The Lumenary · Advanced undergraduate

Partial differential equations and Fourier series

Name: ______________________

Date: ______________________

You will learn to

Derive the heat, wave and Laplace equations, and solve the heat and wave equations on an interval by separation of variables and Fourier sine series
Compute Fourier coefficients, prove Bessel's inequality and state Parseval's identity, and use them to sum series
Prove Dirichlet's convergence theorem at points of differentiability, explain the Gibbs phenomenon with its constant $\tfrac{2}{\pi}\mathrm{Si}(\pi)$ , and recognise when term-by-term differentiation fails
Prove d'Alembert's formula for the wave equation on the line
Prove uniqueness for the heat and wave equations with the maximum principle and energy methods, and state the mean value property of harmonic functions

1.
Which function solves the heat equation $u_t = u_{xx}$ ?
- A. $u = e^{-4t}\sin 2x$
- B. $u = e^{-2t}\sin 2x$
- C. $u = e^{t}\sin x$
- D. $u = \cos t\,\sin x$
2.
Let $f(x) = 3\sin x - 5\sin 4x$ on $[0, \pi]$ . What is its Fourier sine coefficient $b_4 = \frac{2}{\pi}\int_0^\pi f(x)\sin 4x\,dx$ ?
3.
Solve $u_{tt} = 4u_{xx}$ on $\R$ with $u(0, x) = x^2$ and $u_t(0, x) = 0$ . What is $u(1, 0)$ ?
4.
Let $S_N$ be the partial sums of the Fourier series of the square wave $f = \operatorname{sign}(\sin x)$ . Which statements are true? Choose all that apply.
- A. $S_N(x)$ converges for every real $x$ .
- B. $S_N \to f$ uniformly on $\R$ .
- C. $S_N \to f$ uniformly on $[\delta, \pi - \delta]$ for each $\delta > 0$ .
- D. $\max_x S_N(x) \to 1$ .
- E. $\|S_N - f\|_2 \to 0$ .
5.
In the proof of the weak maximum principle (Theorem 6), why do we work with $v = u + \varepsilon x^2$ rather than with $u$ itself?
- A.At an interior maximum of $u$ we only get $u_t \ge 0 \ge k u_{xx}$ , which does not contradict $u_t = ku_{xx}$ ; for $v$ we have $v_t - kv_{xx} = -2k\varepsilon < 0$ , which does.
- B. $u$ might not be differentiable, but $v$ is.
- C.To make $v$ vanish on the boundary.
- D.To make $v$ positive, so that its maximum exists.
6.
Let u : ℝ → ℝ → ℝ take time first, u t x. Which Lean statement says that u satisfies the heat equation u_t = k u_xx at the point (t, x)?
- A.deriv (fun s => u s x) t = k * deriv (fun y => deriv (fun z => u t z) y) x
- B.deriv (fun s => u s x) t = k * deriv (fun y => u t y) x
- C.deriv (fun s => u t s) x = k * deriv (fun y => deriv (fun z => u z x) y) t
- D.deriv (fun s => deriv (fun r => u r x) s) t = k * deriv (fun y => deriv (fun z => u t z) y) x
7.
Apply Parseval's identity to the square wave $f = \operatorname{sign}(\sin x)$ , whose sine coefficients are $b_n = \frac{4}{\pi n}$ for odd $n$ and $0$ for even $n$ . In the real form, Parseval reads $\frac1\pi\int_{-\pi}^{\pi} f^2\,dx = \frac{a_0^2}{2} + \sum_{n\ge1}(a_n^2 + b_n^2)$ . You get $\displaystyle\sum_{n \text{ odd}}\frac{1}{n^2} = \frac{\pi^2}{k}$ . What is $k$ ?
8.
Write sine_coefficients(f, N, M) returning the list [b_1, ..., b_N] of Fourier sine coefficients $b_n = \frac{2}{\pi}\int_0^\pi f(x)\sin(nx)\,dx$ of f on $[0, \pi]$ , each computed by the midpoint rule with M subintervals.
Computational — Practice
9.
On $(-\pi, \pi)$ , $x = 2\sum_{n\ge1}(-1)^{n+1}\frac{\sin nx}{n}$ and $x^2 = \frac{\pi^2}{3} + 4\sum_{n\ge1}(-1)^n\frac{\cos nx}{n^2}$ . Which statements are true? Choose all that apply.
- A.Differentiating the series for $x$ term by term gives a series converging to $1$ .
- B.Differentiating the series for $x^2$ term by term gives the series for $2x$ .
- C.The series for $x^2$ converges uniformly on $[-\pi, \pi]$ .
- D.The series for $x$ converges uniformly on $[-\pi, \pi]$ .
10.
For $n \ge 1$ , $u_n(x, y) = \dfrac{\sin(nx)\sinh(ny)}{n^2}$ solves Laplace's equation with $u_n(x, 0) = 0$ and $\partial_y u_n(x, 0) = \dfrac{\sin nx}{n}$ . Which conclusion is correct?
- A.The initial-value problem for Laplace's equation does not depend continuously on its data, so it is ill posed.
- B.Laplace's equation has non-unique solutions for this data.
- C.No solution exists for this data.
- D.The maximum principle fails for harmonic functions.

Answer key — Partial differential equations and Fourier series

1.
For $u = e^{-at}\sin nx$ : $u_t = -au$ and $u_{xx} = -n^2u$ , so $u_t = u_{xx}$ iff $a = n^2$ . Only $e^{-4t}\sin 2x$ has $a = 4 = 2^2$ . The function $\cos t \sin x$ solves the wave equation instead.
2.
By orthogonality, $\frac2\pi\int_0^\pi 3\sin x\sin 4x\,dx = 0$ and $\frac2\pi\int_0^\pi(-5)\sin^2 4x\,dx = -5\cdot\frac2\pi\cdot\frac\pi2 = -5$ . So $b_4 = -5$ : a finite sine sum is its own Fourier series.
3.
$u(t, x) = \frac{(x - 2t)^2 + (x + 2t)^2}{2} = x^2 + 4t^2$ . Check: $u_{tt} = 8 = 4 \cdot 2 = 4u_{xx}$ , $u(0, x) = x^2$ , $u_t(0, x) = 0$ . So $u(1, 0) = 4$ .
4.
True: pointwise convergence everywhere (to the average $0$ at the jumps), uniform convergence on compact sets avoiding the jumps, and $L^2$ convergence. False: uniform convergence on $\R$ (the limit is discontinuous) and $\max S_N \to 1$ (the maxima tend to $1.17898\ldots$ ).
5.
At an interior (or top-edge) maximum, $u_t \ge 0$ and $u_{xx} \le 0$ ; both could be $0$ , consistent with the equation, so no contradiction arises. For $v = u + \varepsilon x^2$ , $v_t - kv_{xx} = -2k\varepsilon < 0$ , which is incompatible with $v_t \ge 0 \ge kv_{xx}$ . Hence $\max v$ is on the parabolic boundary, and $u \le M + \varepsilon L^2$ for every $\varepsilon$ .
6.
deriv (fun s => u s x) t = k * deriv (fun y => deriv (fun z => u t z) y) x. The other options encode $u_t = ku_x$ , $u_x = ku_{tt}$ and the wave equation.
7.
$2 = \frac{16}{\pi^2}\sum_{n\text{ odd}}\frac1{n^2}$ , so $\sum_{n\text{ odd}}\frac{1}{n^2} = \frac{\pi^2}{8}$ and $k = 8$ . Consistently, the even terms contribute $\frac14\sum\frac1{n^2} = \frac{\pi^2}{24}$ , and $\frac{\pi^2}{8} + \frac{\pi^2}{24} = \frac{\pi^2}{6}$ .
8.
```
import math

def sine_coefficients(f, N, M):
    h = math.pi / M
    xs = [(i + 0.5) * h for i in range(M)]
    return [2 / math.pi * h * sum(f(x) * math.sin(n * x) for x in xs)
            for n in range(1, N + 1)]
```
For the trigonometric polynomial $\sin 3x$ the midpoint rule reproduces the orthogonality relations essentially exactly (it is exact for trigonometric polynomials of low enough degree relative to $M$ ). For $x(\pi - x)$ the computed values match $b_n = \frac{8}{\pi n^3}$ (odd $n$ ) and $0$ (even $n$ ), the coefficients used in the heat-equation example.
9.
The periodic extension of $x^2$ is continuous and piecewise $C^1$ : its series converges uniformly and may be differentiated term by term, giving the (non-uniformly convergent) series for $2x$ . The periodic sawtooth jumps, its series does not converge uniformly, and its term-by-term derivative diverges.
10.
The data $(0, \frac{\sin nx}{n})$ tend to zero uniformly, yet $u_n(\tfrac{\pi}{2n}, 1) = \frac{\sinh n}{n^2} \to \infty$ . Arbitrarily small data produce arbitrarily large solutions: no continuous dependence. Laplace's equation needs boundary data around a closed curve, where the maximum principle gives exactly the stability that fails here.