- Lecturer
- Alessandro Carlotto
- Coordinator
- Federico Glaudo

- Lecture time and place
- The lecture will take place every Thursday at 10-12, in HCI J 7.
- Exercise classes
- For information on the exercise classes, see the section Exercise classes.
- First lecture
- 23.09.2021
- Course Catalogue
- 401-0373-00L Mathematics III: Partial Differential Equations

- Multivariate functions: partial derivatives, differentiability, Jacobian matrix, Jacobian determinant
- Multiple integrals: Riemann integrals in two or three variables, change of variables
- Sequences and series of numbers and of functions
- Basic knowledge of ordinary differential equations

Examples of partial differential equations. Linear partial differential equations. Separation of variables. Fourier series, Fourier transform, Laplace transform. Applications to solving commonly encountered linear partial differential equations (Laplace's Equation, Heat Equation, Wave Equation).

S.J. Farlow. *Partial Differential Equations for Scientists and
Engineers*. Dover Books on Mathematics, NY.

N. Hungerbühler. *Einführung in partielle Differentialgleichungen
für Ingenieure, Chemiker und Naturwissenschaftler*. vdf
Hochschulverlag, 1997.

T. Westermann. *Partielle Differentialgleichungen, Mathematik für
Ingenieure mit Maple*. Band 2, Springer-Lehrbuch, 1997 (chapters
XIII,XIV,XV,XII).

E. Kreyszig. *Advanced Engineering Mathematics*. John Wiley & Sons
(chapters 1,2,11,12,6).

The basic information (e.g. written exam, duration 120 minutes, etc..) is published in the ETH course catalogue.

Additionally, we provide you with:

- A detailed description of the exam structure.
- Some suggestions regarding the preparation for the exam.
- A mock exam (Probeprüfung) with a structure similar to the real exam.
- The solutions of the Probeprüfung.
- A list of training problems to get ready for the exam.
- The official exam of January 2022.
- The solutions of the exam of January 2022.

Date | Content | Notes | Reference | Additional material | |
---|---|---|---|---|---|

1 | 23.09. | Introduction to the course, general information. Recollections about ordinary differential equations. Examples of partial differential equations (linear/non-linear, homogeneous/inhomogeneous, different orders...), classification of second-order linear equations (elliptic, parabolic, hyperbolic). | lecture_notes1.pdf | Farlow, Lesson 1 | Study advice |

2 | 30.09. | Modelling the temperature profile in a metal rod. The heat equation, heuristics and basic facts. Different classes of boundary conditions, and homogeneous special cases (Dirichlet, Neumann, Robin). | lecture notes2.pdf | Farlow, Lessons 2-3-4 | |

3 | 07.10. | The method of separation of variables applied to an IBVP for the heat equation. Review on orthonormal bases. The (real) Fourier series of a piecewise C^1 function, basic setup and convergence results (statements). | lecture notes3.pdf | Farlow, Lesson 5 and Lesson 11 (first part) | |

4 | 14.10. | Quick review. Fourier series of even/odd functions; even/odd extensions and (apparent) ambiguity problems; periodic extension. Complex Fourier series and conversion formulae. Two examples: the square wave, and the (half-) sawtooth wave. | lecture notes4.pdf | ||

5 | 21.10. | Dealing with oblique (Robin) boundary conditions: the associated Sturm-Liouville problem, statements of some basic results. Inhomogeneous problems via eigenfunction expansions: general procedure and application to one concrete example. | lecture notes5.pdf | Farlow, Lesson 7 and Lesson 9 | |

6 | 28.10. | The Fourier transform of an integrable function: motivation, heuristics, key definitions and basic properties (linearity, derivation/multiplication, translation/modulation, convolution/pointwise product). Example: computing the Fourier transform of a Gaussian. | lecture notes6.pdf | Farlow, Lesson 11 and lesson 12 | Cheat sheet on the basic properties of the Fourier transform |

7 | 4.11. | Two explicit computations of Fourier transforms: the characteristic function of an interval, the tent function. The use of Fourier transform to solve an initial-value problem for the heat equation on the real line. Heat kernel and integral representation of the solution. | lecture notes7.pdf | ||

8 | 11.11. | Heat equation on the real line: fast diffusion, L^1-conservation, L^2-dissipation, asymptotics via heat kernel and the convolution formula. The one-dimensional wave equation as a model for a violin string. The form an IBVP for the wave equation, classes of boundary conditions. | lecture notes8.pdf | Farlow, Lesson 16 and lesson 19 | |

9 | 18.11. | The D'Alembert formula for the wave equation, obtained in two different ways (via Fourier transform, or via canonical coordinates). Basic consequences: finite speed of propagation, left- and right-moving trains, domains of dependence. | lecture notes9.pdf | Farlow, Lesson 17 | |

10 | 25.11. | Fundamentals of wave propagation: characteristic lines, domains of dependence, cones of influence. One case study. Duhamel's principle and the D'Alembert formula for the inhomogenous wave equation. | lecture notes10.pdf | Farlow, Lesson 18 | |

11 | 02.12. | Introduction to elliptic boundary value problems. Physical motivations and heuristics. Spectrum of the Laplacian under various boundary conditions. Two model problems in the case of rectangles, solved via separation of variables. | lecture notes11.pdf | Farlow, Lessons 31 and lesson 32 | |

12 | 09.12. | Elliptic boundary value problems on the disc. Integral solution via Poisson kernel. Mean value property, maximum principle. | lecture notes12.pdf | Farlow, Lesson 33 | |

13 | 16.12. | Review on Poisson kernel and consequences. Uniqueness of solutions of the Dirichlet problem. Comments about annuli vs. discs. Solving via Fourier transform the Dirichlet problem on the upper half-space subject to decay conditions at infinity. | lecture notes13.pdf | Farlow, Lesson 34 | |

14 | 23.12. | Spherical harmonics via separation of variables. Euler's equation. Legendre's equation and the Legendre polynomials. An example of Dirichlet BVP in the three-dimensional ball. | lecture notes14.pdf | Farlow, Lesson 35 |

In order to easily interact, we set up a forum for our course at the link Mathematik III (Autumn 2021) - Forum. You have to sign up with your ETH credentials. There you find several topics where you can ask questions and discuss about the lectures, the problem sets, the exam, etc. Use it!

Please register and enroll for a teaching assistant in myStudies. The enrollment is needed to attend the exercise class and to hand in your homework.

The first exercise class will take place on the 30th of September.

Three of the exercise classes take place with phisical attendance, while one is online (via Zoom). The registrations of the online exercise class will be uploaded on this website.

time | room | assistant |
---|---|---|

Th 09-10 | HCI J 7 | Luis Brummet |

Th 09-10 | HCP E 47.1 | Joel Jenny |

Th 09-10 | ETH Zoom | Pieter-Bart Peters |

Th 12-13 | HCP E 47.2 | Samuel Anzalone |

Here is the diary of the exercise classes

Date | Content | Notes | Recordings | Slides | |
---|---|---|---|---|---|

1 | 30.09. | Discussion on the general solution of au''+bu'+cu=0; examples of integration by separation of variables, e.g., Problem 1.2 (d)-(e); classification of 2nd order linear PDE in two variables (elliptic, hyperbolic, parabolic). | Notes 1 | Recording 1 | Slides 1 |

2 | 07.10. | Problem 2.3 (third problem in problem set 2), computation of integrals like \int_{0}^T sin(2\pi m t/T) sin(2\pi n t/T). | Notes 2 | Recording 2 | Slides 2 |

3 | 14.10. | Basic facts about even and odd functions, solutions of problem 3.3 nd 3.4. | Notes 3 | Recording 3 | Slides 3 |

4 | 21.10. | Solution of problem 4.3. | Notes 4 | Recording 4 | Slides 4 |

5 | 28.10. | Solutions of problem 5.1 and 5.2. | Notes 5 | Recording 5 | Slides 5 |

6 | 04.11. | Key properties of the Fourier transform, solutions of problem 6.3 and 6.4. | Notes 6 | Recording 6 | Slides 6 |

7 | 11.11. | Solutions of problem 7.3, 7.4. | Notes 7 | Recording 7 | Slides 7 |

8 | 18.11. | Solution of problem 8.2. | Notes 8 | Recording 8 | Slides 8 |

9 | 25.11. | Solution of problem 9.2. | Notes 9 | Recording 9 | |

10 | 02.12. | Solution of problem 10.2 question (2), solution of problem 10.3, solution of problem 9.2 via d'Alembert formula. | Notes 10 | Recording 10 | |

11 | 09.12. | Review of the spectrum of the Laplacian and discussion on problem 11.1. Solution of problem 11.3. | Notes 11 | Recording 11 | |

12 | 16.12. | Solution of problem 12.1 and 12.3. | Notes 12 | Recording 12 | |

13 | 23.12. | Elliptic problems on radially symmetric domains, review on Fourier transform. | Notes 13, Detailed solution of the problems considered in the exercise class, Review of elliptic problems on balls and annuli | Recording 13 |

Every Friday, a new problem set is uploaded here. You have seven days to solve the problems and hand in your solutions for grading. You can hand in your solution not later than 10pm of the Friday following the publication of the problem set.

You may hand in your solutions in the mailbox of your TA (which you can find in HG F28) or sending them to your TA via email or via the platform SAMUpTool. The solutions will be published on Monday (after the due date for the problem set).

During exercise classes on Thursday some of the problems are discussed (notice that you can hand in your solutions also after the exercise classes).

Assignment date | Due date | Problem set | Solution |
---|---|---|---|

Fri 24.09. | Fri 01.10. | Problem set 1 | Solutions 1 |

Fri 01.10. | Fri 08.10. | Problem set 2 | Solutions 2 |

Fri 08.10. | Fri 15.10. | Problem set 3 | Solutions 3 |

Fri 15.10. | Fri 22.10. | Problem set 4 | Solutions 4 |

Fri 22.10. | Fri 29.10. | Problem set 5 | Solutions 5 |

Fri 29.10. | Fri 05.11. | Problem set 6 | Solutions 6 |

Fri 05.11. | Fri 12.11. | Problem set 7 | Solutions 7 |

Fri 12.11. | Fri 19.11. | Problem set 8 | Solutions 8 |

Fri 19.11. | Fri 26.11. | Problem set 9 | Solutions 9 |

Fri 26.11. | Fri 03.12. | Problem set 10 | Solutions 10 |

Fri 03.12. | Fri 10.12. | Problem set 11 | Solutions 11 |

Fri 10.12. | Fri 17.12. | Problem set 12 | Solutions 12 |

You are free to come and ask questions. The office hours are held via Zoom. The schedule is as follows (up to possible short-term changes, please check for updates).

Date | Time | Zoom link | Assistant |
---|---|---|---|

Tue 28.9. | 16.20-17.50 | ETH Zoom | Federico Glaudo |

Tue 5.10. | 16.00-17.30 | ETH Zoom | Luis Brummet |

Tue 12.10. | 16.00-17.30 | ETH Zoom | Pieter-Bart Peters |

Tue 19.10. | 16.00-17.30 | ETH Zoom | Joel Jenny |

Tue 26.10. | 16.00-17.30 | ETH Zoom | Samuel Anzalone |

Wed 3.11. | 16.15-17.45 | ETH Zoom | Federico Glaudo |

Wed 10.11. | 16.15-17.45 | ETH Zoom | Joel Jenny |

Wed 17.11. | 16.15-17.45 | ETH Zoom | Samuel Anzalone |

Wed 24.11. | 16.15-17.45 | ETH Zoom | Peters Pieter-Bart |

Wed 1.12. | 16.15-17.45 | ETH Zoom | Luis Brummet |

Wed 8.12. | 16.15-17.45 | ETH Zoom | Joel Jenny |

Wed 15.12. | 16.15-17.45 | ETH Zoom | Federico Glaudo |