M8300 Partial differential equations

Faculty of Science
Spring 2025
Extent and Intensity
4/2/2. 10 credit(s). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
In-person direct teaching
Teacher(s)
doc. Phuoc Tai Nguyen, PhD (lecturer)
Tuan Dat Tran (seminar tutor)
Guaranteed by
doc. Phuoc Tai Nguyen, PhD
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Mon 17. 2. to Sat 24. 5. Wed 12:00–13:50 M3,01023, Thu 14:00–15:50 MZAS,02015
  • Timetable of Seminar Groups:
M8300/01: Mon 17. 2. to Sat 24. 5. Tue 18:00–19:50 M6,01011, T. Tran
Prerequisites
! M8110 Partial Diff. Equations && !NOW( M8110 Partial Diff. Equations )
The main pre-requisites are basic concepts in Calculus of one and several variables, Functional Analysis, Measure Theory and Integration, and Theory of ordinary differential equations. It is recommended to start with courses M7300 Global analysis, M7120 Spectral analysis I and M8120 Spectral analysis II before entering this course.
Course Enrolment Limitations
The course is offered to students of any study field.
Course objectives
This is a basic course in Partial Differential Equations, covering both classical and modern methods. Students meet many examples of both linear and nonlinear equations and their use in modelling physical and other phenomena. The first part is devoted to the study of first-order PDEs and classical second-order equations of mathematical physics. In the following parts, modern methods are covered, based on methods of functional analysis. In close connection with modern methods, numerical methods are discussed.
Learning outcomes
At the end of the course, students should be able to:
- know some methods for solving first-order PDEs,
- understand the properties of solutions to simple second-order equations such as the Laplace equation, the heat equation and the wave equation,
- to understand the theory of second-order elliptic and parabolic equations.
- to use different methods to solve second-order equations.
Syllabus
  • Basic notations and motivations.
  • First-order PDEs: transport equations, method of characteristics.
  • Laplace and Poisson equations: harmonic functions, fundamental solutions, energy method.
  • Heat equations: : fundamental solution, initial value problems, nonhomogeneous problems, properties of solutions, energy method.
  • Wave equations: Representation formulas, energy method.
  • Method of separation of variables: Method and applications in solving PDEs.
  • Lebesgue spaces and Sobolev spaces. Definitions and properties of these spaces.
  • Fourier method: definitions and basic properties, applications in solving PDEs.
  • General second-order elliptic equations: existence of weak solutions, regularity results, maximum principle, eigenvalues, and eigenfunctions.
  • General second-order parabolic equations: existence and uniqueness, regularity, maximum principle.
  • Semigroup theory: Definitions and properties, applications.
Literature
    recommended literature
  • BREZIS, Haïm. Functional analysis, Sobolev spaces and partial differential equations. New York: Springer, 2011, xiii, 599. ISBN 9780387709130. info
  • Partial differential equations. Edited by Jürgen Jost. New York: Springer-Verlag, 2002, xi, 325. ISBN 0387954287. info
  • EVANS, Lawrence C. Partial differential equations. Providence, R.I.: American Mathematical Society, 1998, xvii, 662. ISBN 0821807722. info
  • GILBARG, David and Neil S. TRUDINGER. Elliptic partial differential equations of second order. Berlin: Springer-Verlag, 1997, x, 401. ISBN 0387080074. info
Teaching methods
Lectures, exercises, homeworks
Assessment methods
Written and oral exam
Language of instruction
English
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
Listed among pre-requisites of other courses
Teacher's information
The lessons are in English. The target skills of the study include the ability to use the English language passively and actively in their own expertise and also in potential areas of application of mathematics. Assessment in all cases may be in Czech and English, at the student's choice.
The course is also listed under the following terms Spring 2020, Spring 2021, Spring 2022, Spring 2023, Spring 2024.
  • Enrolment Statistics (recent)
  • Permalink: https://is.muni.cz/course/sci/spring2025/M8300