Analysis 3 Autumn 2022

Mikaela Iacobelli
Lauro Silini


Informations in ETHZ Course Catalogue.

Written Exam

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



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.

Schematic Syllabus

  1. General introduction to PDEs and their classification (linear, quasilinear, semilinear, nonlinear / elliptic, parabolic, hyperbolic)
  2. Quasilinear first order PDEs
  3. Hyperbolic PDEs
  4. Parabolic PDEs
  5. Elliptic PDEs


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.

26.091.1-1.5, 2.1, 2.2Introduction, classification of PDEs (order, linearity, quasilinearity, homogeneity), examples, associated conditions to obtain a unique solution. Extra01
03.102.1-2.3First order equations, quasilinear equations, Method of Characteristics, examples. Extra02
10.102.4-2.6 Examples of the characteristics method, and the existence and uniqueness theorem.
17.103.1-3.5Conservation laws and shock waves.
24.103.5-3.6, 6.1Shock waves: the Rankine-Hugoniot condition, and the entropy condition. Classification of second order linear PDEs.
31.104.1-4.2The one-dimensional wave equation, canonical form and general solution. The Cauchy problem and d'Alembert formula.
07.114.3-4.5Domain of dependence. The non-homogeneous one-dimensional wave equation. Nonhomogeneous d'Alembert formula. Separation of variables.
14.115.2-5.3Separation of variables for the heat and wave equation, homogeneous problems. Dirichlet and Neumann boundary conditions.
21.115.3-5.4Separation of variables for non-homogeneous equations. Resonance. The energy method for the wave and heat equation, and uniqueness of solutions.
28.116, 7.1-7.4Elliptic equations. The weak maximum principle. The mean value principle. The strong maximum principle.
05.127.4-8.1Applications of maximum principle (uniqueness). The maximum principle for the heat equation. Separation of variables for elliptic problems.
12.128.1-8.6Separation of variables in rectangles, Dirichlet and Neumann compatibility conditions. The Laplace equation in circular domains.
19.128.6The Laplace equation in circular domains: annulus and sectors.


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 ta
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.

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


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