M5160 Differential Equations and Continuous Models

Faculty of Science
Autumn 2007
Extent and Intensity
4/2/0. 6 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).
Teacher(s)
doc. RNDr. Josef Kalas, CSc. (lecturer)
Guaranteed by
doc. RNDr. Josef Kalas, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Wed 8:00–9:50 UM, Fri 8:00–9:50 UP2
  • Timetable of Seminar Groups:
M5160/01: Fri 10:00–11:50 UP2, J. Kalas
Prerequisites
M3100 Mathematical Analysis III && M2110 Linear Algebra II
Mathematical analysis: Differential calculus of functions of one and several variables, integral calculus, sequences and series of numbers and functions, metric spaces, complex function of a real variable. Linear algebra: Systems of linear equations, determinants, linear spaces, linear transformation and matrices, canonical form of a matrix.
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 / plans the course is directly associated with
Course objectives
The theory of differential equations ranks among basic parts of mathematical analysis. It is utilized by a number of other courses and in many applications. The course is concentrated to the foundations of the theory of ordinary differential equations, introduction to the stability and qualitative theory of differential equations and mathematical modelling in natural sciences.
Syllabus
  • 1. Fundamental notions - ordinary differential equations and their systems, order of an equation, initial value problem, solutions of differential equations and initial value problems. 2. Systems of linear differential equations - existence and uniqueness of solutions, structure of the family of solutions, variation-of-constants method, linear systems with constant coefficients, connection of linear systems with higher-order linear differential equations. 3. Local and global properties of solutions - local existence and uniqueness of solutions of nonlinear initial value problems, global existence and uniqueness, dependence of solutions on initial values and parameters. 4. Introduction to the stability theory - Lyapunov concept of stability, uniform, asymptotic and exponential stability, stability of linear and perturbed linear systems, Hurwitz criterion, direct method of Lyapunov. 5. Autonomous equations - basic notions and properties, elementary types of singular points of two-dimensional systems, classification of singular points of linear and perturbed linear systems, the structure of a limit set in R2, Poincaré-Bendixson theory, Dulac criterion, characteristic directions. 6. Notion of a mathematical model, classification of models, basic steps of the process of mathematical modelling, formulating a mathematical model, dimensional and mathematical analysis of mathematical models. Selected mathematical models in natural sciences.
Literature
  • KALAS, Josef and Miloš RÁB. Obyčejné diferenciální rovnice (Ordinary differential equations). 2nd ed. Brno: Masarykova univerzita, 2001, 207 pp. ISBN 80-210-2589-1. info
  • KALAS, Josef and Zdeněk POSPÍŠIL. Spojité modely v biologii (Continuous models in biology). 1st ed. Brno: Masarykova univerzita v Brně, 2001, 256 pp. ISBN 80-210-2626-X. info
  • KURZWEIL, Jaroslav. Obyčejné diferenciální rovnice : úvod do teorie obyčejných diferenciálních rovnic v reálném oboru. 1. vyd. Praha: SNTL - Nakladatelství technické literatury, 1978, 418 s. info
  • GREGUŠ, Michal, Marko ŠVEC and Valter ŠEDA. Obyčajné diferenciálne rovnice. 1. vyd. Bratislava: Alfa, vydavateľstvo technickej a ekonomickej literatúry, 1985, 374 s. info
  • Hartman, Philip. Ordinary differential equations. Wiley, New York-London-Sydney, 1964.
  • Coppel, W. A. Stability and asymptotic behaviour of differential equations. D. C. Heath and company, Boston, 1965.
  • RÁB, Miloš. Metody řešení diferenciálních rovnic. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1989, 68 s. info
  • RÁB, Miloš. Metody řešení diferenciálních rovnic. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1989, 61 s. info
  • VERHULST, Ferdinand. Nonlinear differential equations and dynamical systems. Berlin: Springer Verlag, 1990, 277 s. ISBN 3-540-50628-4. info
  • Mesterton-Gibbons, M. A. A concrete approach to mathematical modelling. Addison-Wesley Publishing Company, 1989.
  • Edelstein-Keshet, L. Mathematical models in biology. The Ramdom House/Birkhäuser Mathematics Series, New York, 1987.
  • Ponomarev, K. K. Sostavlenie differencial'nych uravnenij. Vyšejšaja škola, Minsk, 1973.
  • BRAUN, Martin. Differential equations and their applications : an introduction to applied mathematics. 2nd ed. New York: Springer-Verlag, 1978, xiii, 518. ISBN 0-387-90266-X. info
Assessment methods (in Czech)
Výuka: přednáška, klasické cvičení. Zkouška: písemná a ústní.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
The course is taught annually.
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 1999, Autumn 2010 - only for the accreditation, Autumn 2001, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.
  • Enrolment Statistics (Autumn 2007, recent)
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