(in Polish) Ewolucyjne równania różniczkowe cząstkowe. Przegląd podstawowych metod ich badania

General data

Course ID:	1000-1M20ERR
Erasmus code / ISCED:	11.1 Kod klasyfikacyjny przedmiotu składa się z trzech do pięciu cyfr, przy czym trzy pierwsze oznaczają klasyfikację dziedziny wg. Listy kodów dziedzin obowiązującej w programie Socrates/Erasmus, czwarta (dotąd na ogół 0) – ewentualne uszczegółowienie informacji o dyscyplinie, piąta – stopień zaawansowania przedmiotu ustalony na podstawie roku studiów, dla którego przedmiot jest przeznaczony. / (0541) Mathematics The ISCED (International Standard Classification of Education) code has been designed by UNESCO.
Course title:	(unknown)
Name in Polish:	Ewolucyjne równania różniczkowe cząstkowe. Przegląd podstawowych metod ich badania
Organizational unit:	Faculty of Mathematics, Informatics, and Mechanics
Course groups:	(in Polish) Przedmioty obieralne na studiach drugiego stopnia na kierunku bioinformatyka Elective courses for 2nd stage studies in Mathematics
ECTS credit allocation (and other scores):	(not available) Basic information on ECTS credits allocation principles: the annual hourly workload of the student’s work required to achieve the expected learning outcomes for a given stage is 1500-1800h, corresponding to 60 ECTS; the student’s weekly hourly workload is 45 h; 1 ECTS point corresponds to 25-30 hours of student work needed to achieve the assumed learning outcomes; weekly student workload necessary to achieve the assumed learning outcomes allows to obtain 1.5 ECTS; work required to pass the course, which has been assigned 3 ECTS, constitutes 10% of the semester student load. view allocation of credits
Language:	(unknown)
Main fields of studies for MISMaP:	mathematics
Type of course:	elective monographs
Mode:	Remote learning Self-reading
Short description:	We shall present a number of methods for studying evolutionary PDE's, they are all with the scope of a basic course. We will present the stability theory of steady states, traveling waves, self-similar solutions and gradient flows. For the purpose of illustration we will use a number of different examples.
Full description:	The course is devoted to the presentation of a selection of methods for studying evolutionary problems. We shall regard them as infinitely dimensional dynamical system. A particularly nice example is the reaction-diffusion equations. It is well-known from the ODE theory that investigating stability of the steady states and orbits connecting them is important. A particular example of such a solution to reaction-diffusion problems are traveling waves. Another important example are self-similar solutions. They appear in different types of nonlinear problems. They are the key ingredients in the construction of shock waves. This is a type of solution to hyperbolic conservation laws. These problems are completely different from the diffusion-reaction equations. The structure of an equation simplifies if we know that it is a gradient flow or at least it has a Lapunov functional. One of important problems of this sort is the Cahn-Hilliard equation, which is of fourth order. Due to the the Lapunov functional we can show existence for all positive times. The lecture is for students interested in PDEs, no prerequisite is required beyond the basic PDE course. More topics than mentioned above will be covered.
Bibliography:	Christian Kuehn, PDE dynamics. An introduction. Mathematical Modeling and Computation, 23. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2019 Alain Miranville, The Cahn-Hilliard equation. Recent advances and applications. CBMS-NSF Regional Conference Series in Applied Mathematics, 95. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2019 Guido Schneider, Hannes Uecker, Nonlinear PDEs: A Dynamical Systems Approach, AMS, Providence, RI, 2017
Learning outcomes:	A Student: 1. knows the importance of the study of stability of the steady states 2. knows importance of the travelling waves and self-similar solutions for the study of the dynamics 3. knows examples of the gradient flows and knows the notion of the omega-limit sets.
Assessment methods and assessment criteria:	A student is supposed to write an essay on a topic related to the course. A final grade is issued after the conversation on this essay.

This course is not currently offered.

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