1. SylabUZ
2. Wydział Matematyki, Informatyki i Ekonometrii
3. semestr zimowy 2022/2023
4. Mathematics - drugiego stopnia z tyt. magistra
5. Mathematical Modelling 1

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Mathematical Modelling 1 - opis przedmiotu

Cel przedmiotu

The aim of the course is to familiarize students (at a basic level) with the nature, scope and stages of mathematical modeling. The lecture will present a broad overview of mathematical models and methods used in technical problems, biology and medicine. The goal of the laboratory classes is to simulate the presented models using a given mathematical package (Matlab, Octave or Scilab). After completing this course, the student should be prepared to create simple mathematical models using computers and their mathematical knowledge.

Wymagania wstępne

Zapisz zmiany

Students should pass: Introduction to Numerical Methods, Differential Equations.

Zakres tematyczny

Lecture

Introduction to mathematical modeling

1. Mathematical modeling as a description of the world. (1 hour)
2. Aim, scope and stages of mathematical modeling. Validation of the model. Application of computer simulations. (1 hr.)
3. Examples of mathematical models. (2 hrs.)
4. Data analysis and visualization - basic methods. (2 hrs.)

Discrete and continuous mathematical models

1. Modeling with linear differential and differential equations. Construction of models. Methods of determining solutions. Examples of mathematical models. (4 hrs.)
2. Modeling with nonlinear differential and differential equations. Construction of models. Methods of determining solutions. Examples of mathematical models. (4 hrs.)
3. Modeling with linear and nonlinear dynamic systems. Construction of models. Methods of determining solutions. Examples of mathematical models. (4 hrs.)
4. Numerical schemes. (4 hrs.)

Mathematical modeling in biology and medicine

1. Modeling of a single population. (2 hrs.)
2. Single population models with age. (2 hrs.)
3. Models of interactions between two populations. (2 hrs.)
4. Epidemiological models. (2 hrs.)

Laboratory

Introduction to mathematical modeling

1. Introduction to a mathematical package (Matlab, Octave or Scilab). (2 hrs.)
2. Data visualization. (4 hrs.)
3. Methods of interpolation and approximation of data. (4 hrs.)

Discrete and continuous mathematical models

1. Discrete mathematical models - construction of discrete models, determination of solutions, interpretation and visualization of results, use of a mathematical package in the modeling process. (5 hrs.)
2. Colloquium (1 hr)
3. Continuous mathematical models - construction of continuous models, determination of solutions, interpretation and visualization of results, use of a mathematical package in the modeling process. (5 hrs.)

Mathematical modeling in biology and medicine

1. Single population modeling - construction of models, determination of solutions, interpretation and visualization of results, use of a mathematical package in the modeling process. (4 hrs.)
2. Models of interactions between two populations - construction of models, determination of solutions, interpretation and visualization of results, use of a mathematical package in the modeling process. (4 hrs.)
3. Colloquium (1 hour)

Metody kształcenia

Lectures using multimedia devices; Laboratory exercises in which students solve computational problems analytically and using a given mathematical package (Matlab, Octave or Scilab).

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

Laboratory grade is based on tests (80%) and class activity (20%). The course grade consists of the laboratory grade (50%) and the exam grade (50%).  The condition for passing the course is positive grades from the laboratory and the exam.

Literatura podstawowa

1. B. Burnes, G. R. Fulford, Mathematical modeling with case studies, Taylor and Francis, 2002.
2. J. D. Murray, Mathematical Biology.  An Introduction, Springer, 2002.
3. G. R. Fulford, P. Forrester, A. Jones, Modelling with Differential and Difference Equations, Cambridge University Press, 1997.
4. D. Kincaid, W. Cheney, Numerical Analysis. Mathematics of Scientific Computing, The University of Texas at Austin, 2002.

Literatura uzupełniająca

1. J. D. Logan,  Applied mathematics, a contemporary approach, John Wiley and Sons, New York, 2001.
2. A. Björck, G. Dahlquist, Numerical Methods in Scientific Computing, SIAM, 2008.
3. G. R. Fulford, P. Broadbridge, Industrial Mathematics, Cambridge University Press, 2002.

Uwagi

Zmodyfikowane przez dr Maciej Niedziela (ostatnia modyfikacja: 07-02-2024 22:16)