Course: Nonlinear Differencial Equations

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Course title Nonlinear Differencial Equations
Course code KMA/NDR
Organizational form of instruction Lecture + Exercise
Level of course Master
Year of study not specified
Semester Summer
Number of ECTS credits 4
Language of instruction Czech, English
Status of course Compulsory-optional
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
Course availability The course is available to visiting students
  • Fürst Tomáš, RNDr. Ph.D.
Course content
A. The language of dynamical systems 1. A summary of linear systems 2. De-motivation: Chaos and the Lorenz attractor 3. Case-study: Driven damped oscillations 4. Relation of non-linear to linear systems: The Hartman-Grobman Theorem 5. Chaos forbidden: The Poincare-Bendixon Theorem B. Fixed-point methods 1. Fixed-point theorems 2. Standard applications 3. BVP for nonlinear ODEs 4. nonlinear PDEs: Classical approach 5. nonlinear PDEs: Modern approach 6. modern approach to nonlinear evolution PDEs. C. Monotonicity methods 1. Monotonicity and the Browder-Minty Theorem 2. Application: The method of lower and upper solutions 3. Pseudo-monotonicity and the Brezis Theorem 4. Application to steady state problems 5. Application to evolution problems

Learning activities and teaching methods
Lecture, Dialogic Lecture (Discussion, Dialog, Brainstorming), Demonstration
  • Preparation for the Exam - 45 hours per semester
  • Attendace - 65 hours per semester
  • Semestral Work - 15 hours per semester
Learning outcomes
Understand the language of dynamical systems and solution methods for nonlinear ODEs and PDEs based on fixed-point principles and monotonicity.
Comprehension Understand the mathematical tools of nonlinear partial differential equations.
Classical and modern theory of ODEs and PDEs, Lebesgue's theory, calculus.

Assessment methods and criteria
Oral exam, Written exam

seminar work
Recommended literature
  • C. Robinson. (1999). Dynamical Systems: Stability, Symbolic Dynamics, and Chaos. CRC Press.
  • E. Feireisl. (2004). Dynamics of viscous compressible fluids. Oxford: Oxford University Press.
  • J. Lukeš. (2001). Zápisky z funkcionální analýzy. MatFyzPress.
  • L.C. Evans. (1998). Partial differential equations. AMS.
  • P. Drábek, J. Milota. (2004). Lectures on Nonlinear Analysis. Plzeň.
  • Roubíček T. (2008). Nonlinear Partial Differential Equations with Applications. Birkhauser.
  • V. M. Aleksejev, V. M. Tichomirov, S. V. Fomin. (1991). Matematická teorie optimálních procesů. Academia, Praha.

Study plans that include the course
Faculty Study plan (Version) Branch of study Category Recommended year of study Recommended semester
Faculty of Science Applied Mathematics (2014) Mathematics courses 1 Summer
Faculty of Science General Physics and Mathematical Physics (2014) Physics courses 1 Summer