Mathematische Modellierung und Übung zu Mathematische Modellierung
University of Vienna, Sommersemester 2016

In the course of this module, the students get to know mathematics in its role as a modeling language for selected applications from physics, natural science, economics, or social sciences.

Schedule
Course: Tue 12.30-14.45 HS13, Oskar-Morgenstern-Platz 1, 2.Stock
Übung: Tue 15.00-15.45 HS13, Oskar-Morgenstern-Platz 1, 2.Stock
First lecture: Tue, March 1, 2016

Outline
Introduction to mathematical modeling: dimensional analysis and scaling, stability analysis, introductory examples; discrete models in finance and population dynamics; algebraic linear systems modeling of electric and mechanical networks; ordinary differential equation models in mechanics and population dynamics; hints on partial differential equation models in physics and natural sciences.

Suggested reading
[EGK] Christof Eck, Harald Garcke, Peter Knabner, Mathematische Modellierung, Springer-Lehrbuch, 2011
[S] Christian Schmeiser, Modellierung (Lecture Notes)
Possible additional material will be distributed during the course.

Evaluation
Course: Final written exam.
Exam dates (registration by email specifying exam date, your last name, first name, and Matrikelnummer):
28.6.2016, h 12:30, HS13 (no registration needed)
08.07.2016, h 13:15, HS13 (registration by July 5)
21.09.2016, h 13:15, HS13 (registration by September 18)
01.02.2017, h 13:15, HS13 (registration by January 29)
Übung: Blackboard presentation of solutions to homework problems; checking of the solutions to homework problems; 2 midterms. Presence, crossing and presentation: 40%, 1st midterm: 30%, 2nd midterm: 30%

Program:
01/03/2016: Introduction; a first example from population dynamics (exponential growth and logistic equation); the SI system; dimension analysis and scaling, the Buckingham Pi theorem ([EGK, Sections 1.2 and 1.3], [S, Section 5])
15/03/2016: Perturbations and asymptotic expansions ([EGK, Section 1.4 nd 1.5], [S, Section 6])
05/04/2016: Asymptotic expansion: perturbed differential problems. Discrete models in finance ([S, Section 2]): capital with compounded interest; loans
12/04/2016: Loans with continuously compounded interest, with constant and non constant interest rate; annuities. Element of calculus of probablities
19/04/2016: Survival function, life expectancy, expected value of the present value of an annuity; discrete models in population dynamics ([S, Section 3]): recursions and solutions to lienar recursions
26/04/2016: Discrete dynamical systems: stationary points and their stability properties
03/05/2016: Periodic points and their stability properties; the vector-valued case
10/05/2016: Examples
24/05/2016: Continuous dynamical systems with examples from population dynamics: exponential growth, logistic equation, prey-predator systems ([EGK, Sections 4.3 and 4.4])
31/05/2016: Systems: stationary points, linearised stabilty, principle of linearised stability ([EGK, Sections 4.5 and 4.6])
07/06/2016: Optimal control of ODEs ([EGK, Section 4.8])
21/06/2016: Pontryagin's maximum principle; examples
28/06/2016: Exam