Assignment 9: Fourier Transforms

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import mmf_setup;mmf_setup.nbinit()
import logging;logging.getLogger('matplotlib').setLevel(logging.CRITICAL)
%matplotlib inline
import numpy as np, matplotlib.pyplot as plt

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Assignment 9: Fourier Transforms#

Due Mon 03 Nov 2025 at the start of class

1. Fourier Series#

Find the Fourier series representation of the periodic function \(f(x+2\pi) = f(x)\) where

\[\begin{gather*} f(x) = \begin{cases} 0 & -\pi < x \leq 0\\ x & 0 \leq x < \pi. \end{cases} \end{gather*}\]

Sketch this function bye hand and check your result numerically.

From this Fourier expansion, show that

\[\begin{gather*} \frac{\pi^2}{8} = 1 + \frac{1}{3^2} + \frac{1}{5^2} + \cdots. \end{gather*}\]

2. Fourier Transform#

Find the Fourier transform of the following triangular pulse:

\[\begin{gather*} f(x) = \begin{cases} h(1-a\abs{x}) & \abs{x} < 1/a,\\ 0 & \text{otherwise}. \end{cases} \end{gather*}\]

Sketch this function by hand.

Show that, setting \(h=a\) and taking the limit \(a\rightarrow \infty\) gives the Fourier transform of the Dirac delta function, hence this provides another sequence that approaches \(f(x) \rightarrow \delta(x)\).

3. Neutron Diffusion#

The 1D neutron diffusion equation with a (plane) source is

\[\begin{gather*} - D\diff[2]{\phi(x)}{x} + K^2 D \phi(x) = Q \delta(x) \end{gather*}\]

where \(\phi(x)\) is the neutron flux, \(Q\delta(x)\) is the (plane) source at \(x=0\), and \(D\) and \(K^2\) are constants. Apply a Fourier transform. Solve the equation in Fourier space, then transform your solution back to position space.

4. Poisson’s Equation#

Solve Poisson’s equation

\[\begin{gather*} \nabla^2 \psi(\vect{r}) = - \frac{\rho(\vect{r})}{\epsilon_0} \end{gather*}\]

by taking the Fourier transform of both sides of the equation, solving for the Fourier transform of \(\psi(\vect{r})\) and then applying the inverse Fourier transform to find the solution in position space.

5. Laplace Transform#

Find the Laplace transform of the square wave (with period \(a\)) defined by

\[\begin{gather*} F(t) = \begin{cases} 1 & 0 < t < \frac{a}{2}\\ 0 & \frac{a}{2} < t < a \end{cases}, \qquad F(t + a) = F(t) \end{gather*}\]
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