| Chapter/Section | Topics |
| Chapter 1 | Where PDEs come from |
| 1.1 | What is a Partial Differential Equation? |
| 1.2 | First-order Linear Equations (Solution in the constant-coefficient case; the variable-coefficient case and characteristic curves. |
| 1.3 | Flows, Vibrations and Diffusions (Derivations of PDEs in various physical situations; e.g., the vibrating string, the vibrating drumhead, diffusion, heat flow, hydrogen atom). |
| 1.4 | Initial and boundary conditions (the Dirichlet, Neumann and Robin conditions and their significance for the vibrating string and diffusion equations. Conditions at infinity.) |
| 1.5 | Well- (and ill-)Posed Problems. |
| 1.6 | Types of second-order equations. |
| Chapter 2 | Waves and Diffusion |
| 2.1 | The Wave Equation (D'Alembert's solution on the line; the plucked string). |
| 2.2 | Causality and Energy. |
| 2.3 | The Diffusion (or Heat) Equation (the maximum principle; uniqueness for the Dirichlet problem). |
| 2.4 | Diffusion on the whole real line (the Gaussian or fundamental solution). |
| 2.5 | Comparison of waves and diffusion. |
| Chapter 4 | Boundary Problems |
| 4.1 | Separation of Variables, the Dirichlet Condition (both for the wave and the diffusion equations). |
| - | Chapter 5 §§ 5.1, 5.2 and 5.3. - These sections will be part of Exam 1. |
| 4.2 | The Neumann Condition. |
| Chapter 5 | Fourier Series |
| 5.1 | The Coefficients (or discrete Fourier transform): formulas for the coefficients, applications to the wave and the diffusion equations. |
| 5.2 | Even, Odd, Periodic and Complex-valued functions. |
| 5.3 | Orthogonality and "General Fourier Series" (orthogonal systems from symmetric boundary conditions; complex eigenvalues), |
| 5.4 | Completeness (three notions of convergence: pointwise, uniform and mean-square: convergence results for Fourier series and their generalizations). |
| ***1st Exam*** | |
| Chapter 6 | Harmonic Functions |
| 6.1 | The Laplace Equation (its physical significance, maximum principle, uniqueness of solutions of the Dirichlet Problem, invariance of the Laplace operator under rigid motions). |
| 6.2 | Rectangles and Cubes. |
| 6.3 | Poisson's Formula |
| Chapter 7 | Green's Identities and Green's Functions |
| 7.1 | Green's First Identity (and some consequences). |
| 7.2 | Green's Second Identity (and some consequences). |
| 7.3 | Green's Functions and the Dirichlet Problem. |
| 7.4 | Half-Spaces and Spheres. |