Instructor: Sigurd Angenent

For information about exams, homework, office hours, etc. see canvas.

Lecture notes, homework assignments and solutions will be posted here during the semester.

Notes

• All notes in one pdf file
• Transcripts of each lecture
• First order equations
• The Wave Equation
• Fourier Series and the Wave Equation

Calculus, linear algebra, and real analysis review

• Differential Equations from Calculus reference
• Real Analysis reference

First order equations

• Solution by characteristics
• solutions without derivatives
• equations for the derivatives of solutions
• nonlinear equations: solutions can become singular—then what?

The wave equation (mostly 1–D)

• d’Alembert solution on the real line; derivation
• non smooth solutions — what is a solution?
• coordinate changes: Lorentz transformations, special relativity
• solution on an interval with boundary conditions; Fourier’s solution
• digression: convergence of Fourier series
• Higher dimensions: what about vibrating membranes?

Elliptic equations

• The Laplacian and Newton’s potential; where they appear, relation to Brownian motion
• Harmonic functions, the Maximum Principle and the Mean Value Property
• Dirichlet’s principle — the need for generalized functions
• $L^2(\Omega)$ and $H_0^1(\Omega)$ — an introduction without measure theory
• Hilbert spaces; Riesz representation theorem
• Weak solutions to Poisson's equation $-\Delta u = f$
• Poincaré inequality, compactness
• Eigenfunctions of the Laplacian (and also of $-\Delta+V(x)$)

Parabolic equations

• Derivation, relation with random walks, Brownian motion, Diffusion
• Solution on $\mathbb{R}^n\times[0, T)$; the heat kernel; semigroup property
• Well-posed initial value problems --- smoothing property and why you can't go back in time
• A finite domain, Fourier's solution --- the origin of Fourier series
• Duhamel's principle and the inhomogeneous equation $u_t=\Delta u + f(x, t)$
• The Maximum Principle
• Semilinear equation, $u_t=\Delta u + f(u, \nabla u)$ and Picard iteration
• Systems of equations, reaction diffusion equations
• Turing instability and the spots on a leopard