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Differential Equations Unit 10: Fourier series and partial differential equations

Fourier series write periodic functions as sums of sines and cosines, which lets us solve the heat and wave equations by separation of variables.

Covers: Periodic functions, Fourier series coefficients, even and odd extensions, convergence, the heat equation, the wave equation, separation of variables

Study guide

A function is periodic with period P if f(t + P) = f(t) for all t. A Fourier series writes a periodic function with period 2L as a sum of cosines and sines: f(t) = a₀/2 + Σ [aₙ cos(nπt/L) + bₙ sin(nπt/L)]. The coefficients come from integrals over one period, using the orthogonality of sines and cosines: aₙ = (1/L)∫₋ᴸᴸ f(t) cos(nπt/L) dt and bₙ = (1/L)∫₋ᴸᴸ f(t) sin(nπt/L) dt. For period 2π, these simplify to aₙ = (1/π)∫ f(t) cos(nt) dt and bₙ = (1/π)∫ f(t) sin(nt) dt over [−π, π]. The constant a₀/2 is the average value of f. Even functions have only cosine terms (bₙ = 0), and odd functions have only sine terms (aₙ = 0), which saves work. A standard example is the square wave, whose Fourier series contains only odd sine harmonics with coefficients 4/(nπ).

At points where f is continuous, the Fourier series converges to f(t); at a jump discontinuity, it converges to the average of the left and right limits. Near jumps, partial sums overshoot by about 9% (the Gibbs phenomenon). A function defined only on [0, L] can be extended to an even function (giving a cosine series) or an odd function (giving a sine series). Fourier series also solve linear ODEs with periodic forcing: write the input as a Fourier series, find the response to each harmonic, and add them; a harmonic near the system's natural frequency produces resonance.

Partial differential equations (PDEs) involve functions of several variables. The heat equation uₜ = k uₓₓ describes temperature in a rod, and the wave equation uₜₜ = c² uₓₓ describes a vibrating string. Separation of variables assumes u(x, t) = X(x)T(t), which splits the PDE into two ODEs linked by a separation constant. With boundary conditions like u(0, t) = u(L, t) = 0, the spatial equation X'' + λX = 0 has nonzero solutions only when λ = (nπ/L)², giving Xₙ = sin(nπx/L). For the heat equation, each mode decays like e^(−k(nπ/L)²t), so higher modes die out fastest and the temperature approaches equilibrium. For the wave equation, each mode oscillates with frequency nπc/L. The initial condition is matched by expanding it in a Fourier sine series, and superposition combines the modes into the full solution.

Key ideas

• Fourier series write periodic functions as sums of sines and cosines.
• Coefficients come from integrals using orthogonality.
• Even functions have only cosines; odd functions only sines.
• At a jump, the series converges to the average of the one-sided limits.
• Separation of variables turns the heat and wave equations into ODEs.

Formulas and rules

Common mistakes

• Forgetting that a₀ is divided by 2 in the series.
• Expecting the series to equal f at a jump; it converges to the midpoint.

Words to know

Periodic function

A function that repeats every period P.

Fourier series

A sum of sines and cosines representing a periodic function.

Fourier coefficients

The aₙ and bₙ values found by integration.

Orthogonality

The integral of the product of different sines/cosines over a period is zero.

Even function

f(−t) = f(t); its series has only cosines.

Odd function

f(−t) = −f(t); its series has only sines.

Square wave

A periodic function alternating between two constant values.

Gibbs phenomenon

Overshoot of Fourier partial sums near a jump.

Even extension

Extending a function from [0, L] symmetrically to get a cosine series.

Odd extension

Extending a function from [0, L] antisymmetrically to get a sine series.

Partial differential equation

An equation with partial derivatives of a multivariable function.

Heat equation

uₜ = k uₓₓ, modeling heat flow.

Wave equation

uₜₜ = c² uₓₓ, modeling vibrating strings.

Separation of variables

Assuming u = X(x)T(t) to split a PDE into ODEs.

Boundary conditions

Conditions on the solution at the edges of the domain.

Eigenfunction

A nonzero solution of the spatial equation for an allowed λ.

Sample practice questions

5 more questions like these, from level 1 to 5, are in the practice set.