**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