Introduction to Partial Differential Equations
Undergraduate exercise sessions for the ETH Zurich course on first and second order PDEs, covering the method of characteristics and the Laplace, heat, and wave equations.
Instructor: Prof. Dr. Mikaela Iacobelli
Term: Fall
Location: ETH Zurich, Department of Mathematics
I was a teaching assistant responsible for holding weekly exercise sessions. The course is instructed by Prof. Dr. Mikaela Iacobelli.
Content
The course provides a general overview of first and second order PDEs, with a focus on the solution of quasilinear first order PDEs with the method of characteristics, and on the study of three fundamental types of second order partial differential equations: the Laplace equation, the heat equation, and the wave equation.
My role
I taught and led a class of 40 engineering students weekly throughout the semester. I also designed weekly exercise sheets and corrected student submissions.
Schedule
| Week | Date | Topic | Materials |
|---|---|---|---|
| 1 | 30.09.2022 | Introduction and classification of PDEs Order, linearity, quasilinearity, homogeneity, examples, and associated conditions to obtain a unique solution. | |
| 2 | 07.10.2022 | First order equations and the method of characteristics Quasilinear equations, the method of characteristics, and worked examples. | |
| 3 | 14.10.2022 | Existence and uniqueness for the characteristics method Further examples of the characteristics method and the existence and uniqueness theorem. | |
| 4 | 21.10.2022 | Conservation laws and shock waves Conservation laws, formation of shocks, and weak solutions. | |
| 5 | 28.10.2022 | Shock waves and classification of second order PDEs The Rankine-Hugoniot and entropy conditions; classification of second order linear PDEs. | |
| 6 | 04.11.2022 | The one-dimensional wave equation Canonical form and general solution. The Cauchy problem and d’Alembert’s formula. | |
| 7 | 11.11.2022 | Non-homogeneous wave equation and separation of variables Domain of dependence. The non-homogeneous one-dimensional wave equation. Nonhomogeneous d’Alembert formula. Separation of variables. | |
| 8 | 18.11.2022 | Separation of variables for heat and wave equations Homogeneous problems; Dirichlet and Neumann boundary conditions. | |
| 9 | 25.11.2022 | Non-homogeneous equations, resonance, and the energy method Separation of variables for non-homogeneous equations. Resonance. The energy method for the wave and heat equations and uniqueness of solutions. | |
| 10 | 02.12.2022 | Elliptic equations and maximum principles The weak maximum principle. The mean value principle. The strong maximum principle. | |
| 11 | 09.12.2022 | Applications of the maximum principle Uniqueness via the maximum principle. The maximum principle for the heat equation. Separation of variables for elliptic problems. | |
| 12 | 16.12.2022 | Separation of variables in rectangles and circular domains Dirichlet and Neumann compatibility conditions. The Laplace equation in circular domains. | |
| 13 | 23.12.2022 | Laplace equation on annulus and sectors The Laplace equation in circular domains continued: annulus and sectors. |