Informations in ETHZ Course Catalogue.
The Exam will take place Friday, 28 January 2022, from 3pm to 6pm.
Random checks of the above rules will be made on the day of the exam.
The lecture takes place in room HG G3 every Monday at 08-10 a.m.
Live Streaming is available for members of ETH. A video recording of the lecture is uploaded every Tuesday under password protection.
In this lecture we treat problems in applied analysis. The focus lies on the solution of quasilinear first order PDEs with the method of characteristics, and on the study of three fundamental types of partial differential equations of second order: the Laplace equation, the heat equation, and the wave equation.
The aim of this class is to provide students with a general overview of first and second order PDEs, and teach them how to solve some of these equations using characteristics and/or separation of variables.
Y. Pinchover, J. Rubinstein, "An introduction to Partial Differential Equations", Cambridge University Press (12. Mai 2005).
Analysis I and II, Fourier series (Complex Analysis).
Here is a first version of the complete LECTURE NOTES.
Date | Summaries | Corresponding chapters in the literature | Hand-written notes/Extras |
---|---|---|---|
27.09 | Introduction, classification of PDEs (order, linearity, quasilinearity, homogeneity), examples, associated conditions to obtain a unique solution. | 1.1, 1.2, 1.3, 1.4.3, 1.5.1, 1.5.2. | Lecture 01/ Extra 01 |
04.10 | First order equations, quasilinear equations, Method of Characteristics, examples. | 2.1, 2.2, 2.3 (up to example 2.2 included) | Lecture 02/ Extra 02 |
11.10 | Examples of the characteristics method, and the existence and uniqueness theorem. | 2.3, 2.4 (examples 2.3, 2.5, 2.6), 2.5 (Theorem 2.10) | Lecture 03/ Extra 03 |
18.10 | Conservation laws and shock waves | 2.7 | Lecture 04 |
25.10 | Shock waves: the Rankine-Hugoniot condition, and the entropy condition. Classification of second order linear PDEs. | 2.7, 3.2 | Lecture 05 |
01.11 | The one-dimensional wave equation, canonical form and general solution. The Cauchy problem and d'Alembert formula. | 4.1, 4.2, 4.3 | Lecture 06 |
08.11 | Domain of dependence. The non-homogeneous one-dimensional wave equation. Nonhomogeneous d'Alembert formula. Separation of variables. | 4.4, 4.5, 5.2 | Lecture 07 |
15.11 | Separation of variables for the heat and wave equation, homogeneous problems. Dirichlet and Neumann boundary conditions. | 5.2, 5.3 | Lecture 08 |
22.11 | Separation of variables for non-homogeneous equations. Resonance. The energy method for the wave and heat equation, and uniqueness of solutions. | 5.4, 5.5 (Example 6.45, 5.3) | Lecture 09 |
29.11 | Elliptic equations. The weak maximum principle. The mean value principle. The strong maximum principle. | 7.1, 7.2, 7.3 | Lecture 10 |
6.12 ONLY RECORDED |
Applications of maximum principle (uniqueness). |
7.4, 7.5, 7.6, 7.7. | Lecture 11 |
13.12 | Separation of variables in rectangles, Dirichlet and Neumann compatibility conditions. The Laplace equation in circular domains. | 7.7.1, 7.7.2 | Lecture 12 |
20.12 |
🎄 The Laplace equation in circular domains: annulus and sectors. | 7.7.2 | Lecture 13 |
Every Monday after the lecture the corresponding exercise list will be uploaded. We encourage the students to attempt solving the exercises. The first session will be on Friday October 1st . Please do not forget to register to one of the following exercise groups via myStudies.
exercise sheet | solutions | Comments |
---|---|---|
Exercise sheet 1 | Solutions 1 | |
Exercise sheet 2 | Solutions 2 |
Fixed typo in 2.3: Cross the |
Exercise sheet 3 | Solutions 3 | |
Exercise sheet 4 | Solutions 4 | |
Exercise sheet 5 | Solutions 5 |
Corrected empty reference in part (b), exercise 2. In exercise 5.4 (a) substitute |
Exercise sheet 6 | Solutions 6 | |
Exercise sheet 7 | Solutions 7 | Solution of 7.4 is more detailed now. |
Exercise sheet 8 | Solutions 8 | |
Exercise sheet 9 | Solutions 9 | |
Exercise sheet 10 | Solutions 10 | |
Exercise sheet 11 | Solutions 11 | |
Exercise sheet 12 | Solutions 12 | |
Exercise sheet 13 | Solutions 13 | |
MOCK EXAM | SOLUTIONS | Two typos corrected: ex 1 (c) we ask for the maximal domain where the PDE is elliptic, not just a subset of it. In 1 (f) there was a missing square root in the definition of D. |
time | room | assistant |
---|---|---|
Fr 10-12 | CAB G 56 | Marina Fernandez Garcia |
Fr 10-12 | CLA E 4 | Salmane El Messoussi |
Fr 10-12 | ETZ E 7 | Henry Marius |
Fr 10-12 | ETZ J 91 | Luca Krebs |
Fr 10-12 | ETZ K 91 | Anthony Salib |
Fr 10-12 | LEE C 114 | Jean Mégret |
Fr 10-12 | LFV E 41 | Eric Soriano Baguet |
Fr 10-12 | online | Jan-Marco Haldemann |
Fr 10-12 | online | Arielle Rüfenacht |