Analysis 3 Autumn 2023

Mikaela Iacobelli
Vikram Giri


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 NO C 60 every Friday at 10.15-12.00.



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 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.


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.

exercise sheet solutions comments
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.

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


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