θ, \sin^4θ, {3 \over 3 – {\rm e}^{\rm i θ}}, {1 \over 1 – 2{\rm e}^{\rm i θ}}
Hint: You may wish to try the change of variables $z = {\rm e}^{-{\rm i}θ}$.
Problem 3.2⋆ (B) Prove that if the first $λ-1$ derivatives $f(θ), f'(θ), …, f^{(λ-1)}(θ)$
are 2π-periodic and $f^{(λ)}$ is uniformly bounded that
|f̂_k| = O(|k|^{-λ})\qquad \hbox{as $|k| → ∞$}
Use this to show for the Taylor case ($0 = f̂_{-1} = f̂_{-2} = ⋯$) that
|f(θ) – ∑_{k=0}^{n-1} f̂_k {\rm e}^{{\rm i}kθ}| = O(n^{1-λ})
Problem 3.3⋆ (C)
If $f$ is a trigonometric polynomial ($f̂_k = 0$ for $|k| > m$) show
for $n ≥ 2m+1$ we can exactly recover $f$:
f(θ) = \sum_{k=-m}^m f̂_k^n {\rm e}^{{\rm i} k θ}
Problem 3.4⋆ (B) For the general (non-Taylor) case and $n = 2m+1$, prove convergence for
f_{-m:m}(θ) := ∑_{k=-m}^m f̂_k^n {\rm e}^{{\rm i} k θ}
to $f(θ)$ as $n \rightarrow ∞$.
What is the rate of convergence if the first $λ-1$ derivatives $f(θ), f'(θ), …, f^{(λ-1)}(θ)$
are 2π-periodic and $f^{(λ)}$ is uniformly bounded?
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