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Lecture:
Tuesdays 10:15 - 11:45, Teams
Practice:
Tuesdays 14:15 - 15:45, Teams
Materials: 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, which you should solve at home, and then send your solutions via e-mail (you can take a photo of it or scan it).
There are also some bonus problems at the end of each practice part, which can be solved at home and then sent to me via e-mail.
Grades at the end of the semester: (the points are subjects 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.
The exam contains only some shorter question, and then you should answer the question in a similar way as the midterms. Sample test
Last modified: 11 May 2021 (These are being corrected continuosly, so please always use the most up-to-date version.)
Schedule for the semester:
| 9 February 2021 |
Lecture
|
Introduction. Physical examples: heat conductivity
Sections 1.1, 1.2.1. Video |
| Practice |
Simple equations, Video |
| 16 February 2021 |
Lecture
|
Physical examples: wave equation. Classification of 2nd order linear PDEs, canonical form
Sections 1.2.2-1.2.3, 2.1-2.3.1. Video |
| Practice |
First order linear and quasilinear equations, Video |
| 23 February 2021 |
Lecture
|
The class of smooth functions with compact support (C^{\infty}_0)
Sections 2.3.2., 3.1-3.4. Video |
| Practice |
Classification of 2nd order PDEs, canonical form, Video |
| 2 March 2021 |
Lecture
|
Distributions: basic concepts, examples
Sections 3.5, 4.1, Video |
| Practice |
Distributions I., Video |
| 9 March 2021 |
Lecture
|
Distributions: equivalence, support, operations, differentiation
Sections 4.2-4.4, Video |
| Practice |
Distributions II., Video |
| 16 March 2021 |
Lecture
|
Distributions: Cartesian product, convolution (part 1).
Sections 4.5-4.6.1, Video |
| Practice |
First midterm Exercises, Solutions,
2020 Midterm (Note that the midterm this year will be easier since last year students had 3 weeks to solve them.) |
| 23 March 2021 |
Lecture
|
Distributions: convolution (part 2). Fundamental solutions.
Sections 4.6.2-4.6.3, Chapter 5, Video |
| Practice |
Parabolic fundamental solutions, Video |
| 30 March 2021 |
Lecture
|
Cauchy problem of the wave equation.
Chapter 6, Video |
| Practice |
Parabolic Cauchy problems, Video |
| 6 April 2021 |
Lecture
|
Spring break |
| Practice |
|
| 13 April 2021 |
Lecture
|
Cauchy problem of the heat equation
Chapter 7, Video |
| Practice |
Hyperbolic Cauchy problems, Video |
| 20 April 2021 |
Lecture
|
Boundary value problems
Chapter 8, Video |
| Practice |
Elliptic boundary-value problems, Video |
| 27 April 2021 |
Lecture
|
Green functions. Poisson's formula on a half-space and on a ball
Chapter 9, Video |
| Practice |
Eigenvalues, parabolic problems, Video |
| 4 May 2021 |
Lecture
|
Sobolev spaces
Chapter 10, Sections 10.1-10.3, Video |
| Practice |
Second midterm Exercises, Solutions,
2020 Midterm (Note that the midterm this year will be easier since last year students had 3 weeks to solve them.) |
| 11 May 2021 |
Lecture
|
Weak solution of boundary value problems
Sections 10.4-10.5, Chapter 11, Video |
| Practice |
(No practice this week.) |
