# Mathematical modelling

2021/2022
Programme:
Mathematics Education
Year:
4 year
Semester:
second
Kind:
optional
Group:
B
ECTS:
5
Language:
slovenian
Course director:
Lecturer (contact person):
Hours per week – 2. semester:
Lectures
2
Seminar
0
Tutorial
2
Lab
0
Prerequisites

Completed courses Analysis 1 and Algebra 1.

Content (Syllabus outline)

Problem solving using Matlab package: introduction into Matlab package, manipulation of matrices and arrays, graphics, writing scripts and functions, overview of basic Matlab toolboxes (numerical solution of sytems of linear and nonlinear equations, optimization, numerical integration and numerical solution of ordinary differential equations, sparse matrices), Matlab as a tool for solving some simple problems.
Optimization: solving problems based on constrained optimization (discrete catenary, symmetric discrete catenary, symmetric discrete catenary having an odd or even number of segments, truss oscillation).
Calculus of variations: brachistochrone problem, catenary, minimal rotational surface.
Statistics: χ2 test (chi square test), statistical simulations, simulation of games.

E. Zakrajšek: Matematično modeliranje, DMFA-založništvo, Ljubljana, 2004.
D. J. Higham, N. J. Higham: Matlab Guide, 2nd edition, SIAM, Philadelphia, 2005.
B. Jurčič Zlobec, A. Berkopec: Matlab z uvodom v numerične metode, Založba FE in FRI, Ljubljana, 2005.
V. M. Tikhomirov: Stories About Maxima and Minima, AMS, Providence, 1991.
D. E. Knuth: The Art of Computer Programming II : Seminumerical Algorithms, Addison-Wesley, Reading, 1981.

Objectives and competences

A student is faced with bacis concepts of problem solving, particularly those arising from mathematical modelling. She or he is able to use matlab as a tool and learns how to evaluate obtained results. Some deeper skills are obtained in solving problems based on finding extrema, calculus or variations and statistical simulations.

Intended learning outcomes

Knowledge and understanding: Basic programming in Matlab. Capability of solving some simple problems of mathematical modelling using Matlab. Understandig of theoretical fundamentals to solve problems involving scalar field extrema, capability of solving problems in calculus of variations and skills in implementation of statistical simulations.
Application: Using Matlab package as a tool for solving some simple problems arising from mathematical models.
Reflection: Understanding theory through practical experiments (computer programme coding).
Transferable skills: Capability of using computer software, particularly Matlab package. Understanding of basic approaches for solving mathematical problems and evaluation of results. The subject upgrades the knowledge obtained from several other subjects of mathematical studies (analysis, algebra, programming,…)

Learning and teaching methods

Lectures, exercises, homework, laboratory work, consultations, individual projects

Assessment

2 homeworks and a project instead of written exam, written exam
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

George Mejak:
MEJAK, George. Eshebly tensors for a finite spherical domain with an axisymmetric inclusion. European journal of mechanics. A, Solids, ISSN 0997-7538. [Print ed.], 2011, vol. 30, iss. 4, str. 477-490. [COBISS-SI-ID 16025177]
MEJAK, George. Two scale finite element method. V: 21st International congress of theoretical and applied mechanics, August 15-21, 2004, Warsaw, Poland. ICTAM04 : abstracts and CD-ROM proceedings. Warszawa: IPPT PAN, 2004, str. 209. [COBISS-SI-ID 13216857]
MEJAK, George. Finite element solution of a model free surface problem by the optimal shape design approach. International journal for numerical methods in engineering, ISSN 0029-5981. [Print ed.], 1997, vol. 40, str. 1525-1550. [COBISS-SI-ID 9983833]
Emil Žagar:
JAKLIČ, Gašper, KOZAK, Jernej, KRAJNC, Marjetka, VITRIH, Vito, ŽAGAR, Emil. An approach to geometric interpolation by Pythagorean-hodograph curves. Advances in computational mathematics, ISSN 1019-7168, 2012, vol. 37, no. 1, str. 123-150. [COBISS-SI-ID 16051289]
JAKLIČ, Gašper, ŽAGAR, Emil. Curvature variation minimizing cubic Hermite interpolants. Applied mathematics and computation, ISSN 0096-3003. [Print ed.], 2011, vol. 218, iss. 7, str. 3918-3924. [COBISS-SI-ID 16049241]
JAKLIČ, Gašper, KOZAK, Jernej, KRAJNC, Marjetka, VITRIH, Vito, ŽAGAR, Emil. Hermite geometric interpolation by rational Bézier spatial curves. SIAM journal on numerical analysis, ISSN 0036-1429, 2012, vol. 50, no. 5, str. 2695-2715. [COBISS-SI-ID 16449369]