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 plan  

Introduction
Complement 1. 3D fractals
Complement 2. Geocentric and Heliocentric systems

Population dynamics

Topics:
Malthusian equation, nonlinear models of population growth, Discrete delay model, harvesting,
Complement 1.
Complement 2. Population models with age distribution.
Lotka-Volterras model
  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), see Introduction

  2. HW 2. Suggest an algorithm of dividing the class into working groups of three.
  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)

  2. Project 2. Population dynamics
    Simulate the population dynamics of three countries using data available at Internet. Approximate the unavailable data and justify.
    Compute asymptotic behavior of the model. Suggest stabilizer and improve the model. Justify
    Simulate the result of a sudden change in the population distribution ("an earth quack") over next 30 years.
    Build a model of evolution of population as a continuous function of time and age.

  3. Project 3. Traffic modeling
    Review one of two suggested papers on traffic modeling and control. Discuss the possible ways to optimize traffic flow, the needed simulation, sensors, and controllers.
    Models for traffic control, by Tom Bellemans, Bart De Schutter, Bart De Moor. Journal A vol:43 issue:3-4 pages:13-22, date:2002
    The formation and structure of vehicle clusters in the Paine-Whitham traffic law mode, by W.L.Jin and H.M.Zhang. Transportation research, Part B 37 (2003) 207-223

  4. Project 4. Falling dominos
    Construct a model that describes a wave of falling domino train. Assume that all geometric parameters are given. Use Matlab or Maple to simulate the wave of falling pieces.

  5. Project 5. Optimal Cleat hitch.
    Consider a convex cylinder and a rope around it. Assume that the rope is stretched at one end and is fasten to the surface at the other end. The tension force is proportional to the normal component of the force (Coulomb/Amonton friction law).

    Compute the force in the rope and the normal force.
    Design a Cleat hitch which has a constant normal force everywhere under the rope.

  6. Project 6. Euler column
    Find the variable thickness of the highest column made from a fixed amount M of a material with limited resistance to stress. On top of the column, there is a weight m.

  7. Project 7. System with multiple equilibria
    Consider a massless wheel of radius R that can rotate on its axes with a mass m attached to its rim and a rope of the length L that runs around the wheel as in a pulley or in a reel. One end of the rope is attached to the rim of the wheel, and a mass M is attached to the other end.
    If the mass M is released free, will the wheel evolve? Find the equilibria of the system.
    Assume (a) no losses and (b) a viscous resistance to the motion.

  8. Project 8. Dynamics of damage propagation
    Consider a chain of N masses m joined by a system of springs. There are three springs between neighboring masses. Each spring is (linear elastic)-brittle, it breaks after elongation reaches a threshold y1. The first family of springs is in equilibrium at the beginning, the other two families are slack: the springs in them start to work when elongation exceeds thresholds y2 and y3, respectively. First mass is attached to a wall, and the last "large" mass M initially moves with a constant speed v. Model and simulate the spread of damage.

Resume

Some principles of math. modeling