Partial differential equations

BMETE92AM45

Mathematics BSc, MSc

Lecture:

Tuesdays 14:15 - 15:55, H507 (with a 10-minute break)

Practice:

Tuesdays 10:15 - 11:55, E501

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: 20 May 2025 (These are being corrected continuously, so please always use the most up-to-date version.)


Schedule for the semester:


17 February 2026 Practice
Simple equations
Lecture Introduction. Physical examples: heat equation, wave equation.
24 February 2026 Practice
First order linear equations
Lecture Classification of 2nd order linear PDEs. The class of smooth functions with compact support.
3 March
2026
Practice
First order quasilinear equations, Classification of 2nd order PDEs
Lecture The applications of mollifiers. Smooth partition of unity. Definition of a distribution.
10 March
2026
Practice
Classification of 2nd order PDEs, Distributions I.
Lecture Distributions: equivalence, support, operations, derivative, direct product.
17 March 2026 Practice
Distributions I. (ending)
Lecture Distributions: convolution. Fundamental solutions.
24 March 2026 Practice

First midterm

2025 Midterm, 2024 Midterm, 2024 Mock Midterm , 2020 Midterm
Lecture Cauchy problem of the wave equation.
31 March 2026 Practice
Distributions II.
Lecture Cauchy problem of the heat equation.
7 April
2026
Practice
Spring break
Lecture
14 April
2026
Practice
Hyperbolic Cauchy problems (wave equation)
Lecture Maximum principle for the heat equation.
21 April
2026
Practice
Parabolic Cauchy problems (heat equation)
Lecture Boundary value problems
28 April
2026
Practice
Elliptic boundary-value problems
Lecture Eigenvalue problems
5 May
2026
Practice
Eigenvalues, method of Fourier
Lecture Green functions.
12 May
2026
Practice

Second midterm

2025 Midterm, 2024 Midterm, 2024 Mock Midterm , 2020 Midterm
Lecture Sobolev spaces.
19 May
2026
Practice

Midterm retakes

Lecture Weak solution of boundary value problems.
26 May
2026
Practice
To be announced...
Lecture To be announced...


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.