 Introduction to Mathematical Modeling
V63.0251 Spring 2005

 Lectures: Mon/Wed 3:30:4:45pm, Room 813 WWH
 Office hours: Hours 3:303:30 Mon/2:303:30 Wed (Room 713 WWH) or
by appointment (childress@cims.nyu.edu).
 This course will treat various examples of mathematical modeling taken
from
 various scientific and industrial disciplines.
 Both linear and nonlinear problems will be considered. Specific
applications will be selected based upon
 the interests of the class. Homework will be assigned, collected,
and graded, and there will be a final examination.
 The course will be largely selfcontained. The calculus through Calculus
III makes up the prerequisite, and some linear algebra
will be needed. The necessary mathematics, physics, and biology
will be developed as needed.

Textbooks and Software
The lecture notes will provide
the main basis of the course, to be supplemented by
a textbook and handouts and reserve books.
One text has been ordered for the course:
Richard Haberman Mathematical Models. S.I.A.M.,
Philadelphia (1998).
Reserve books: TBA
Although it is not going to be required for the course, students may want
to have MATLAB
on their computers. This is a
good course for starting to use the program. The student
version is available at the campus Computer Store. A
primer for MATLAB is available here.
Tentative syllabus: Population dynamics and mathematical ecology. Introduction
to traffic flow. A selection of models from operations
research, financial mathematics, and biology.
Nonlinear oscillators and models of clocks.
Week 1
Introduction to mathematical
modeling. The modeling process. The mothball problem.
A frictiondriven oscillator. Reading:
320 of text. Problems (These are not to be handed in.) 2.1,5.2,5.7,7.2.
Week 2
Begin population dynamics
and mathematical ecology. Exponential growth, discrete
and
continuous, in a onespecies
population model. Densitydependent
growth. Begin the continuous logistic equation.
Reading: Sections
3034, 37 of text. Problems (to be handed in Monday Jan.
31): 32.2,32.3 (Hint for part b: Try N_m=A+B alpha^m), 33.3,
34.5(parts a,c,d,)
Note: problems 37.2, 37.5 moved to
week 3, alonq with reading of sections 38,39
Week 3
The continuous logistic equation. Phase plane
and solution by quadrature. Stability of equilibria. The
discrete logistic equation. Period doubling
as a route to chaos. The butterfly
effect.
Reading: Sections
38,39. Problems due February 7: Get pdffile.
Week 4
Discrete one species model with age distribution.
US census data and modeling by age groups. Continuous and
discrete logistic models with time delay. Begin study of two
species models.
Reading: Sections
35,40 (pp. 162165),41,43. Problems due February 14: Get
pdffile.
US census data for use with this problem set: Get it
here.
NOTE: The
problem session will meet Thursday's 56pm in room 407
of the Silver Building (formerly Main building). The first session
will be February 10. Frederic will schedule an office hour which
we hope be available to students not able to make the problem session.
Week 5
Two species
models. The LotkaVolterra model of hostparasite and preypredator
interaction. Analysis in
the phase plane. Equilibria and linearization
around equilibria Discrete analogs.
Reading: Sections 43,44, 45,48, 49, beginning
of 50. Problems due February 23: Get pdffile.
Note: Owing to the holiday
Monday, Feb. 21 Homework 5 is due Feb. 23. The TA's office hours
will be 23pm Tuesday and Wednesday, room 807 WWH.
In addition I will try to have office hours
10:3012 Tuesday mornings.
Week 6 (one class)
Two species
models cintinued. Analysis pf 2X2 linear systems with constant
coefficient. Application to stability of equilibria. The LotkaVolterra
model of twospecies competition. Competitive exclusion and stable
coexistence.
Reading: Sections 45,46,54. Optional: 47
(some of this material will be discussed in class). Sketch of the
four cases of species competition in the phase plane (note there
are two pages, but second page is repeated in this file): get pdffile.
Problems due March 2: Get pdffile.
Note: Problem set 6 due Wednesday,
March 2. From now on problem sets will be collected Wednesday
instead of Monday. I will try to get new problem sets
online by Monday nevertheless.
Week 7
Finsh analysis of twospecies competition in
the phase plane, determining the four cases. Case study 1: The
bucketbrigade production line.
ReadingDescription of Case Study 1: pdf file. Handout on the bucket brigade problems:
pdf file. (1995 paper by Bartholdi et al.):
pdf file.
Problems due March 9: Get pdffile.
Week 8
The modeling of vehicle traffic. The continuum
model. The velocity field. Traffic density and flux. Conservation
of vehicles. The velocitydensity relation. Linearization and traffic
waves.
ReadingSections 56 through 61. My notes on Traffic Flow: pdf file.
There will be no problems assigned this week
. However you should carefully read the handout notes which contain
some answered problems to study.
Student paper: homework 6 get pdf file.
Week 9
The modeling of vehicle traffic continued. The
linear and nonlinear traffic wave. Characteristics and their use in
solving firstorder PDEs. Solution of the initialvalue problem for
the nonlinear traffic flow equation. Traffic flow when a red light turns
green. The expansion fan. Motion of a car in the pack.
Reading The material discussed this week appears in section Sections
62 through 72. My updated notes on Traffic Flow: pdf file.
NOTE: An online monograph on traffic flow: go to pdf files of chapters.
Problems due March 30: pdf file.
Student paper: homework 7 get pdf file.
Week 10
The modeling of vehicle traffic continued. Motioon
of cars in a fan. Discontinuous traffic and the shock wave. Calculation
of shock velocity from the global conservation law. Example of shock formation.
The greenredgreen traffic light problem. Modification of shock velocity
by an expansion fan. The effect of a change of road conditions.
Reading Some material discussed this week appears in section
Sections 77 and 82. My updated notes on Traffic Flow: pdf file.
Problems due April 6: pdf file.
Student paper: homework 8 get pdf file.
NOTE: TWO CORRECTION TO PROBLEM SET 9:
IN PROBLEM 2 FIRST LINE, THE DENSITY IN X < 0 SHOULD BE 50 CARS/MILE.
IN PROBLEM 3, LAST LINE, THE FLOW RATE (NOT DENSITY) IS 6000 CARS/MILE.
Week 11
Case Study 2: Turing's model of chemical morphogenesis.
Outline of the ideas. The ODEs of chemical reactions. The isolated
cell and its linear stability. The model of tissue and diffusive communication
between cells. Analysis of the diffusive, pattern forming instability
in the case of a ring of cells. The conditions needed for pattern formation.
Reading My notes on the problem pdf file.
Problems due April 13 : pdf file.
Student paper: homework 9 get pdf file.
Week 12
Mechanical vibrations: Newton's second law. The linear spring
and simple harmonic motion. Phase plane analysis of the oscillation.
Kinetic energy and work. Oscillation of two coupled masses. Nonlinear
oscillators and EV analysis. The simple pendulum and its phase plane.
The effect of friction. Qualitative description of a limit cycle and application
to clocks.
Reading My notes on mechanical vibrations pdf file.
Problems due April 20 : pdf file.
Student paper: homework 10 get pdf file.
Week 13
Mechanical vibrations continued: Newton's second law. The effect
of wall friction on a simple harmonic oscillator. The simple pendulum,
equation and phase plane. How to make a pendulum's period be independent
of amplitude. Charcteristics of a good clock. The limit cycle and the PoincaréBendixson
theorem.
Reading Second part of my notes on mechanical vibrations pdf file.
NOTE: The final examination will be Wednesday,
May 4, 4:005:50 pm, room to be announced.
Review problems for the final will eventually be obtainable here.
I will be adding to them so check every so often: Get the pdf file.
I will be posting here eventually the answers to the review problems. answers pdf file.
NOTE: Although the final exam is closed
book you may bring an 81/2x11 page of notes (both sides may be used).
Student paper: homework 11 get pdf file.
Week 14
Case study 3: Waiting in line: A simple deterministic flow
model of a server. The need for a stochastic model. The Poisson process.
The M/M/1 queuing model, single server and line. Analysis of the ODE model.
The steady starte queue.
Reading My notes on queuing theory pdf file.
NOTE: The final examination will be Wednesday,
May 4, 4:005:50 pm, room to be announced.
Additional review problems: Get the pdf file.
NOTE: THE FINAL EXAM WILL BE IN OUR NORMAL CLASSROOM 813 WWWH.
OFFICE HOURS 10 TO 12 AM TUES. MAY 3 AND WED. MAY 4 .