Mathematical modelling

Completed courses Analysis 1 and Algebra 1.

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 systems 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.

Numerical solution of ordinary and partial differential equations:
solution of initial value problems (simulation of a free fall of a parachuter), solution of boundary value problems (bending of a radial symmetric membrane, bubble over a wire loop).

Optimization:
solving problems based on constrained optimization (discrete catenary, symmetric discrete catenary, symmetric discrete catenary having an odd or even number of segments).

Calculus of variations:
brachistochrone problem, catenary, minimal rotational surface.

Statistical simulations:
basics of pseudorandom number generation and simple simulations (using Monte Carlo in prediction models).

Basics of computer-aided geometric design:
parametric polynomial curves, Bernstein-Bezier representation, de Casteljau algorithm, Bezier spline curves, approximation of circular arcs.

1. D. J. Higham, N. J. Higham: Matlab Guide, 2nd ed., Philadelphia : Society for Industrial and Applied Mathematics, cop. 2005.
2. B. Jurčič-Zlobec, A. Berkopec: Matlab z uvodom v numerične metode, Ljubljana : Fakulteta za elektrotehniko, 2005.
3. D. E. Knuth: The art of computer programming. Vol. 2, Seminumerical algorithms, 3rd ed., Reading : Addison-Wesley, 1997, cop. 1998.
4. B. Plestenjak: Razširjen uvod v numerične metode, DMFA-založništvo, Ljubljana, 2015.
5. V. M. Tikhomirov: Stories about maxima and minima, Providence : American Mathematical Society : Mathematical Association of America, cop. 1990.
6. E. Zakrajšek: Matematično modeliranje, Ljubljana : DMFA - založništvo, 2004.

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.

Knowledge and understanding: Basic programming in Matlab. Capability of solving some simple problems of mathematical modelling using Matlab. Understanding 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 program 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,…)

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

Homework and computer-based exam (50 %)
Project defence (50 %)
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Tadej Kanduč:
DEGLI ESPOSTI, Bruno, FALINI, Antonella, KANDUČ, Tadej, SAMPOLI, Maria Lucia, SESTINI, Alessandra. IgA-BEM for 3D Helmholtz problems using conforming and non-conforming multi-patch discretizations and B-spline tailored numerical integration. Computers & mathematics with applications. ISSN 0898-1221, 2023, vol. 147, str. 164-184 [COBISS-SI-ID 178695171]
KANDUČ, Tadej. Isoparametric singularity extraction technique for 3D potential problems in BEM. Computer methods in applied mechanics and engineering. ISSN 0045-7825, 2022, vol. 398, 115271, 25 str. [COBISS-SI-ID 139374339]
GIANNELLI, Carlotta, KANDUČ, Tadej, MARTINELLI, Massimiliano, SANGALLI, Giancarlo, TANI, Mattia. Weighted quadrature for hierarchical B-splines. Computer methods in applied mechanics and engineering. ISSN 0045-7825, 2022, vol. 400, 115465, 24 str. [COBISS-SI-ID 178708995]

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]