PřF:M8300 Part. diff. equations - Course Information

M8300 Partial differential equations

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

Prerequisites

!M8110 Partial Differential Equations && !NOW(M8110 Partial Differential 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.

Abstract

The course aims to explore fundamental aspects of the theory of partial differential equations (PDEs). Various types of linear and nonlinear equations arising in physics and biology are addressed, including first-order equations and second-order elliptic and parabolic equations. Several classical and modern methods based on functional analysis and operator theory, with applications to PDE theory, are discussed. Concrete examples are also provided to illustrate the applicability of these methods.

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.

Key topics

- Basic notations and motivations.
- First-order PDEs: method of characteristics.
- Laplace and Poisson equations: harmonic functions, mean-value formulas, fundamental solutions, Green function, Poisson kernel, energy method.
- Heat equations: fundamental solution, initial value problems, Duhamel principle, maximum principle, energy method.
- Wave equations: formulas for solutions, domains of dependence, and energy method.
- Lebesgue spaces: definitions, Holder inequality, completeness, duality, convolutions.
- Sobolev spaces: weak derivative, approximation by smooth functions, continuous embeddings, compact embedding.
- Second-order elliptic equations: weak solutions, Lax-Milgram theorem, local and global regularity results, maximum principle, eigenvalues, and eigenfunctions.
- Second-order parabolic equations: weak solutions, Galerkin method, regularity, maximum principle.
- Spectral-based Method: method and applications in solving PDEs.
- Fourier method: definitions and basic properties, applications in solving PDEs.
- Semigroup theory: definitions and properties, applications.
- Nonlinear equations: introduction of some nonlinear equations, methods for existence, properties of solutions.

Study resources and literature

recommended literature
• Gilbarg, David and Trudinger, Neil S. Elliptic partial differential equations of second order. Reprint of the 1998 edition Classics Math. Springer-Verlag, Berlin, 2001. xiv+517 pp.
• 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

not specified
• Evans, Lawrence C. Partial differential equations. Second edition Grad. Stud. Math., 19 American Mathematical Society, Providence, RI, 2010. xxii+749 pp.

Approaches, practices, and methods used in teaching

Lectures, exercises, homeworks

Method of verifying learning outcomes and course completion requirements

- Assessment and grading: The final grade is based on the midterm (written) exam (30%) and the final (written) exam (70%).
- Grading scale: A (90–100%), B (80–89%), C (70–79%), D (60–69%), E (50–59%), and F (below 50%).
- Eligibility for the final exam requires the submission of at least 50% of the assigned homework.
- Bonus points are awarded for active participation in lectures and exercise classes.

Language of instruction

English

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.

Further comments (probably available only in Czech)

The course is taught annually.
The course is taught every week.

Listed among pre-requisites of other courses

• M8110 Partial Differential Equations
  !(M8300 || NOW(M8300))  

• Enrolment Statistics (Spring 2027, recent)
  
• Permalink: https://is.muni.cz/course/sci/spring2027/M8300