Mathematical Modeling

MATH 5740/MATH 6870 001 MATH MODELING
MWF 09:40 AM-10:30 AM JWB 208
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

Note 1. Population dynamics models
Note 2. Leslie Population dynamics model http://www.math.utah.edu/~cherk/teach/12mathmodel/leslie-model.pdf

  1. 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. 14.

    Research the models of population dynamics (see Note 1). Simulate population dynamics (you may use the codes in note 1) in different models as differential and difference equations. Simulate the diffusive population growth using difference scheme. Assume periodic boundary conditions,
    Due Wednesday, Jan. 16.

  1. 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, writer-presenter)
    Due Friday, Jan. 18

    Project 2 "Population dynamics"

    2.1. Find the data: age-dependent rate of birth and surviving in a specific community (city, state, country) Determine the asymptotic age distribution, based on the Leslie matrix analysis. Run the model, suggest an adjustment to account for limiting habitat capacity. Suggest a modele and simulate the age-dependent migration between two communities.

    2.2. Suggest a predator-prey model for two predator species and one prey species. Find stable points. Are they stable? Simulate a spatial migration of species in a Lotka-Volterra model. Use 1d diffision in a ring.

    2.3. Investigate and simulate epidemics models, model a vaccination effect http://en.wikipedia.org/wiki/Epidemic_model. Simulate epidemics of a desease. Write a model of spatial spread of an epidemics.

    Additional sources
    Equilibria, Stability of equilibria points
    MATHEMATICAL MODELS OF SPECIES INTERACTIONS. IN TIME AND SPACE. JON C. ALLEN
    http://chesterrep.openrepository.com/cdr/bitstream/10034/118016/3/chapter%202.pdf
    https://www.math.duke.edu//education/postcalc/

    A more complex model:
    A Mathematical Model of Three-Species Interactions in an Aquatic Habitat by J. N. Ndam, J. P. Chollom, and T. G. Kassem
    http://www.hindawi.com/isrn/appmath/2012/391547/

    Epidemic models: starting page for the search
    http://en.wikipedia.org/wiki/Epidemic_model

    Chaotic behavior
    Designing Chaotic Models EDWARD N. LORENZ
    http://journals.ametsoc.org/doi/pdf/10.1175/JAS3430.1
    Chaos in a long-term experiment with a plankton community. Beninca E, Huisman J, Heerkloss R, Johnk KD, Branco P, Van Nes EH, Scheffer M, Ellner SP.
    Mathematical models predict that species interactions such as competition and predation can generate chaos. However, experimental demonstrations of chaos in ecology are scarce ...
    http://www.ncbi.nlm.nih.gov/pubmed/18273017

    Rroject 3. Traffic: Lighthill, Whitham, and Richards (LWR) model
      Steven Childres (CIMS) Notes on traffic flow (The main text)
      http://www.math.nyu.edu/faculty/childres/traffic3.pdf

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

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

      Textbooks:
      Richard Haberman. Mathematical models . Mechanical Vibrations, Population dynamics and Traffic Flow. SIAM 1998 Chapter 6.\
      C.Dym. Principles of mathematical modeling. Elsevier 2004. Chapter 6. \

    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(pho) (endorse regulations) to influence traffic jams.

      2. Model and simulate traffic on a road with entries (and exits). Regulate the entry density to avoid jams. Estimate the effect of regulation on the travel time.

      3. Model and simulate traffic at a two-line road when one line is closed. How the speed limit before the traffic jam will affect the jam and the travel time.

    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

    Working groups for the projects: (the scheme is suggested by Kyle Zortman)

    Equilibrium and dynamics of time-variable multistable systems Equilibria of a damaged frame. Modeling of equilibrium states of a frame with a number of damageable rods

    Project: Waves, collisions, and coherent motion of assemblages.
    Three models of nonlinear waves and multiple equilibria