- Lecturer
- Mikaela Iacobelli
- Coordinator
- Vikram Giri

Informations in ETHZ Course Catalogue.

- 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.

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.

- 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

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

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.

Date | Chapters | Summaries | Extra | Remarks |
---|---|---|---|---|

22.09 | 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 | |

29.09 | 2.1-2.3 | First order equations, quasilinear equations, Method of Characteristics, examples. | Extra02, Extra on method of characteristics | |

06.10 | 2.4-2.6 | Examples of the characteristics method, and the existence and uniqueness theorem. | ||

13.10 | 3.1-3.5 | Conservation laws and shock waves. | ||

20.10 | 3.5-3.6, 6.1 | Shock waves: the Rankine-Hugoniot condition, and the entropy condition. Classification of second order linear PDEs. | ||

27.10 | 4.1-4.2 | The one-dimensional wave equation, canonical form and general solution. The Cauchy problem and d'Alembert formula. | ||

03.11 | 4.3-4.5 | Domain of dependence. The non-homogeneous one-dimensional wave equation. Nonhomogeneous d'Alembert formula. Separation of variables. | ||

10.11 | 5.2-5.3 | Separation of variables for the heat and wave equation, homogeneous problems. Dirichlet and Neumann boundary conditions. | ||

17.11 | 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. | ||

24.11 | 6, 7.1-7.4 | Elliptic equations. The weak maximum principle. The mean value principle. The strong maximum principle. | ||

01.12 | 7.4-8.1 | Applications of maximum principle (uniqueness). The maximum principle for the heat equation. Separation of variables for elliptic problems. | Extra11 | |

08.12 | 8.1-8.6 | Separation of variables in rectangles, Dirichlet and Neumann compatibility conditions. The Laplace equation in circular domains. | ||

15.12 | 8.6 | The 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 |

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 |
---|---|---|

Mo 0815-1000 | HG E 33.1 | S. Sherif Azer |

Mo 0815-1000 | HG E 33.3 | C. Bugnon |

Mo 0815-1000 | HG F 26.5 | J. Baumann |

Mo 1215-1400 | HG E 33.3 | M. Gwozdz |

Mo 1215-1400 | HG E 33.5 | C. Sacan |

Mo 1215-1400 | ML F 40 | N. Bianchi |

Mo 1415-1600 | GLC E 29.2 | M. Noseda |

Mo 1415-1600 | HG E 21 | B. Lezon |

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