Fourier Series: Why Your Approximation Always Fails?! (The Math Behind PDEs Ep5)

Three of the most important partial differential equations in physics — the heat equation, the wave equation, and Laplace's equation — look like completely different problems. But a single idea solves all three: expand the solution in the eigenfunctions of the Laplacian. This is the story of separation of variables, Fourier series, and why eigenfunctions turn hard PDEs into simple lists of ordinary differential equations. We start from the guess u(x,t) = X(x)T(t), discover the separation constant, let the boundary conditions quantize the spectrum (for a pinned interval, the Dirichlet eigenvalues are λₙ = n²), reinterpret Fourier coefficients as orthogonal projection, confront the Gibbs phenomenon, and finally see — through Sturm–Liouville theory — why this machinery always works, on drums, spheres, and even graphs. ⏱️ Chapters 0:00 Three equations, one hidden idea 1:31 Separation of variables 3:14 Boundary conditions build the spectrum 5:06 Fourier series as projection 7:08 Convergence and the Gibbs phenomenon 8:33 Sturm–Liouville theory: why this always works 10:30 One eigenbasis for all three 📚 Key concepts Separation of variables and the separation constant The Dirichlet eigenvalue problem on an interval (eigenvalues λₙ = n² for L = π) Orthogonality of {sin(nx)} on [0, π] and the Fourier sine series Fourier coefficients as orthogonal projection; Parseval's identity (energy conservation in L²) The Gibbs phenomenon: a persistent ≈8.95% overshoot at a jump discontinuity, with pointwise vs. L² convergence Sturm–Liouville theory and self-adjoint (symmetric) operators The spectral theorem for a real symmetric operator (real eigenvalues, orthogonal eigenfunctions, completeness) Eigenfunctions of the Laplacian in other geometries: Bessel, Legendre, and Hermite functions 📖 References Walter A. Strauss, Partial Differential Equations: An Introduction — separation of variables and Fourier series (Ch. 4–5), higher-dimensional eigenvalue problems (Ch. 10) Richard Haberman, Applied Partial Differential Equations with Fourier Series and Boundary Value Problems — heat equation & separation (Ch. 2), Fourier series (Ch. 3), Sturm–Liouville eigenvalue problems (Ch. 5) Brown & Churchill, Fourier Series and Boundary Value Problems — Fourier series and Sturm–Liouville expansions Stein & Shakarchi, Fourier Analysis: An Introduction — the wave equation as genesis (Ch. 1), convergence and the Gibbs phenomenon (Ch. 2–3) 🛠️ Tools used in this video: Animation: Manim (Python) Voice: ElevenLabs Manim Starter Pack (31 ready-to-use scenes): https://axiommotion.gumroad.com/l/drhyqd