This page merges details from the Fall 2004 semester with general information about the course because a general course page was created only after the end of the course. The general information will appear in both places since it would be troublesome to remove it from this page without leaving obvious gaps.

*Richard Haberman*; **Mathematical Models:
Mechanical Vibrations, Population Dynamics, and Traffic Flow**; S. I. A. M. 1998 (402
pp.); (ISBN# 0-89871-408-7)

It is intended to cover the whole book. This requires a sustained pace of 4 or 5 sections per lecture. This seems unrealistic, so some details will need to be skipped.

The course aims to introduce students to the art of building and refining mathematical models. This is also the main theme of the exercises in the textbook, which will be used as a framework for class discussion. However, this noble aim is difficult to measure, so exams will concentrate on mathematical techniques. To practice these current techniques, there will be a quiz in each class (unless there is an exam scheduled for that date).

The instructor is Prof. Bumby (follow this link for home page). Contact information and office hours can be found on his home page.

Useful fragments that are relevant to the course will be linked to this section.

- An outline of the use of differential
equations to obtain the shape of a hanging cable. This was prepared
as part of an introductory lecture in 244, and retains signs
of that heritage, but it is a good illustration of the process
of creating differential equations to model physical systems.
There will be some discussion of this in the second or third
class meeting. The method of solving this equation is similar
to the use of the
**energy integral**in section 19, but there was no time to describe this example in lecture. - A discussion of methods for extracting information about
trajectories, including periods of closed orbits, in the form of a project used in Math 252 in Spring 2002. I
hope to produce my own
*answers*to the issues raised in the project, but this will serve as an introduction. This suggests a method for using Green's theorem as an alternative to the energy equation to derive formulas for the period of a closed orbit. Only brief mention was made in the lecture on October 4. - Notes and a Maple worksheet used in Math 244 as a shortcut for solving two dimensional linear homogeneous systems with constant coefficients. While not required for this course, these notes may help avoid tedious computations connected with problems mentioned in the textbook.
- Notes on nonlinear systems and phase planes prepared for Math 244. This collects many definitions and techniques in one place.

- September 01: Practice quiz on an important prerequisite from Differential Equations. Sections 1-4 done, with 5 begun.
- September 08: Quiz on the reduction of equation (4.2) to equation (4.1). Sections 5-8 done, with 9 begun.
- September 13: Finished section 9 including exercise 9.3. Sections 10 and 11 can be considered finished based on a brief discussion.
- September 15: Most time spent
on section 12, especially exercise 12.7. Some discussion of section 13
to identify what is
**critical**about "critical damping". Some discussion of exercise 13.1. Brief consideration of sections 14 and 15. - September 20: The derivation of the
equation of the pendulum in section 14 was the main focus of the
lecture. This illustrated the use of a
**moving coordinate system**, which in this case was given by a vector in the radial direction (along the shaft of the pendulum) and another in the direction tangent to the circular path of motion. Resolution of forces in this coordinate system allows the unknown tension of the pendulum shaft to be eliminated from the equations resulting from Newton's laws of motion to leave an equation describing the behavior of the angular displacement theta as a function of t. Exercise 14.3 is another application of this method leading to the physical*law of conservation of angular momentum*. This leads into exercises 14.4 and 14.7, which should be fairly easy. A more elaborate use of this moving coordinate system can be found in exercises 14.2 and 14.6, but there is not enough time to include those exercises in the course. A brief treatment of sections 15 through 17 followed. - September 22: After a brief treatment of
**stability**from section 18 (including exercise 18.3), the lecture moved on to section 19. The main**method**of that section was the solution of equations like (19.1) expressing the second derivative of position as a function of position alone. After expressing the equation as a separable equation relating position and velocity, one integration leads to equation (19.2a) that is a first order autonomous equation for position as a function of time. This means that the solution of the original equation can be found by integration. This method of solving equation (19.1) is the main mathematical content of the lecture; the principle of**conservation of energy**is a paraphrase of the intermediate step (19.2a) in the solution. - September 27: Concentrated on trajectories in the phase plane for both linear and nonlinear oscillators, with emphasis on finding periods of closed orbits. Finished section 21, with some material from later sections.
- September 29: Finished through section 25. A simpler example of the Taylor series method (p. 89) was described. In that example, the trajectories of the pendulum equation in the phase plane were shown to resemble the ellipses of the linear oscillator for small energy.
- October 4: Concentrated on the pictures
in section 28 to finish part 1. The quiz on October 11 will
concentrate on the concept of
**isocline**from that section. - October 6: Exam on
**Part 1**of the text. Content will stay close to the quizzes, but there will be an attempt to bring the main ideas together. - October 11: A quiz on
**isoclines**will reinforce a key link between Part 1 and Part 2, although the phase plane won't return until section 38. Sections 30 through 34 were covered. The analogy between difference equations and differential equations was strengthened by the appearance of exponential functions as solutions of both types of equations when the change in a variable is proportional to its size. Fairly straightforward problems exploring this connection are numbers 2, 4, 5, 6, 10, 11 in section 32 and numbers 1, 3, 6, 7 in section 34. The claim that the simplest number greater than 1 is 2 (see (1) J.H. Conway: "On numbers and games"; (2) D. E. Knuth: "Surreal numbers : how two ex-students turned on to pure mathematics and found total happiness : a mathematical novelette"; or (3) Harry Gonshor: "An introduction to the theory of surreal numbers" for details) was used to justify the use of**doubling rate**as a method of describing exponential growth (with**half-life**being the corresponding quantity for describing decay). - October 13: Section 35 was emphasized, followed by a brief treatment of section 36. There are many subtle features in section 36 and we don't have the time to do it justice, but some of the high points were noted. The theory behind section 35 assures us that computers will have no trouble working with large matrices of this type, but even the limited exposure to linear algebra in 244 should allow properties of a small example to be found.
- October 18: Sections 37 through 39 on
the
**logistic equation**were covered. The key section was section 38, where exercise 38.4 was done in lecture. The methods of this section illustrate the ability to discover more information, and to discover it more easily, using the equation rather than the explicit solution found in section 39. - October 20: Most time in lecture was spent on section 40. There are a lot of very useful techniques in this section. In the end, it leads (via a perturbation analysis) to second order linear difference equations with constant coefficients, which is the subject of section 41. A brief treatment of this section was enough to reveal things that could be solved easily, so the quiz on Oct. 25 will be taken from this section. Section 42, which got brief mention, applies section 41 to identifying stable equilibrium values for these equations and for the differential-delay equations that led to them.
- October 25: Section 43 provided some of
the
**cultural background**for this subject. Section 44 illustrated the use of**isoclines**for identifying the**equilibrium points**in the phase plane of a two dimensional autonomous system. The direction fields along**arcs of isoclines**between equilibrium points are**strictly parallel**(i.e. parallel as vectors, not just as lines), so**one arrow**on each of these arcs gives a lot of information about the trajectories in the phase plane. The examples in exercise 44.3 were attempted, but the left sides of part (i) were matched with wrong right sides, so an entirely different example was described. Using this approach for the rest of this exercise gives twice as many examples to try while practicing this technique. You can get even more examples by changing the sign of an expression in these systems. Although these changes look minor, the effect on the equations and their solutions can be dramatic. - October 27: Section 45B was used to get
solutions of linear systems. A helpful observation was that the
coefficients of the characteristic polynomial of a 2 by 2 matrix are
(with suitable signs) the trace and determinant of the matrix. This
leads directly to the trace-determinant plane considered in section
46. The axes in this plane are supplemented by the parabola where the
**discriminant**is zero. These curves mark the boundaries between qualitatively different solutions to a system. The solutions described in section 47 were mentioned in connection with some explicit systems. If there are**real eigenvalues**, the eigenvectors determine**straight-line solutions**of the system. Combining this with the easily identified**isoclines**gives a useful base for sketching several solutions to the system. If the eigenvalues are a**complex conjugate pair**, the solutions will be spirals (or closed curves if the eigenvalues are pure imaginary). This fact, combined with the isoclines allow the solutions to be sketched. - November 01: Discussion of the
**predator-prey**model, sections 48 through 50. Emphasis was on the location of stationary points, as shown in figure 50-1, and the linearization about the equilibrium point interior to the first quadrant. The closed-form solution derived at the end of section 50 was mentioned, but details were skipped. - November 03: Finish part 2. The main topic of this class will not appear on the November 8 exam, but will be used for a quiz on November 10.
- November 08: Exam on
**Part 2**of the text. Content will stay close to the quizzes, but there will be an attempt to bring the main ideas together. - November 10: The class will begin with a
quiz on the
**competing species**model from the November 03 class. Then, we aim to finish section 58, with emphasis on exercises 1 and 2 from section 57. - November 15: General discussion on sections 59 and 60 deriving properties of continuous models of traffic flow leading to the partial differential equation 61.1. Section 61 then proposed an additional assumption that the velocity field is a function of density. The validity of this assumption was discussed in section 62 using observations from two highways.
- November 17: Discussed sections 63 through 65 (although not much was said about section 64). Emphasis was on the easily solved initial value problems of section 65.
- November 22: Discussed through section 68. Several important ideas were met here. (1) If density is almost constant, a perturbation analysis linearizes the flow equation (up to a small error). (2) A linear partial differential equation with constant coefficients admits a change of variables that allows the equation to be simplified to one solved in section 65. (3) The solutions of these equations are functions of (x-ct), so they appear to be traveling waves. (4) In heavy traffic, the motion of these waves is opposite to the flow of individual cars.
- November 29: Discussed through section
70, including filling in section 64, which was omitted earlier. The
main point was that, for light traffic, boundary conditions at x=0
**and**t=0 determine a solution for flow on a semi-infinite highway. - December 01: Discussed through the
middle of section 73. This gives an example of how the
**method of characteristics**can be used to solve an equation that models a traffic light turning green. - December 06: The derivation of equation 73.7 was emphasized. Overview of material through section 77.
- December 08: Covered through section 79 (with gaps). Although the example in section 79 seems trivial, it is essential that the theory give an easy treatment of this example. A result different from what is expected, or an overly complicated way of reaching the obvious conclusion would be a flaw in the theory. Such tests are an important part of development of new applications.
- December 13: Plan to cover remainder of text, lightly. Nothing from this lecture will appear on the final exam.

The aim is to have an hour exam at the end of the first two
**Parts** of the textbook. The **first
exam** is now **definitely scheduled** for
**Wednesday, October 6** (originally planned for Monday,
October 4, but postponed by popular request). The date for the second
exam is now **definitely scheduled** for **Monday,
November 8** (originally planned for Wednesday, November 3, but
postponed by popular request). The final exam (scheduled for
**Thursday, December 16, 8 - 11 AM in SEC-220**, and
**not subject to change**) will cover the entire course,
revisiting the first two parts and relating them to the third
part. Extra weight will be given to the third part. All questions
should look familiar, but sloppiness in previous versions should have
been cleaned up, and details may be modified.

The final exam has been graded and course grades submitted. You can get your grade from the FAS Gradebook.

Page started by RT Bumby on August 30, 2004

Last revised by
RT Bumby on September 06, 2006