Haberman Traffic Flow Homework Solution 6.1

Introduction to Mathematical Modeling    V63.0251     Spring 2005

Lectures: Mon/Wed 3:30:4:45pm, Room 813 WWH
Office hours: Hours 3:30-3:30 Mon/2:30-3:30 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 self-contained. 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 friction-driven oscillator. Reading: 3-20 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 one-species population model.
Density-dependent growth. Begin the continuous logistic equation.

Reading: Sections 30-34, 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 pdf-file.

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. 162-165),41,43. Problems due February 14: Get pdf-file.

US census data for use with this problem set: Get it here.

NOTE: The problem session will meet Thursday's  5-6pm 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 Lotka-Volterra model of host-parasite and prey-predator 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

Note: Owing to the holiday Monday, Feb. 21 Homework 5 is due Feb. 23. The TA's office hours
will be 2-3pm Tuesday and Wednesday, room 807 WWH. In addition I will try to have office hours
10:30-12 Tuesday mornings.

Week 6 (one class)

Two species models cintinued.  Analysis pf 2X2 linear systems with constant coefficient. Application to stability of equilibria. The Lotka-Volterra model of two-species 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 pdf-file.

Problems due March 2: Get

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 two-species competition in the phase plane, determining the four cases. Case study 1: The bucket-brigade production line.

Reading-Description 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

Week 8

The modeling of vehicle traffic. The continuum model. The velocity field. Traffic density and flux. Conservation of vehicles. The velocity-density relation. Linearization and traffic waves. 

Reading-Sections 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 first-order PDEs. Solution of the initial-value 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 green-red-green 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.


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 9get 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 E-V 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 10get 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 Poincaré-Bendixson theorem.

Reading- Second part of my notes on mechanical vibrations pdf file.
NOTE: The final examination will be Wednesday, May 4, 4:00-5:50 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 11get 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, 4:00-5:50 pm, room to be announced.
Additional review problems: Get the
pdf file.



Mathematics 142, Winter Quarter, 1999

Mathematical Modeling
Instructor:Christian Ratsch, MS7619c

Phone: 825-4127 (M,W,F) or 317-5852 (Tu,Th)


Meeting Time: M,W,F, 10:00 - 10:50, MS5147

Office Hours: M 11:00 - 12:00, W 9:00 - 10:00, W 1:00 - 2:00

Assistant: Frederic Gibou (fgibou@math.ucla.edu), Office Hours: tba

Discussion: Tu 10:00 - 10:50, MS5147

Textbook: Richard Haberman, Mathematical Models



Prerequisites: Lower division mathematics course sequence

Course Description Mathematical Modeling is part of Applied Mathematics. It is the goal to illuminate problems from real life by abstracting its essence to a mathematical model. It is then hoped that the mathematical solution to the problem yields useful information to the original problem. Thus, the follwing parts are involved:

Analyze the original problem.This step might not at all be mathematical. It requires substantial understanding of the field of the problem. Next, the essence of the problem needs to be translated into a mathematical problem, which is then solved. This might be an exact, analytic solution, or more commonly (for more complicated problems) a numerical solution. Last, the solution that has been obtained needs to be analyzed and interpreted.

This course consists of four parts: The first part is about mechanical vibrations. In this part, we will study simple linear and nonlinear ordinary differential equations, focusing on the motion of a mass-spring system, and a pendelum. In the second part, we will study population dynamics. In a simple two species model, this involves coupled ordinary differential equations. In the third part of the course, simple models of traffic flow will be discussed, which involves using partial differential equations. In the last part of the class, I will discuss some ideas of modeling epitaxial growth, which is an area of current research in the department.

Grading Policy

The grade of this course will be determined on the basis homeworks, one midterm, and a final. The homeworks will count 25% toward your final grade, the midterm 25% as well, and the final examination will count 50%. If you miss any of the exams or homeworks, it will be given a score of 0. In order to accommodate some problems with the collection of the homeworks, the lowest homework score will not be counted (in other words: you can afford to miss one homework assignment).

Midterm and Final Exam There will be one midterm and one final exam. The miderm will be given on February 16, during class; Please note: due to some travel I might have to do in February, I might reschedule the midterm !. This will be announced as soon as possible ! The contents of the midterm will be the material taught until the lecture before the midterm. The final exam will be given during exam period 3, March 21, 3-6, and will cover the material of the whole class (but with more emphasis on parts 2 to 4, which were taught after the midterm).

It is your responsibility to be there on time. Potential time conflicts need to be discussed with the instructor at the beginning of the course. If you do not show up to an exam, the score will be counted as 0. If you do have an admissible, official excuse, for the midterm, the final grade will be scaled, based on all the other grades. However, you must take the final exam.

All midterms and finals will be taken without the use of books or notes. Calculators will be allowed.


Homeworks are required from every student. They are essential if you want to understand the material that is taught in the class. The homework assignments will typically be handed out on Fridays during class, and are due on Fridays, one week later. They will be collected during class as well. No late homeworks will be accepted !! If you miss a Friday class, you may turn in your homework earlier. It is your responsibility to make sure that all the pages have your name on it, and that they are securely fastened together.

Mathematics 142, Winter 2000, Schedule
Day Date ChapterTopics
M Jan 10 1 Class overview; some general remarks
W Jan 12 2,3,4 Newton's laws, Hooke's law
F Jan 14 5,6,7 Solution to second order ODE, harmonic oscillator
W Jan 19 8,9,10,11 Qualitative and quantitative behaviour of harm. osc.
F Jan 21 12,13 Damped oscillations
M Jan 24 14,15,16 A pendelum
W Jan 26 17,18 Linear stability analysis of a pendelum
F Jan 28 19,20 Conservation of energy, energy curves
M Jan 31 21 Phase planes, linear oscillator
W Feb 2 22,23,24 Phase planes, pendelum
F Feb 4 30,31,32,33 Population models; linear difference equations
M Feb 7 34,37 Exponential growth or Midterm
W Feb 9   Midterm or Exponential growth
F Feb. 11 38,39 Logistic equations
M Feb 14 40 Growth with time delays
W Feb 16 41,42 Linear difference equations; effect of time delays
F Feb 18 43 Two species
W Feb 23 48,49 Predator Prey Models
F Feb 25 50 Solutions of Lotka Volterra equations
M Feb 28 56,57,58,59 Traffic flow; velocity fields; traffic density
W Mar 1 60,61 Conservation of cars
F Mar 3 63 Traffic flow
M Mar 6 65,66 Partial differential equations: linearization
W Mar 8 67,68 A linear partial differential equation
F Mar 10 71 Nonuniform traffic - method of characteristics
M Mar 13 72 After a traffic light turns green
W Mar 15   Epitaxy 1
F Mar 17  Epitaxy 2

Please note: This is just a tentative schedule. The exact topic for each lecture might change, as the pace of the class might get adjusted. There is no textbook for the last two lectures.

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