Partial differential equations and Fourier series

Builds on

Required prerequisites, all the way down: 38

You will learn to

• Solve the heat and wave equations on an interval by separation of variables and Fourier series.
• State convergence results for Fourier series and compute examples, including the Gibbs phenomenon.
• Use the maximum principle and energy methods to prove uniqueness of solutions.

Definition

• The heat equation is u_t = k u_xx, the wave equation is u_tt = c² u_xx, and Laplace's equation is Δu = 0.
• The Fourier series of an integrable 2π-periodic function f is Σ cₙ e^(inx) with cₙ = (1/2π) ∫_{−π}^{π} f(x) e^(−inx) dx.

Theorem

• Parseval: for square-integrable f, Σ |cₙ|² = (1/2π) ∫_{−π}^{π} |f(x)|² dx.
• Harmonic functions have the mean value property and satisfy the maximum principle.
• d'Alembert: every twice continuously differentiable solution of the wave equation on ℝ has the form f(x − ct) + g(x + ct).

Common misconceptions

• Thinking the Fourier series of every continuous function converges at every point; du Bois-Reymond constructed a counterexample.
• Assuming a Fourier series can always be differentiated term by term.

Optional discoveries

• Functional analysis Fourier series are expansions in an orthonormal basis of the Hilbert space L².
• ← Trigonometry

Learning pathways

• Applied mathematics

Further reading

• Walter A. Strauss, Partial Differential Equations: An Introduction
• Elias M. Stein and Rami Shakarchi, Fourier Analysis: An Introduction (2003)