Partial differential equations

BMETE92AM45

Mathematics BSc

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Lecture:

Tuesdays 10:15 - 11:45, Building T, Room T603

Practice:

Tuesdays 14:15 - 15:45, Building T, Room T603

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.

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

Points


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.

Topics of the exam

Mock exam

Lecture notes

Last modified: 15 March 2023 (These are being corrected continuously, so please always use the most up-to-date version.)


Schedule for the semester:


28 February 2023 Lecture
Introduction. Physical examples: heat equation
Practice Simple equations
7 March 2023 Lecture
Physical examples: wave equation. Classification of 2nd order linear PDEs.
Practice First order linear and quasilinear equations
14 March 2023 Lecture
The class of smooth functions with compact support. The applications of mollifiers.
Practice Classification of 2nd order PDEs
21 March 2023 Lecture
Smooth partition of unity. Distributions: basic concepts.
Practice Break (issued by the dean)
28 March 2023 Lecture
Distributions: examples, equivalence, support, operations.
Practice Distributions I.
4 April 2023 Lecture
Distributions: differentiation, Cartesian product.
Practice Distributions II.
11 April 2023 Lecture
Spring break
Practice
18 April 2023 Lecture
Distributions: Convolution.
Practice

First midterm

Solutions, 2020 Midterm
25 April 2023 Lecture
Fundamental solutions. Cauchy problem of the wave equation (part 1).
Practice Parabolic fundamental solutions
2 May 2023 Lecture
Cauchy problem of the wave equation (part 2). Cauchy problem of the heat equation.
Practice Parabolic Cauchy problems
9 May 2023 Lecture
Maximum principle of the heat equation.
Practice Hyperbolic Cauchy problems
16 May 2023 Lecture
Boundary value problems.
Practice Elliptic boundary-value problems
23 May 2023 Lecture
Sobolev spaces.
Practice Eigenvalues, parabolic problems
30 May 2023 Lecture
Weak solution of boundary value problems.
Practice

Second midterm

Solutions, 2020 Midterm


Budapest University of Technology and Economics
Faculty of Science, Institute of Mathematics
Department of Analysis and Operations Research
1111 Budapest, Egri József Street 1. (Building H) Room H.668 (6th floor)
E-mail: takacsbm (at) math (dot) bme (dot) hu
Office Hours: Mondays and Tuesdays 1 pm - 2 pm.