PřF:M5160 Differential Eqs.&Cont. Models - Course Information
M5160 Differential Equations and Continuous Models
Faculty of ScienceAutumn 2009
- 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 13:00–14:50 M3,01023, Fri 8:00–9:50 M3,01023
- Timetable of Seminar Groups:
- 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
- Mathematics (programme PřF, M-MA)
- Mathematics (programme PřF, N-MA)
- Course objectives
- The theory of differential equations ranks among fundamental parts of
mathematical analysis. It is utilized by a number of other courses and
in many applications. The basic aim of the course is to familiarize students with the fundamentals of
the theory of ordinary differential equations, with the basic parts of the stability and qualitative theory of differential equations and
mathematical modelling in natural sciences.
After passing the course, the student will be able:
to define and interpret the basic notions used in the fields mentioned above;
to formulate relevant mathematical theorems and statements and to explain methods of their proofs;
to use effective techniques utilized in these subject areas;
to apply acquired pieces of knowledge for the solution of specific problems;
to analyse selected mathematical dynamic deterministic models. - 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
- Teaching methods
- lectures and class exercises
- Assessment methods
- Teaching: lecture 4 hours a week, class exercises 2 hours a week. Examination: written and oral.
- Language of instruction
- Czech
- Follow-Up Courses
- Further comments (probably available only in Czech)
- Study Materials
The course is taught annually.
- Enrolment Statistics (Autumn 2009, recent)
- Permalink: https://is.muni.cz/course/sci/autumn2009/M5160