# Analysis 3 Autumn 2023

Lecturer
Mikaela Iacobelli
Coordinator
Vikram Giri

## 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 NO C 60 every Friday at 10.15-12.00.

## 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

1. General introduction to PDEs and their classification (linear, quasilinear, semilinear, 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 (Complex Analysis).

## Lecture Summaries

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

Please alse see: Additional notes for some excellently written notes and study maps.

22.091.1-1.5, 2.1, 2.2Introduction, classification of PDEs (order, linearity, quasilinearity, homogeneity), examples, associated conditions to obtain a unique solution. Extra01
29.092.1-2.3First order equations, quasilinear equations, Method of Characteristics, examples. Extra02, Extra on method of characteristics
06.102.4-2.6 Examples of the characteristics method, and the existence and uniqueness theorem.
13.103.1-3.5Conservation laws and shock waves.
20.103.5-3.6, 6.1Shock waves: the Rankine-Hugoniot condition, and the entropy condition. Classification of second order linear PDEs.
27.104.1-4.2The one-dimensional wave equation, canonical form and general solution. The Cauchy problem and d'Alembert formula.
03.114.3-4.5Domain of dependence. The non-homogeneous one-dimensional wave equation. Nonhomogeneous d'Alembert formula. Separation of variables.
10.115.2-5.3Separation of variables for the heat and wave equation, homogeneous problems. Dirichlet and Neumann boundary conditions.
17.115.3-5.4Separation of variables for non-homogeneous equations. Resonance. The energy method for the wave and heat equation, and uniqueness of solutions.
24.116, 7.1-7.4Elliptic equations. The weak maximum principle. The mean value principle. The strong maximum principle.
01.127.4-8.1Applications of maximum principle (uniqueness). The maximum principle for the heat equation. Separation of variables for elliptic problems.Extra11
08.128.1-8.6Separation of variables in rectangles, Dirichlet and Neumann compatibility conditions. The Laplace equation in circular domains.
15.128.6The Laplace equation in circular domains: annulus and sectors.

## Exercises

Every Friday after the lecture the corresponding exercise list will be uploaded. We encourage the students to attempt solving the exercises. First exercise session: 25.09.2023.

Sheet 1 Solns 1
Sheet 2 Solns 2
Sheet 3 Solns 3
Sheet 4 Solns 4
Sheet 5 Solns 5
Sheet 6 Solns 6
Sheet 7 Solns 7
Sheet 8 Solns 8
Sheet 9 Solns 9
Sheet 10 Solns 10
Sheet 11 Solns 11
Sheet 12 Solns 12

## 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.

timeroomassistant
Mo 0815-1000HG E 33.1S. Sherif Azer
Mo 0815-1000HG E 33.3C. Bugnon
Mo 0815-1000HG F 26.5J. Baumann
Mo 1215-1400HG E 33.3M. Gwozdz
Mo 1215-1400HG E 33.5C. Sacan
Mo 1215-1400ML F 40N. Bianchi
Mo 1415-1600GLC E 29.2M. Noseda
Mo 1415-1600HG E 21B. Lezon

## Literature

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