Math 321 - Introduction to Applied Mathematics

General information about the course can be found in the general course page and won't all be repeated here.

Textbook

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

Plan for course

It is intended to cover the whole book.  The 2004 version of the course contains an outline of the approximate syllabus to be followed this year. It will be modified based on current experience.

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).

Two exams are planned, one at the end of the first two parts of the text. The final exam will revisit parts 1 and 2 as well as including problems based on part 3. Tentative dates for the midterm exams are Wednesday, October 04 and Monday, November 06. The first exam was held as originally scheduled, but the second exam will be postponed until Wednesday, November 08.

Supplements

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

• None at this time.

Lecture details

During the semester, a log was kept on the Sakai site. Here is a copy of that log.

1. (Sep. 06) Sections 1 - 4. Part of Section 5 was covered in the discussion of quiz 0. A discussion of some features of the answers to the quiz was prepared.
2. (Sep. 11) Sections 5 - 8, including exercises 5.1, 5.2, 7.2.
3. (Sep. 13) Sections 9 - 11, including exercise 9.3. The alternate forms of damping mentioned in section 10 were ignored since there is only time to discuss linear damping in detail. Section 11 is devoted to showing that there are qualitative differences in the solution as the damping constant changes. The different types of solution will be examined in the next two sections.
4. (Sep. 18) Sections 12 - 13, including exercises 12.1, 12.7, and 13.1. The discussion of 12.7 was skimpy, but more details about resonance may be found in textbooks on Differential Equations or Mechanics. We do not need more here. The discussion of exercise 13.1 is completed in notes.
5. (Sep. 20) Sections 14 -15, including exercises 14.2 and 14.3. The project description of a Maple lab from Math 244 (Spring 2006 version) has been attached as a guide to obtaining graphs comparing solutions of a pendulum equation to its linear approximation.
6. (Sep. 25) Sections 16 - 19, including exercise 19.1. The main topic was section 19. Exercise 19.6 was mentioned since it shows that the theory becomes more difficult in higher dimensions. There was not enough time to say much at this time, but you should have already met the idea of a conservative vector field. Exercise 19.7 was also recommended as an interesting of conservation of energy.
7. (Sep. 27) Sections 20 - 25. Section 20 introduce the phase plane for second order autonomous systems. The coordinates in the plane are x (position) and v (velocity). For equations in which force depends only on position, conservation of energy applies and there are curves of constant energy. The motion of such a system has time increasing with x when v is positive and decreasing when v is negative (because v is velocity). For the linear oscillator (section 21), the curves are ellipses and the direction of increasing time is clockwise. The remaining sections deal with the various curves that describe motion of a pendulum. Elapsed time along a trajectory is found as the line integral of dx/v. For the linear oscillator, this integral around a closed trajectory is evaluated (with some difficulty) and shown to agree with the previous value of the period. Section 25 develops some property of the period of the pendulum from this line integral. The value is not expressible in terms of elementary functions, but it can be evaluated efficiently (by methods not mentioned in the text).
8. (Oct. 02) Sections 26 -29. The direction field introduced in section 26 has become a standard part of differential equations courses. The isoclines at which the direction field is horizontal or vertical (called nullclines in a differential equations course) are also familiar. The generalization to any fixed slope is natural. Exercise 26.1 identifies cases in which an isocline is also a trajectory. Isoclines giving different directions can intersect only at equilibrium points. Exercises 26.6 and 26.7 explore nonlinear equations for which the isoclines give useful information about solutions that are unlikely to be given by formulas. Section 27 introduces the general use of linearization at an equilibrium point to study stability of such points. This is currently done by writing the linearization as a matrix and finding the eigenvalues of the matrix, but the book was written before that approach was a standard part of the curriculum. Section 28 returns to the study of the pendulum to give the effect of damping. This completes part 1. These sections will not be part of the first exam, but will be covered in a quiz on the lecture after the exam.
9. (Oct. 04) Exam on part 1.
10. (Oct. 09) Begin part 2 after a quiz on sections 26 - 29. Then did sections 30 - 34. Section 32 sets up a difference equation model for a simple growth process in which the change from one time to the next is proportional to the current size of the quantity. Exercise 32.1 deals with the effect of the sign of this proportionality constant. Exercise 32.2 gives a numerical example of an equation of this type and its solution. Exercise 32.4 indicates that the same model applies to interest on bank accounts or loans., Subsequent exercises trace the transition to continuous compounding and the appearance of the exponential function with base e in the analogous differential equation. Section 34 introduces the idea of doubling time as a means of easily describing exponential growth.
11. (Oct. 11) Section 35 introduces the Leslie matrix that refines birth and death rates in a population by allowing those rates to depend on age. A finite number of age ranges, of equal length, are used. In a time step equal to the length of an age range, the distribution vector changes by multiplication by this matrix. Whether the population as whole grows, remains stable, or declines depends on the positive eigenvalue of this matrix. It is easily seen that there is only one positive eigenvalue in this case, but there is a more general theory that applies to any matrix with no negative entries. Exercise 35.2 sketches the role of eigenvalues. Exercise 35.3 gives some simple numerical examples. Computers are required to deal with large models of this type, but examples with only three or four age ranges and rates given by simple fractions (as in exercise 35.3) can be worked easily by hand.
12. (Oct. 16) Section 36 was skipped. It is tangential to the main flow of this part and introduces some tricky methods, so we can do without it. Sections 37 - 39 on the logistic equation were done. Exercise 37.5 shows how the model is constructed from experimental results. Exercise 38.1 has a similar connection to experiment, but is less precise. Although a phase plane appears in Section 38, this is not completely appropriate since the equation is first order, so that N' is dependent on N. In the 252 text, this subject is discussed using a phase line. Stability can be determined directly by determining the direction of flow between equilibrium values or by studying the derivative with respect to N of the expression for N' as was done in Section 18. In Section 39, an explicit solution of the logistic equation was found, but it was noted that the formula tells us little about the nature of the solution that could not be deduced more easily from the equation itself.
13. (Oct. 18) Sections 40 -42. Section 40 raises the question of changing the model to allow behavior other than the monotonic solutions forced by autonomous first order differential equations, and proposes that the quantities giving the rate of growth act with a delay. This is plausible from the point of view of the application, but the mathematics becomes more difficult. To simplify the mathematics, a discrete version of the problem is studied. The simulation on page 166 shows a stable value, and it is easily shown that two consecutive vales equal to that quantity lead to a constant solution. To investigate the stability of such solutions, a linearization is introduced, and Section 41 solves the linearized problem. These results are interpreted in Section 42. There should have been exercises on the solution of explicit Linear Difference equations of the type studied in Section 41.
14. (Oct. 23) Sections 43-47.The data that Volterra was asked to explain, as described in Section 43, is in the attachment. Most attention in lecture was on Exercise 44.3 (parts (i) and (ii) were done in lecture) and the nature of the linearization at each equilibrium point, using methods of the later sections, was studied. Previous courses should have given a thorough treatment of the different types of equilibria using the methods of linear algebra, so an independent study is not required here. You should review the generalities of this treatment as we move on to deal with details of predator-prey and competing species models.
15. (Oct. 25) Sections 48-53. Emphasis was on Section 50. The logistic term limiting the fish population was included, and two cases were identified. In one, the fish population never got large enough to provide enough food for the sharks, and there was a stable equilibrium in which sharks were not present. In the other case, that was a stable equilibrium with positive sizes for both species. Curiously, the size of the shark populations depended only on the parameters related to the change in fish population and vice versa. This model explained the observations presented to Volterra. The text, along with most textbook treatments of this model, gave most attention to the case in which the logistic term is not present. Here, the equilibrium point is a center. A special argument is able to produce closed orbits around this center for all positive initial conditions.
16. (Oct. 30) Section 54. A second classical system arises from competing species. Here both variables have logistic growth modified by an additional adverse effect of the other species. Figure 54-1 shows the four possible arrangements of isoclines for such systems. The cases of most interest are those for which there is an equilibrium point with both coordinates positive, and there are two types of such systems. In one, this positive equilibrium is stable with two real negative eigenvalues and the equilibrium points on the axes are both saddle points, so that for all initial values (except those in which one of the variables is zero) the limit for large time is the positive equilibrium point. In the other, the positive equilibrium is a saddle point, so (except for initial values on a separatrix) one of the species most eventually disappear.
17. (Nov. 01) Sections 56-58. We only finished the definition of flow from Section 58; density will wait for the next lecture. Most of the effort was on Section 57, especially exercises 57.2 and 57.5. An important observation was that the formula for the velocity field is nothing but the right side of a differential equation and the curves where velocity is constant are isoclines. The special case given by equation (57.2) is one in which these isoclines are also solutions of the equation, but exercise 57.2 gives a similar formula for the velocity field where the isoclines are straight lines without being solutions of the equation.
18. (Nov. 06) Sections 58 - 60. Density was described, completing Section 58 and the formula paraphrased in the title of Section 59 was established. Exercises 58.1 and 59.1 were used to illustrate definitions of flow and density and the relation shown in Section 59. Then, the condition expressing conservation of cars was introduced with a sketch of one of the proofs from the text. This law gives a partial differential equation involving density and flow, which could be restated in terms of density and velocity.
19. (Nov. 08) Exam on Part 2.
20. (Nov. 13) Sections 60 - 63. Including exercises 60.2 and 61.1. Starting with this lecture, we assume that the speed u(x,t) is a function only of density. To justify this, the result of observations of particular highways was examined. Although it is not difficult to imagine cases that this will not cover, it appears to be suitable for studying some features of traffic flow. We are not aiming to explain everything. Rather, we look for models that allow some limited properties to be treated mathematically. That model will be used to make predictions that we will evaluate.
21. (Nov. 15) Sections 64 - 66. Section 64 deals with the acceleration of cars in traffic, which is not used in later sections. It was skipped. In section 65, the principle was proposed that a partial differential equation should have a unique solution corresponding to initial values at t=0 if the partial derivative with respect to t is to equal an expression containing all variables, both dependent and independent, and partial derivative of the dependent variable with respect to the independent variables other than t. Three examples were given in the text and exercises 65.1, 65.3, 65.5 were done in class. Section 66 considered the traffic flow equation and demonstrated that, with constant initial density, there is a solution in which that density holds for all time. Then, it was suggested that solutions that are approximately constant could be considered by a process that leads to a linearization of the equation. In the next section, the resulting linear equation will be solved.
22. (Nov. 20) Sections 67 - 71. Section 70 was skipped. The linearized equation (67.1) was solved in two different ways. First, the left side was identified as the dot product of the gradient of rho with a fixed vector in the (t,x) plane. Thus the derivative in this direction is zero and rho is constant on lines in this direction. Such lines have x-ct constant. When the equation is combined with a value of rho(0,x) of f(x), the solution is f(x-ct). The second solution, discussed in detail in the text, introduces e new coordinate system of time and initial position for motion at speed c. The chain rule for this change of coordinates shows that the solution is a constant function of time in these coordinates, which is the same as what we got from the other method. These lines where x-ct is constant, on which the solution is forced to be constant are called characteristics of the equation. In section 71, it is shown that similar lines can be found for the general equation 71.1. The lines are parallel in the linear model, but may have different slopes in general. Exercise 69.1 was done as preparation for section 71.
23. (Nov. 27) Sections 72 - 73. Section 72 applies the method of characteristics to the equation that models a traffic light turning green after a large number of cars have accumulated behind it while the road ahead is completely free of cars. Although this initial condition isn't even continuous, the method of solution gives a reasonable answer. If the initial distribution and the assumed dependence of u on rho are approximated by smooth functions, the solution will be differentiable. All such solutions are close together, so a limit can be formed that will be taken as the solution for the given data. In section 73, the case in which speed is a linear function of density is considered. A simple dependence allows formulas for everything, including the paths of individual cars, to be found. A solution in Maple was used to give an alternate plot of the result shown in Figure 73-8.
24. (Nov. 29) Sections 74 - 76. Section 74 provides the details of the use of characteristics to solve a traffic flow equation with a variable initial density. Section 75 relates the result that the wave velocity is negative in heavy traffic to the phenomenon of brake lights propagating back a line of cars. Section 76 notes that solutions of the type described in Section 74 require that there be only one characteristic through each point in the (t,x) plane, but this assumption fails when traffic density is allowed to increase -- exactly the scenario described in Section 75. A method to handle this possibility by allowing a discontinuous density function while preserving a form of the conservation law will be treated next.
25. (Dec. 04) Sections 77 - 78. The graph for quiz 22 has been added as an attachment to this section, along with the related graph produced by Maple from the data of Figure 73-8 In Section 77, the integral conservation law is used to produce a condition that must be satisfied by the path of a discontinuity of the density function in the (t,x) plane. Such a discontinuity in the solution of a partial differential equation is called a shock. Two derivations of the condition are given in the text and Figure 77-3 shows how the slope of the shock can be found in flow-density picture, relating it to the slopes of the characteristics on either side of the shock. Section 78 applies this analysis to traffic building up behind a red light. An alternate view based on when the individual cars must stop is shown to lead to the same formula for the line separating moving cars approaching the light from the cars that have already stopped.
26. (Dec. 06) Sections 80 - 82. We will say a few words about section 79 next time. Section 80 illustrates how shocks are generated whenever density is increasing in the direction of motion. Formulas (80.2a) and (80.2b) give explicit times that these shocks first appear. In most cases, this time is bounded away from zero, so there will be a solution free of shocks for some time. Section 81 exploits this to show that this time is quite long if the density is almost constant, so that the linear approximation is valid for some time. The weakness of this approach is that no recipe is given to locate more than the start of the shock. Part of the reason for this is that density distributions tend to become triple-valued with the middle value marked by a pair of shocks. This means that examples that might occur in nature are not likely to lead to functions having agreeable algebra or calculus, since agreeable functions don't usually have three roots. Section 82 is devoted to an exception that can be analyzed and is also a natural application of the theory. Traffic that is originally at constant density is stopped by a traffic light and starts again a short while later. The initial distribution at the time the light turns green is known from Section 78. For the green light, we return to Section 72, with the original distribution modified to have two places where density increases in the direction of motion. This introduces shocks, but the functions are so simple that a complete analysis is possible.
27. (Dec. 11) Sections 83 - 85. In Section 83, we allow traffic to enter or exit, and the equation of conservation of cars is modified to become inhomogeneous with the new flow appearing on the right side of equation (83.1). Retaining the meaning of characteristic as a curve on which dx/dt is equal to the partial derivative of q with respect to rho, we find that density is no longer constant on characteristics; rather its derivative is the flow. In order to get equations that we can solve, constant flow is assumed in Section 84. The density is then a linear function of time on characteristics. If speed is a linear function of density, the characteristics are seen to have an equation in which x is a quadratic function of t. Finally, Section 85 analyzes the case in which cars enter an initially empty road at a constant rate through a bounded interval of space.
28. (Dec. 13) Review. Response to student questions.

Page started by RT Bumby on September 05, 2006
Last revised by RT Bumby on December 18, 2006