Lecture:
Tuesdays 10:20 - 12:00, R510 (with a 10-minute break)
Practice:
Tuesdays 14:15 - 15:45, R510
Material: The lecture does not follow any (English) book, but different topics can be found in the following books:
Lawrence C. Evans, Partial Differential Equations, AMS, Providence, 2002.
Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2010.
Vladimir Arnold, Lectures on Partial Differential Equations, Springer, 2004.
Exercises can be found at the end of each above-mentioned book.
Requirements:
1. For the practice part:
During the semester, there are going to be two midterms.
There are also some bonus problems at the end of each practice part, which can be solved at home and then submitted on the next practice session.
Grades at the end of the semester: (the points are subject to modification, but only downwards)
40-59: grade 2
60-79: grade 3
80-99: grade 4
above 100: grade 5
2. For the lecture part:
The course ends with a written exam.
It has two parts: the first part consists of small questions, like stating a theorem, or some easy questions which will be answered during the semester as "remarks" (or they are trivial consequences of some theorems).
In the second part you have to describe a part of the material thoroughly, but you will be guided by some helping questions.
Last modified: 30 April 2024
Last modified: 21 May 2024 (These are being corrected continuously, so please always use the most up-to-date version.)
Schedule for the semester:
| 13 February 2024 |
Lecture
|
Introduction. Physical examples: heat equation |
| Practice |
Simple equations |
| 20 February 2024 |
Lecture
|
Physical examples: wave equation. Classification of 2nd order linear PDEs. |
| Practice |
First order linear and quasilinear equations |
| 27 February 2024 |
Lecture
|
Break (issued by the Dean) |
| Practice |
|
| 5 March 2024 |
Lecture
|
The class of smooth functions with compact support. |
| Practice |
Classification of 2nd order PDEs |
| 12 March 2024 |
Lecture
|
The applications of mollifiers. Smooth partition of unity. The D(Omega) convergence. |
| Practice |
Classification of 2nd order PDEs, Distributions I. |
| 19 March 2024 |
Lecture
|
Distributions: basic concepts, examples, equivalence. |
| Practice |
Distributions I. (ending) |
| 26 March 2024 |
Lecture
|
Distributions: support, operations, differentiation, Cartesian product. |
| Practice |
First midterm Solutions, 2024 Mock Midterm , 2020 Midterm |
| 2 April 2024 |
Lecture
|
Spring break |
| Practice |
|
| 9 April 2024 |
Lecture
|
Distributions: Convolution. |
| Practice |
Distributions II. |
| 16 April 2024 |
Lecture
|
Fundamental solutions. Cauchy problem of the wave equation (part 1). |
| Practice |
Hyperbolic Cauchy problems (wave equation) |
| 23 April 2024 |
Lecture
|
Cauchy problem of the wave equation (part 2). Cauchy problem of the heat equation. |
| Practice |
Parabolic Cauchy problems (heat equation) |
| 30 April 2024 |
Lecture
|
Boundary value problems. |
| Practice |
Elliptic boundary-value problems |
| 7 May 2024 |
Lecture
|
Eigenvalue problems. |
| Practice |
Eigenvalues, method of Fourier |
| 14 May 2024 |
Lecture
|
Sobolev spaces. |
| Practice |
Second midterm Solutions, 2024 Mock Midterm , 2020 Midterm
|
| 21 May 2024 |
Lecture
|
Weak solution of boundary value problems. |
| Practice |
Midterm retakes |
