Partial differential equations

BMETE92AM45

Mathematics BSc, MSc

Lecture:

Tuesdays 10:15 - 11:55, H405A (with a 10-minute break)

Practice:

Tuesdays 14:05 - 15:35, H406

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 in these books.

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 definition/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.

Mock exam

Lecture notes

Last modified: 25 March 2025 (These are being corrected continuously, so please always use the most up-to-date version.)


Schedule for the semester:


11 February 2025 Lecture
Introduction.
Practice Simple equations
18 February 2025 Lecture
Physical examples: heat equation, wave equation. Classification of 2nd order linear PDEs (part 1).
Practice First order linear equations
25 February 2025 Lecture
Classification of 2nd order linear PDEs (part 2). The class of smooth functions with compact support.
Practice First order quasilinear equations, Classification of 2nd order PDEs
4 March
2025
Lecture
The applications of mollifiers. Smooth partition of unity. The D(Omega) convergence.
Practice Classification of 2nd order PDEs, Distributions I.
11 March 2025 Lecture
Distributions: basic concepts, examples, equivalence.
Practice Distributions I. (ending)
18 March 2025 Lecture
Distributions: support, operations, differentiation, Cartesian product.
Practice

First midterm

2025 Midterm, 2024 Midterm, 2024 Mock Midterm , 2020 Midterm
25 March 2025 Lecture
Distributions: Convolution.
Practice Distributions II.
1 April
2025
Lecture
Fundamental solutions. Cauchy problem of the wave equation (part 1).
Practice Hyperbolic Cauchy problems (wave equation)
8 April
2025
Lecture
Cauchy problem of the wave equation (part 2). Cauchy problem of the heat equation.
Practice Parabolic Cauchy problems (heat equation)
15 April
2025
Lecture
Boundary value problems.
Practice Elliptic boundary-value problems
22 April
2025
Lecture
Spring break
Practice
29 April
2025
Lecture
Eigenvalue problems.
Practice Eigenvalues, method of Fourier
6 May
2025
Lecture
Sobolev spaces.
Practice

Second midterm

2024 Midterm, 2024 Mock Midterm , 2020 Midterm
13 May
2025
Lecture
Weak solution of boundary value problems.
Practice

Midterm retakes

20 May
2025
Lecture
???
Practice

Midterm re-retakes (?)


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: Tuesdays 1 pm - 2 pm, Wednesdays 1-2 pm.