Topic 6 of 6

Fourier series and the heat equation

Step 20 · Differential equations topics

Build periodic functions out of sines and cosines, compute Fourier coefficients, and use them to solve the heat equation for a rod.

Fourier series

A function with period 2L can be written as a sum of sines and cosines that fit evenly into that period. Each coefficient measures how much of that wave the function contains.

f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty}\left(a_n\cos\frac{n\pi x}{L} + b_n\sin\frac{n\pi x}{L}\right)

a_n = \frac{1}{L}\int_{-L}^{L} f(x)\cos\frac{n\pi x}{L}\,dx \qquad b_n = \frac{1}{L}\int_{-L}^{L} f(x)\sin\frac{n\pi x}{L}\,dx

Example: A square wave

Find the Fourier series of the square wave that is −1 for −π < x < 0 and 1 for 0 < x < π.

It is odd, so every $a_n$ is 0.
$b_n = \frac{2}{\pi}\int_{0}^{\pi} \sin nx\,dx$ , which is $\frac{4}{n\pi}$ for odd n and 0 for even n.

Answer $\frac{4}{\pi}\left(\sin x + \frac{\sin 3x}{3} + \frac{\sin 5x}{5} + \cdots\right)$

The square wave (dashed) and the sum of its first three sine terms. Adding more terms hugs the corners more closely.

Sine and cosine series

On 0 < x < L, a function can be extended to be odd, giving a series of sines only, or even, giving cosines only. Sine series suit problems where the ends are held at 0.

Example: A sine series

Find the sine series of f(x) = 1 on 0 < x < π.

$b_n = \frac{2}{\pi}\int_{0}^{\pi} \sin nx\,dx$ .
These are the square wave's coefficients again.

Answer $\frac{4}{\pi}\left(\sin x + \frac{\sin 3x}{3} + \cdots\right)$

The heat equation

The temperature u(x, t) along a thin rod obeys $u_t = k\,u_{xx}$ : a spot hotter than its neighbors cools. With the ends held at 0, separation of variables gives a sum of sine waves, each fading exponentially.

u(x, t) = \sum_{n=1}^{\infty} b_n e^{-k(n\pi/L)^{2}t}\sin\frac{n\pi x}{L}

Example: A single wave

A rod of length π with k = 1 starts at u(x, 0) = 3 sin 2x, with both ends held at 0. Find u(x, t).

The starting shape is already one sine wave, with n = 2.
It fades at rate n², which is 4.

Answer $u = 3e^{-4t}\sin 2x$

Example: Which fades first

Starting from sin x + sin 5x, with k = 1 and length π, which part fades faster?

The sin 5x part fades like $e^{-25t}$ , and the sin x part like $e^{-t}$ .

AnswerThe sin 5x part: sharp wiggles smooth out first

Try it yourself

Work each one on paper first, then open the answer to check.

Is f(x) = x on −π < x < π odd or even, and which coefficients vanish?
Show answer
Odd: every $a_n$ is 0
Which waves appear in the Fourier series of an even function?
Show answer
Cosines only, plus the constant
What does the square wave's Fourier series give at the jump x = 0?
Show answer
0, the midpoint of the jump
Find u(x, t) for $u_t = u_{xx}$ on 0 < x < π, with the ends at 0 and u(x, 0) = 5 sin x.
Show answer
$u = 5e^{-t}\sin x$
In the heat equation, how fast does the n = 3 wave fade compared with n = 1?
Show answer
9 times as fast
What is $b_1$ for the square wave?
Show answer
$\frac{4}{\pi}$