M8300 Partial differential equations

Faculty of Science
Spring 2024
Extent and Intensity
4/2/2. 10 credit(s). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. Phuoc Tai Nguyen, PhD (lecturer)
Sushmita Rawat, M.Sc., Ph.D. (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 19. 2. to Sun 26. 5. Mon 14:00–15:50 M3,01023, Wed 10:00–11:50 M3,01023
  • Timetable of Seminar Groups:
M8300/01: Mon 19. 2. to Sun 26. 5. Thu 8:00–9:50 M3,01023, S. Rawat
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:
- master techniques for solving first order PDEs,
- understand properties of solutions to equations such as Laplace equation, heat equation and wave equation,
- use partial differential equations for modelling various phenomena,
- know numerical methods for solving concrete equations.
Syllabus
  • Notations, definitions and examples of linear and nonlinear partial differential equations.
  • First-order partial differential equations: transport equations, method of characteristics.
  • Laplace and Poisson equations: harmonic functions, fundamental solutions, energy method, discretizations method.
  • Heat equations: : fundamental solution, initial value problems, nonhomogeneous problems, properties of solutions.
  • Wave equations: Representation formulas, energy method.
  • Method of separation of variables.
  • Cauchy-Kovalevskaya theorem.
  • Lebesgue spaces and Sobolev spaces.
  • Fourier method: definitions and basic properties, applications in solving differential partial equations.
  • Semigroup theory: Definitions and properties, applications, Brownion motions.
  • General second-order elliptic equations: existence of weak solutions, regularity results, maximum principle, eigenvalues and eigenfucntions.
  • General second-order parabolic equations: existence and uniqueness, regularity, maximum principle.
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 2025.
  • Enrolment Statistics (recent)
  • Permalink: https://is.muni.cz/course/sci/spring2024/M8300