# 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.

## Plan for course

It is intended to cover the whole book.  The pace of the course should be similar to the 2006 version.

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.

## Lecture details

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

1. September 03: Sections 1–4. Using physical laws to write equation for forces.  Newton's laws connect forces with acceleration, which is the second derivative of displacement, so differential equations are inevitable.  When the spring is vertical, gravity appears in the equation giving a nonhomogeneous equation.  We appealed to section 6 to test that all terms in an equation represented forces.
2. September 08: Sections 5–9. Various background topics, including exercise 8.2 leading up to a study of spring-mass systems with one or two springs following the example in the text.
3. September 10: Sections 10–13. Damped oscillations. If there is a a force proportional to velocity and opposite to the direction of motion, the equation remains linear but the solution contains an exponential factor with a negative exponent. For small values of the resisting force, there is still oscillation, but with a longer period than in the undamped case.Exercise 12.3 was discussed: it dealt with the periodicity of relative maxima,.For large values of the resisting force, the solution is a sum of two exponentials, with the slower decay dominating. Exercise 13.4 was discussed: it dealt with the behavior as the resisting force goes to infinity.
4. September 15: Sections 14–16.Basic properties of the pendulum. Setting up the problem is easiest in a coordinate system that moves with the pendulum and has one axis that points in the direction of the rod connecting the weight to the pivot. The slightly more general approach of studying an arbitrary motion using polar coordinates is sketched in exercise 14.2. Exercise 15.1 was discussed: it used Taylor's theorem to estimate the error in replacing the sine of theta by theta.
5. September 17: Sections 17, 18 and a look ahead to 22. There was also a brief discussion about whether a small relative error in the function of x giving d2x/dt2 means that the solutions x(t) of an equation with the error will approximate one without. In some places, like exercise 15.1, the text suggests this. However, a more careful study of differential equations suggests that this is only a local truncation error, and the compounding effect of previous errors needs to be considered. This second type of error typically increases exponentially with time. In section 17, constant solutions of a differential equation were considered, and the question of stability was introduced. In section 18, a linear approximation was proposed as a test for stability. This is best considered by introducing a phase plane, which appears in section 22. Eliminating t from the system leads to an equation relating displacement and velocity that is separable for any equation for which acceleration is a function only of position. Solving this equation gives a relation between position and velocity that is itself a differential equation. This equation may be interpreted as conservation of energy as in sections 19 and 20.
6. September 22: Sections 19, 20. Conservation of energy as a unifying idea for describing the trajectories in the phase plane. Introducing integrals to measure time along a trajectory.
7. September 24: Sections 26–28. Basic properties of damping in the nonlinear case. Energy in the system is no longer conserved; it decreases because some escapes as heat. Energy curves are still useful although they are no longer trajectories. Isoclines are defined and used to get some qualitative properties of trajectories.
8. September 29: Guest lecture by Professor Wheeden featuring Exercise 19.7 on Escape Velocity. Otherwise an opportunity for a fresh view of all of part 1.
9. October 01: Sections 30–34. The first two of these sections give an overview of problems of population growth. The mathematics begins when we explore the consequences of assuming that the change in population over a fixed time interval is strictly proportional to the population at the start of that time interval. This is a simple example of a difference equation. There are many common features between difference equations and differential equations including the ability to find solutions by assuming an exponential solution and exploiting linearity. Other topics mentioned were compound interest and the Fibonacci sequence. The quiz on this material will be on October 8 because of the exam on October 6.
10. October 06: Exam on part 1.
11. October 08: Sections 35–36. Two variations on the simple population model are considered. In one, the population is classified by age with different birth and death rates for different ages (in the case of birth rates, the age of the parent is considered). In the second model, probabilistic considerations are assumed.
12. October 13: Sections 37–38. The logistic equation. To avoid the problem that the simple model has an unstable balance between unsustainable exponential growth and exponential decay leading to extinction, we seek a model showing exponential growth when the population is small, but with a growth rate that decreases as the population decreases. The simplest example has a rate that is a linear function of population. This is the logistic model. By choosing a linear function that is zero for the expected maximum population, you get an equation that is sure to justify the limit you expect. For a first order autonomous equation, there is a phase line. The equilibrium points are identified, and between them, solutions to the equation are either steadily increasing or steadily decreasing. To find out which, simply evaluate the right side of the equation at an interior point of each interval between equilibrium points. Alternatively, the equation can be linearized at each equilibrium point. If this shows the point to be attracting, the arrows on adjacent segments point toward the point; if repelling, they point away. The two methods must give consistent results. In higher dimensions, only the linearization at equilibrium points will be available, so it is useful to relate this method to a more elementary method in a simpler case so you can begin to trust it.
13. October 15: Sections 39–40.Just to show that it is possible, we obtain the explicit solution of the logistic equation. One feature that is easily seen from this is that the decreasing solutions all have vertical asymptotes.
14. October 20: Sections 41–42. In order to learn more what to expect about the time-delay model, we linearize at the equilibrium point. The corresponding linear difference equation can be solved exactly, and the question of stability can be answered using this solution.
15. October 22: Sections 43–47. These sections review properties of a system of two linear differential equations.
16. October 27: Sections 48–53. The predator-prey model.
17. October 29: Section 54. The competing species model.
18. November 03: Sections 56–57. Introduction to traffic flow and properties of a velocity field.
19. November 05: Sections 58–59. Traffic flow and traffic density.
20. November 10: Exam on Part 2.
21. November 12: Sections 60–63. Two fundamental principles of the study of traffic flow are introduced. First, the assumption that cars are neither created nor destroyed in the daily commute leads to a conservation law that can be expressed as a partial differential equation (parsed as an equation involving partial derivatives). Second, the mindless nature of the daily commute is modeled with an assumption that the speed of a car is determined only by the density of traffic. This leads to a velocity field describing the speed of a car in terms of its position in time and space. As we have seen, a velocity field is an ordinary differential equation whose solution describes the behavior of individual cars. The traffic flow also becomes a a function of density that is zero at both zero density and the maximum (bumper to bumper) density). We confine attention to the case in which this function is concave downward, so that it has a unique maximum, although no explanation is proposed for why the relation between density and speed would lead to this behavior of the flow.
22. November 17: Sections 65–67. Section 64 was skipped since it is not used later, and the details of Section 66 were postponed. Some general properties of partial differential equations are given, and the nature of the solution of some linear equations arising in the study of traffic flow is described.
23. November 19: Sections 66–69. The fundamental equation of traffic flow has the property that any constant function is a solution. By considering small perturbations of such solutions, we find that the perturbations must approximately satisfy a linear equation. The solution described in Section 67 produces density waves that travel forward in light traffic and backward in heavy traffic. The wave pattern would seem fixed to an external observer moving at the wave velocity although different cars would be in the wave.
24. November 24: Sections 70–73. The semi-infinite highway gives an example in which there is a boundary condition in the space variable as well as the initial condition in time. In light traffic, the solution to the equation splits into two cases: at any time, the more distant densities are the evolution of the initial density, while those near the entrance are determined by the boundary condition. The solutions of the linearized equation were constant on certain lines, corresponding to an external observer moving at constant velocity. These lines are called characteristics. In general, a characteristic is a curve on which the partial differential equation reduces to an ordinary differential equation. The characteristics in this example are more special: the solution is constant of the characteristic and the characteristic is a straight line. In the example of a traffic light turning green after a long stream of cars has built up behind it, the characteristics that don't correspond to maximum velocity or zero velocity all pass through the origin. For any assumption of the dependence of velocity on density, this allows the density at each point to be determined. Once the density is known, methods used at the beginning of this part of the text determine the motion of individual cars. In Section 73, this solution is found when a linear relation between density and velocity is assumed.
25. December 01: Sections 77–78. The examples of sections 74–76 will be skipped to get directly to the idea of a shock. In previous examples, we were careful to make sure that characteristics did not intersect, but we can't always be so lucky. In particular, traffic stopping for a red light gives an important example in which the characteristics from the original motion of traffic and those from the condition at x=0 for positive t overlap in a large portion of the (x,t) plane. One one set of characteristics, density is the original uniform density on the highway; on the other set, it is the maximum density. Only one of these can be valid at any given point. To resolve this ambiguity, the integral form of the conservation law is considered instead of the differential form that we have been using. This leads to discontinuities in density being allowed only when the resulting ratio of the change in flow to the change in density is equal to dx/dt on the curve, called a shock, on which the value changes. In the case of a red light, there are two constant values to separate, so we get a line of fixed slope as the shock. Among all such lines, the shock is the one passing through tho origin, since this in the only place where the given boundary data is discontinuous.

Page last revised by RT Bumby on June 19, 2009