Analysis 3 Autumn 2022

Lecturer: Mikaela Iacobelli
Coordinator: Lauro Silini

Information

Informations in ETHZ Course Catalogue.

Written Exam

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. They can use markers for the pages, underline/highlight important formulae/concepts, as long as they do not add written text to it (on the sides, over post-its etc.).
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. The only exception are summaries handwritten on a tablet, that are allowed to be printed as long as they remain personal and as long as the font size is comparable to a Summary handwritten directly on paper.
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 of the above rules will be made on the day of the exam.

Time and room

The lecture takes place in room HG E7 every Monday at 08-10 a.m.

First lecture: 26.09.2022

Content

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.

Objective

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.

Schematic Syllabus

General introduction to PDEs and their classification (linear, quasilinear, semilinear, nonlinear / elliptic, parabolic, hyperbolic)
Quasilinear first order PDEs

Solution with the method of characteristics
Conservation laws

Hyperbolic PDEs

Wave equation
D'Alembert formula in (1+1)-dimensions
Method of separation of variables

Parabolic PDEs

Heat equation
Maximum principle
Method of separation of variables

Elliptic PDEs

Laplace equation
Maximum principle
Method of separation of variables
Variational method

Prerequisites

Analysis I and II, Fourier series (Complex Analysis).

Lecture Summaries

See: LECTURE NOTES. Please, communicate all typos to Lauro Silini by sending an e-mail with object: typo+chapter number. Thank you.

Extra material kindly offered by Jean Megrét: MIND MAP and ILLUSTRATIONS AND SUMMARIES.

Date	Chapters	Summaries	Extra
26.09	1.1-1.5, 2.1, 2.2	Introduction, classification of PDEs (order, linearity, quasilinearity, homogeneity), examples, associated conditions to obtain a unique solution.	Extra01
03.10	2.1-2.3	First order equations, quasilinear equations, Method of Characteristics, examples.	Extra02
10.10	2.4-2.6	Examples of the characteristics method, and the existence and uniqueness theorem.
17.10	3.1-3.5	Conservation laws and shock waves.
24.10	3.5-3.6, 6.1	Shock waves: the Rankine-Hugoniot condition, and the entropy condition. Classification of second order linear PDEs.
31.10	4.1-4.2	The one-dimensional wave equation, canonical form and general solution. The Cauchy problem and d'Alembert formula.
07.11	4.3-4.5	Domain of dependence. The non-homogeneous one-dimensional wave equation. Nonhomogeneous d'Alembert formula. Separation of variables.
14.11	5.2-5.3	Separation of variables for the heat and wave equation, homogeneous problems. Dirichlet and Neumann boundary conditions.
21.11	5.3-5.4	Separation of variables for non-homogeneous equations. Resonance. The energy method for the wave and heat equation, and uniqueness of solutions.
28.11	6, 7.1-7.4	Elliptic equations. The weak maximum principle. The mean value principle. The strong maximum principle.
05.12	7.4-8.1	Applications of maximum principle (uniqueness). The maximum principle for the heat equation. Separation of variables for elliptic problems.
12.12	8.1-8.6	Separation of variables in rectangles, Dirichlet and Neumann compatibility conditions. The Laplace equation in circular domains.
19.12	8.6	The Laplace equation in circular domains: annulus and sectors.

Exercises

Every Monday after the lecture the corresponding exercise list will be uploaded. We encourage the students to attempt solving the exercises. First session: 30.09.2022.

exercise sheet	solutions	comments
Serie 01	Solutions 01
Serie 02	Solutions 02
Serie 03	Solutions 03
Serie 04	Solutions 04
Serie 05	Solutions 05
Serie 06	Solutions 06
Serie 07	Solutions 07
Serie 08	Solutions 08	Typo 8.2(a): integral is multiplied and not divided by t^a
Serie 09	Solutions 09
Serie 10	Solutions 10
Serie 11	Solutions 11
Serie 12	Solutions 12
Serie 13	Solutions 13
Mock Exam	Solutions Mock Exam	Corrected solution of Ex 4

Exercise classes

We use the SAMup tool for corrections. Be careful to be connected at the ETH Network, or use a proper VPN, like Cisco.

time	room	assistant
Fr 10-12	CAB G 56	M. Noseda
Fr 10-12	CLA E 4	S. Sherif Azer
Fr 10-12	ETZ E 7	Q. Wu
Fr 10-12	ETZ K 91	A. Kirchgessner
Fr 10-12	ETZ J 91	C. Sonnenschein
Fr 10-12	LEE C 114	Z. Lang
Fr 10-12	LFV E 41	M. Stoll
Fr 10-12	LFW B 3	H. Yu

Literature

Y. Pinchover, J. Rubinstein, "An introduction to Partial Differential Equations", Cambridge University Press (12. Mai 2005).