Mathematical Modeling

MATH 5740/MATH 6870 001 MATH MODELING
MWF 09:40 AM-10:30 AM AEB 310
Three credit hours

Instructor:
Professor Andrej Cherkaev

Department of Mathematics
Office: JWB 225
Email: cherk@math.utah.edu
Tel : +1 801 - 581 6822

Preliminary syllabus

Introduction
Complement. Geocentric and Heliocentric systems

HW 1. Write a short essay (one or two pages) about development of the model of Universe (geocentral- heliocental - Newton's gravity- Einstein's general relativity. Discuss the motivation of the model development, types of the model (data-fitting, equation-solving), Discuss thought model and experimental data in the modeling. See Introduction for hints and references.
Due Monday, Jan. 12.

Project 1. Class division
Suggest an algorithm of division of the class into working groups for a number of projects. Each student should work with different people on different projects, and should assume different roles (researcher, programmer-webmaster, writer-presenter).
Due Wednesday, Jan. 15

Note 1. Population dynamics models
Note 2. Leslie Population dynamics model http://www.math.utah.edu/~cherk/teach/12mathmodel/leslie-model.pdf
HW2
Project: Dynamics of Species.
Working groups for the projects (Thanks to Stephen Durtschi)
HW3 (due Wednesday, February 12)

Traffic flow: Main model: Lighthill, Whitham and Richards (LWR) model (Google it!)

The main text: Stephen Childress (NYU) Notes on traffic flow:
http://www.math.nyu.edu/faculty/childres/traffic3.pdf

More simple presentation: Kurt Bryan (Rose-Hulman).
Traffic flow I http://www.rose-hulman.edu/~bryan/lottamath/traffic1.pdf
Traffic flow II http://www.rose-hulman.edu/~bryan/lottamath/traffic2.pdf

Engineering textbook exposition L.H. Immers and S. Logghe. (KATHOLIEKE UNIVERSITEIT LEUVEN) Traffic Flow Theory 2002
https://www.mech.kuleuven.be/cib/verkeer/dwn/H111part3.pdf

Discussion of traffic shock waves: Christopher Lustri (Oxford, UK) Continuum Modelling of Traffic Flow 2010 http://people.maths.ox.ac.uk/lustri/Traffic.pdf

Projects

1. Consider a circular road with initial distribution of the cars. Model and simulate traffic starting from an inhomogeneous density distribution.
Try to change the speed function u(rho) (endorse regulations) to influence traffic jams: Assume that u depends on rho and x, u=u(rho, x). In a point of the road, the speed must be reduced to the half of the maximal speed. (Group 1)

2. Add the dissipation to the traffic model. Investigate the dependence of close-to-shock wave motion on the dissipation parameter D. (Group 2)

3. Model and simulate traffic on a road with one entry and one exit. Regulate the entry density to avoid jams. Estimate the effect of regulation on the travel time.(Group 3)

4. Model and simulate traffic at a two-line road when one line is closed. Change the speed limit before the point of closure. How this affects the traffic jam and the travel time. (Groups 4, 5)

5. Impement Rule 184 model for 12 cars in a circular road of 80 cells. (Group 6)

Additional sources (numerics and modeling)

1. Modeling and Numerical Approximation of Traffic Flow Problems Prof. Dr. Ansgar Jungel A Universitaat Mainz Lecture Notes (preliminary version) http://asc.tuwien.ac.at/~juengel/scripts/trafficflow.pdf numeric, burger's equation
2. http://www.mat.univie.ac.at/~obertsch/literatur/burgers_equation.pdf Theory and simulation.
3. http://digitalscholarship.unlv.edu/cgi/viewcontent.cgi?article=1087&context=ece_fac_articles derivation, numerical advices
4. Google "Rule 184"

Home work: Evolutionary games Due Friday March 7.
Original formulation of the evolutionary problem

Project: Chain reaction List of projects

Projects: Design List of projects