Fourier Series and Partial Differential Equations

Fourier series

A periodic function f(x) of period 2L can be written as

f(x) = a₀/2 + Σ (aₙ cos(nπx/L) + bₙ sin(nπx/L)), n = 1, 2, 3, …

with

• aₙ = (1/L) ∫_−L^L f(x) cos(nπx/L) dx (a₀ is the same with n = 0, so a₀/2 is the average value);
• bₙ = (1/L) ∫_−L^L f(x) sin(nπx/L) dx.

Symmetry saves work.

• An even function (f(−x) = f(x)) has bₙ = 0: a cosine series.
• An odd function (f(−x) = −f(x)) has aₙ = 0 (and a₀ = 0): a sine series.
• Half-range series: a function given on (0, L) can be extended as even (cosine series) or odd (sine series).

Dirichlet conditions. If f is periodic, piecewise continuous with finitely many maxima, minima and discontinuities in a period, its Fourier series converges to f(x) where f is continuous, and to the average of the left and right limits at a jump.

Example. The square wave f(x) = −1 on (−π, 0) and +1 on (0, π) is odd, with bₙ = 4/(nπ) for odd n and 0 for even n:

f(x) = (4/π)(sin x + sin 3x/3 + sin 5x/5 + …)

At x = 0 (a jump from −1 to 1) the series gives 0, the average. Putting x = π/2 gives the Leibniz series 1 − 1/3 + 1/5 − … = π/4.

Classifying second-order PDEs

For A u_xx + B u_xy + C u_yy + (lower-order terms) = 0, the discriminant B² − 4AC decides the type:

Separation of variables

Assume u(x, t) = X(x)T(t). The PDE splits into two ODEs linked by a separation constant, and the boundary conditions pick out the allowed values.

One-dimensional heat equation u_t = c²u_xx on 0 < x < L with u(0, t) = u(L, t) = 0:

u(x, t) = Σ Bₙ sin(nπx/L) e^{−(cnπ/L)²t},

where the Bₙ are the Fourier sine coefficients of the initial temperature u(x, 0). Each mode decays, the higher ones faster.

Wave equation u_tt = c²u_xx for a string fixed at both ends: u(x, t) = Σ sin(nπx/L)[Cₙ cos(cnπt/L) + Dₙ sin(cnπt/L)]. The n-th mode vibrates at frequency nc/(2L). D'Alembert's solution: u = F(x + ct) + G(x − ct), waves travelling in both directions.

Laplace equation u_xx + u_yy = 0: steady-state temperature or potential; its solutions (harmonic functions) have no interior maximum or minimum.

Worked example

Compute the Fourier coefficients of the square wave numerically, check the series at x = π/2, and find how much of the first heat mode remains after t = 1 for c = 1 and L = π.

import math

def b(n, N=20000):
    # bn = (1/pi) * integral over (-pi, pi) of f(x) sin(nx), with f the square wave
    h = 2 * math.pi / N
    total = 0.0
    for i in range(N):
        x = -math.pi + (i + 0.5) * h
        total += (1 if x > 0 else -1) * math.sin(n * x) * h
    return total / math.pi

print("b1..b5:", [round(b(n), 4) + 0.0 for n in range(1, 6)])
print("4/(n pi) for odd n:", [round(4 / (n * math.pi), 4) for n in (1, 3, 5)])
s = sum(4 / (n * math.pi) * math.sin(n * math.pi / 2) for n in range(1, 20001, 2))
print(f"series at x = pi/2 with 10000 terms: {s:.4f}")
print(f"first heat mode after t = 1 (c = 1, L = pi): e^-1 = {math.exp(-1):.4f} of its start")

b1..b5: [1.2732, 0.0, 0.4244, 0.0, 0.2546]
4/(n pi) for odd n: [1.2732, 0.4244, 0.2546]
series at x = pi/2 with 10000 terms: 1.0000
first heat mode after t = 1 (c = 1, L = pi): e^-1 = 0.3679 of its start

Practice questions