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PřF:M8101 An introduction to PDE - Course Information

## M8101 An introduction to partial differential equations

**Faculty of Science**

spring 2018

**Extent and Intensity**- 2/2. 3 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
**Teacher(s)**- dr. Phuoc Tai Nguyen, Ph.D. (lecturer)
**Supervisor**- dr. Phuoc Tai Nguyen, Ph.D.

Department of Mathematics and Statistics - Departments - Faculty of Science

Supplier department: Department of Mathematics and Statistics - Departments - Faculty of Science **Timetable**- Tue 16:00–17:50 MS1,01016
- Timetable of Seminar Groups:

**Prerequisites**- There is no strict pre-requisites. In general, it would be an advantage if students know some basic concepts in Functional Analysis and Measure Theory. However, all necessary backgrounds will be explained quickly in the course.
**Course Enrolment Limitations**- The course is also offered to the students of the fields other than those the course is directly associated with.
**Fields of study the course is directly associated with**- Mathematical Analysis (programme PřF, N-MA)
- Mathematical Modelling and Numeric Methods (programme PřF, N-MA)

**Course objectives**- The goal of this course is to present an overview of partial differential equations which arise in many different contexts such as potential theory and stochastic processes. In the first part, we will study second order elliptic equations, in particularly Laplace equation and Poisson equation. We will focus on important topics such as harmonic functions, Harnack inequality, Liouville theorem, classical solution, weak solutions and regularity. The second part provides basic background and some methods to solve heat equations. Some applications will be also discussed.
**Learning outcomes**- After completing the course, a student will be able to master standard concepts in the theory of partial differential equations and know some methods to solve elliptic and parabolic equations.
**Syllabus**- 1. Laplace equation, Poisson equation, fundamental solutions. 2. Mean value formulas, properties of harmonic functions. 3. Green kernel and Poisson kernel. 4. Energy method. 5. Sobolev spaces and weak solutions. 6. Regularity and maximum principle. 7. Heat equations. References: - Brezis, Haim Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. xiv+599 pp. ISBN: 978-0-387-70913-0. - Evans, Lawrence C., Partial differential equations. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998. xviii+662 pp. ISBN: 0-8218-0772-2. - Gilbarg, David; Trudinger, Neil S., Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. xiv+517 pp. ISBN: 3-540-41160-7.

**Teaching methods**- Lectures, tutorial/exercise discussion and homework
**Assessment methods**- There will be two mandatory homework assignments, each of them contributes 25% of the total grade, and one final written examination which contributes 50% of the total grade.
**Language of instruction**- English
**Further comments (probably available only in Czech)**- Study Materials

The course can also be completed outside the examination period.

The course is taught annually.

- Enrolment Statistics (spring 2018, recent)
- Permalink: https://is.muni.cz/course/sci/spring2018/M8101