M8300 Partial differential equations

Faculty of Science
Spring 2020
Extent and Intensity
4/2/2. 10 credit(s). Type of Completion: zk (examination).
Teacher(s)
doc. Phuoc Tai Nguyen, PhD (lecturer)
Guaranteed by
doc. RNDr. Martin Kolář, Ph.D.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Mon 14:00–15:50 MP2,01014a, Thu 10:00–11:50 MP2,01014a
  • Timetable of Seminar Groups:
M8300/01: Thu 15:00–16:50 MS1,01016, P. Nguyen
Prerequisites
! M8110 Partial Diff. Equations && !NOW( M8110 Partial Diff. Equations )
In general, it would be an advantage if students know some basic concepts in Calculus of one and several variables, Functional Analysis and Measure Theory.
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 equations of mathematical physics. In the following part, 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 the connection between solutions of PDEs and properties of corresponding stochastic processes,
- know numerical methods for solving concrete equations.
Syllabus
  • Partial differential equations of first order
  • Method of characteristics,
  • Examples of linear and nonlinear equations,
  • Fourier method,
  • Comparing PDE and ODE,
  • Analytic solutions,
  • Cauchy-Kovalevski theorem
  • Equations of mathematical physics
  • Laplace and Poisson equation
  • Heat equation, wave equation
  • harmonic functions, maximum principle
  • Semigroups and Brownian motion,
  • Discretization and standard numerical methods
  • Sobolev spaces,
  • variational formulation, generalized formulation of počátečních-okrajových úloh,
  • traces, weak solutions,
  • regularity, variational methods, finite elements method
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)
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 2021, Spring 2022, Spring 2023, Spring 2024, Spring 2025.
  • Enrolment Statistics (Spring 2020, recent)
  • Permalink: https://is.muni.cz/course/sci/spring2020/M8300