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.
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.
Fall 2006 Information
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
During the semester, a log was kept on the Sakai site. Here is a copy
of that log.
- (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.
- (Sep. 11) Sections 5 - 8, including exercises 5.1, 5.2, 7.2.
- (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.
- (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.
- (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.
- (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.
- (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).
- (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.
- (Oct. 04) Exam on part 1.
- (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.
- (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.
- (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
- (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.
- (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.
- (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
- (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.
- (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.
- (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.
- (Nov. 08) Exam on Part 2.
- (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.
- (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
- (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
- (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
- (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.
- (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.
- (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.
- (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.
- (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