Analysis 3 Autumn 2019

Lecturer
Mikaela Iacobelli
Coordinator
Xavier Fernandez-Real Girona

Information

Informations in ETHZ Course Catalogue

Q&A Session

There will be a Q&A session before the exam, on the 24.01, 13-15 PM.

Place: Room HG G26.1.

Time and Room

The lecture will take place at 8-10a.m. on Monday in HG G3. The first lecture will be on 23.09.

For the exercise classes, see the section Exercise classes.

Prüfungseinsicht

The Prüfungseinsicht for the Analysis 3 exam will take place in the following days:

Tuesday 25th February, 7:45-8:45, Room: HG G19.1.

Wednesday 4th March, 12:15-13:15, Room: HG G19.1.

Exam Information

The exam will take place on the 29.01 at 9 AM, in HIL G41. The duration of the exam will be of 3 hours (180 minutes).

Students will be able to take the exam either in German or in English (and bring a dictionary if needed).

Electronic devices (such as, but not limited to, smartphones, calculators, tablets, laptops, etc.) are NOT allowed to use during the exam.

Students are allowed to bring the book (Pinchover-Rubinstein) to the exam, either the original book, or a printed (total or partial) version.

Students are allowed to bring a summary of the lectures. Summaries can be at most 4 pages long on a DIN A4 paper size (297x210 mm) (that is, either 2 sheets of paper two-sided, or 4 sheets of paper one-sided). Summaries must be personal and handwritten. Photocopies of summaries, or computer typed summaries are NOT allowed.

Students are NOT allowed to bring the exercises and the solutions from this course to the exam (even as part of the summary). Random checks will be made on the day of the exam.

Abstract

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.

Schematic Syllabus

1.) General introduction to PDEs and their classification (linear, quasilinear, nonlinear / elliptic, parabolic, hyperbolic)

2.) Quasilinear first order PDEs
- Solution with the method of characteristics
- Conservation laws

3.) Hyperbolic PDEs
- wave equation
- d'Alembert formula in (1+1)-dimensions
- method of separation of variables

4.) Parabolic PDEs
- heat equation
- maximum principle
- method of separation of variables

5.) Elliptic PDEs
- Laplace equation
- maximum principle
- method of separation of variables
- variational method

Prerequisites

Analysis I and II, Fourier series (Komplexe Analysis)

Literature

Date Summaries of the lectures Chapters of the Literature Lecture Notes Comments
23.9 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. L.01 / ExtL.01 Lecture Notes L.01 covered. Page 11 is an EXTRA example.
Last update: 29.09
30.9 First order equations, quasilinear equations, Method of Characteristics, examples 2.1, 2.2, 2.3 (up to example 2.2 included) L.02 / ExtL.02 Lecture Notes L.02 covered.
Last update: 30.09
7.10 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) L.03 / ExtL.03
ExtraClarifications
Lecture Notes L.03 covered.
Last update: 12.10
14.10 Conservation laws and shock waves 2.7 L.04 Lecture Notes L.04 covered.
Last update: 14.10
21.10 Shock waves: the Rankine-Hugoniot condition, and the entropy condition. Classification of second order linear PDEs. 2.7, 3.2 L.05 Lecture Notes L.05 covered.
Last update: 21.10
28.10 The one-dimensional wave equation, canonical form and general solution. The Cauchy problem and d'Alembert formula. 4.1, 4.2, 4.3 L.06 Lecture Notes L.06 covered.
Last update: 28.10
4.11 Domain of dependence. The non-homogeneous one-dimensional wave equation. Nonhomogeneous d'Alembert formula. Separation of variables. 4.4, 4.5, 5.2 L.07 Lecture Notes L.07 covered.
Last update: 8.11
11.11 Separation of variables for the heat and wave equation, homogeneous problems. Dirichlet and Neumann boundary conditions. 5.2, 5.3 L.08 Lecture Notes L.08 covered.
Last update: 11.11
18.11 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) L.09 Lecture Notes L.09 covered.
Last update: 25.11
25.11 Elliptic equations. The weak maximum principle. The mean value principle. The strong maximum principle. 7.1, 7.2, 7.3 L.10 Lecture Notes L.10 covered.
Last update: 25.11
2.12 Applications of maximum principle (uniqueness). Green's identities. The maximum principle for the heat equation. Separation of variables for elliptic problems. 7.4, 7.5, 7.6, 7.7. L.11/ ExtL.11 Lecture Notes L.11 covered.
Last update: 5.12
9.12 Separation of variables in rectangles, Dirichlet and Neumann compatibility conditions. The Laplace equation in circular domains. 7.7.1, 7.7.2 L.12 Lecture Notes L.12.
Last update: 9.12
16.12 The Laplace equation in circular domains: annulus and sectors. A taste of the Calculus of Variations. Overview of the course. 7.7.2 L.13 Lecture Notes L.13.
Last update: 14.12

Every Monday after the lecture the corresponding exercise list will be uploaded. We encourage the students to attempt solving the exercises BEFORE going to the exercise classes. It is possible to hand in the exercises in order to have a feedback from the TAs. Solutions to the corresponding exercise class will be uploaded after the class.

The students can find the exercise box in the room HG F 28. Please hand in the exercise in the corresponding box of your assistant before the deadline showed below, or during the corresponding Exercise class.

The students are also encouraged to attend the Präsenz offered by D-MATH assistant group 6.

The first exercise class will be on 27.9.

ZeitRaumTutorSprache
Fr 10-12CAB G 56Göktug Alkande/en
Fr 10-12CLA E 4Aurelio Dolfinide/en
Fr 10-12LFV E 41Onur Fisekde/en
Fr 10-12LEE C 114Matteo Guscettide/en
Fr 10-12ETZ G 91Jens Hauserde/en
Fr 10-12ETZ H 91Markus Kiserde
Fr 10-12ETZ E 7Zeqiri Mustafade/en
Fr 10-12LFW B 3Alvin Pyngottude/en