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1983 | Buch

Setting Up First-Order Differential Equations from Word Problems Kapitel lesen Erstes Kapitel lesen

Differential Equation Models

herausgegeben von: Martin Braun, Courtney S. Coleman, Donald A. Drew

Verlag: Springer New York

Buchreihe : Modules in Applied Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

The purpose of this four volume series is to make available for college teachers and students samples of important and realistic applications of mathematics which can be covered in undergraduate programs. The goal is to provide illustrations of how modem mathematics is actually employed to solve relevant contemporary problems. Although these independent chapters were prepared primarily for teachers in the general mathematical sciences, they should prove valuable to students, teachers, and research scientists in many of the fields of application as well. Prerequisites for each chapter and suggestions for the teacher are provided. Several of these chapters have been tested in a variety of classroom settings, and all have undergone extensive peer review and revision. Illustrations and exercises are included in most chapters. Some units can be covered in one class, whereas others provide sufficient material for a few weeks of class time. Volume 1 contains 23 chapters and deals with differential equations and, in the last four chapters, problems leading to partial differential equations. Applications are taken from medicine, biology, traffic systems and several other fields. The 14 chapters in Volume 2 are devoted mostly to problems arising in political science, but they also address questions appearing in sociology and ecology. Topics covered include voting systems, weighted voting, proportional representation, coalitional values, and committees. The 14 chapters in Volume 3 emphasize discrete mathematical methods such as those which arise in graph theory, combinatorics, and networks.

Inhaltsverzeichnis

Frontmatter

Differential Equations, Models, and What to do with Them

Frontmatter
Chapter 1. Setting Up First-Order Differential Equations from Word Problems
Abstract
“Word problems” are sometimes troublesome; but you have learned that most noncalculus applied problems can be conquered with careful translating and attention to the kinds of units involved. A trivial illustration of this type is as follows.
Beverly Henderson West
Chapter 2. Qualitative Solution Sketching for First-Order Differential Equations
Abstract
Qualitative solution sketching, as explained in this module, can yield very useful information about the solutions y = f(x) to a given differential equation y′ = g(x, y). In particular, it usually allows you to examine the limiting or long-range behavior for y as x → ∞ without actually coming up with an explicit expression for the solution. Frequently, an explicit solution is unnecessary or technically difficult, or it might not exist at all in terms of elementary functions (polynomial, trigonometric, logarithmic, exponential). In these cases the qualitative approach may be a lifesaver—or at least a worksaver.
Beverly Henderson West
Chapter 3. Difference and Differential Equation Population Growth Models
Abstract
Ordinarily, the derivative is defined by the following limit:
$$ \frac{{dy}}{{dx}} = \mathop{{\lim }}\limits_{{h \to 0}} \frac{{y(x + h) - y(x)}}{h} $$
(1a)
Similarly, when a computer is used to “solve” a differential equation numerically, derivatives are ordinarily replaced by finite difference approximations such as
$$ \frac{{dy}}{{dx}} \simeq \frac{{y(x + h) - y(x)}}{h} $$
(1b)
These two operations are really just inverses of one another. At times, the conversion of a difference equation into the analogous differential equation is convenient because the calculus can be employed, so the finite interval of the independent variable is made to vanish. At other times, this limit is “undone” so that numerical methods can be used on the difference equation analog of a differential equation. Unfortunately, these inverse operations have a profound effect upon the nature of the solutions found. This frequently neglected point is the main topic of this chapter.
James C. Frauenthal

Growth and Decay Models: First-Order Differential Equations

Frontmatter
Chapter 4. The Van Meegeren Art Forgeries
Abstract
After the liberation of Belgium in World War II, the Dutch Field Security began its hunt for Nazi collaborators. They discovered, in the records of a firm which had sold numerous works of art to the Germans, the name of a banker who had acted as an intermediary in the sale to Goering of the painting ‘Woman Taken in Adultery’ by the famed 17th-century Dutch painter Jan Vermeer. The banker in turn revealed that he was acting on behalf of a third-rate Dutch painter H. A. Van Meegeren, and on May 29, 1945, Van Meegeren was arrested on the charge of collaborating with the enemy. On July 12, 1945, Van Meegeren startled the world by announcing from his prison cell that he had never sold Woman Taken in Adultery to Goering. Moreover, he stated that this painting and the very famous and beautiful Disciples at Emmaus, as well as four other presumed Vermeers and two de Hooghs (another 17th-century Dutch painter) were his own works. Many people, thought that Van Meegeren was lying to save himself from the charge of treason. To prove his point, Van Meegeren began, while in prison, to forge the Vermeer painting Jesus Amongst the Doctors to demonstrate to the skeptics just how good a forger of Vermeer he really was. The work was nearly completed when Van Meegeren learned that a charge of forgery had been substituted for that of collaboration.
Martin Braun
Chapter 5. Single Species Population Models
Abstract
In this module we will study first-order differential equations which govern the growth of various species. At first glance it would seem impossible to model the growth of a species by a differential equation since the population of any species always changes by integer amounts. Hence the population of any species can never be a differentiable function of time. However, if a given population is very large and it is suddenly increased by one, then the change is very small compared to the given population. Thus we make the approximation that large populations change continuously and even differentiably with time.
Martin Braun
Chapter 6. The Spread of Technological Innovations
Abstract
Economists and sociologists have long been concerned with how a technological change or innovation spreads in an industry. Once an innovation is introduced by one firm, how soon do others in the industry come to adopt it, and what factors determine how rapidly they follow? In this chapter we construct a model of the spread of innovations among farmers, and then show that this same model describes the spread of innovations in such diverse industries as bituminous coal, iron, and steel, brewing, and railroads.
Martin Braun

Higher Order Linear Models

Frontmatter
Chapter 7. A Model for the Detection of Diabetes
Abstract
Diabetes mellitus is a disease of metabolism which is characterized by too much sugar in the blood and urine. In diabetes, the body is unable to burn off all its sugars, starches, and carbohydrates because of an insufficient supply of insulin. Diabetes is usually diagnosed by means of a glucose tolerance test (GTT). In this test the patient comes to the hospital after an overnight fast and is given a large dose of glucose (sugar in the form in which it usually appears in the bloodstream). During the next three to five hours, several measurements are made of the concentration of glucose in the patient’s blood, and these measurements are used in the diagnosis of diabetes. A very serious difficulty associated with this method of diagnosis is that no universally accepted criterion exists for interpreting the results of a glucose tolerance test. Three physicians interpreting the results of a GTT may come up with three different diagnoses. In one case recently, a Rhode Island physician, after reviewing the results of a GTT, came up with a diagnosis of diabetes. A second physician declared the patient to be normal. To settle the question, the results of the GTT were sent to a specialist in Boston. After examining these results, the specialist concluded that the patient was suffering from a pituitary tumor.
Martin Braun
Chapter 8. Combat Models
Abstract
During the first World War, F. W. Lanchester outlined several tentative mathematical models of the fledgling art of air warfare [9], [10]. These models have since been extended to represent a variety of competitions ranging from isolated battles to entire wars. We shall outline and “solve” some simple models, commenting on a mixed conventional-guerrilla combat such as Vietnam and studying in some depth the battle of Iwo Jima during World War II.
Courtney S. Coleman
Chapter 9. Modeling Linear Systems by Frequency Response Methods
Abstract
In his classic book Cybernetics [1], Norbert Wiener viewed the human being from an information processing and control system point of view. Such a viewpoint has led to the development of communications and control system oriented mathematical models for human reactions and for physiological components of the human body. Much of this work has involved frequency response techniques, and this module will be concerned with the use of such techniques in mathematical modeling.
William F. Powers

Traffic Models

Frontmatter
Chapter 10. How Long Should a Traffic Light Remain Amber?
Abstract
Let us consider the problem of calculating how long a traffic light should remain amber before turning red. Essentially, the amber cycle exists to allow vehicles in the intersection, or those too close to stop, to clear the intersection. Thus the light should remain amber long enough so that all drivers who cannot stop have a chance to pass through the intersection on the amber. A driver approaching an intersection should never be in the dilemma of being too close to stop safely and yet too far away to pass through the intersection before the red phase starts.
Donald A. Drew
Chapter 11. Queue Length at a Traffic Light via Flow Theory
Abstract
Let us consider the following design problem. Suppose we have a road which, under normal operation, has a traffic flow q of 1000 veh/h at a flow concentration k of 20 veh/mi. There are two intersections on this road, 0.2 mi miles apart. We wish to install a stoplight at one of these intersections. Our primary interest is whether the queue which forms at the downstream light backs up to the upstream intersection (see Figure 11.1). Clearly, if the downstream queue does interfere with the operation of the upstream intersection, then other considerations must be made, possibly installing a synchronized light at the upstream intersection.
Donald A. Drew
Chapter 12. Car-Following Models
Abstract
The automobile is a pervasive feature of modern technological societies despite its accompanying problems of pollution, accidents, and congestion. In the last 25 years a vast amount of literature has been published which we might classify as “traffic science.” This science has attempted to understand through modeling and data gathering the traffic processes and how to modify, optimize, and control them.
Robert L. Baker Jr.
Chapter 13. Equilibrium Speed Distributions
Abstract
One of the fundamental processes in traffic flow, which leads to delays and frustration for individual drivers is that of overtaking (that is, approaching from behind) a slower vehicle. Trucks, busses, and sightseers (“Sunday drivers”) can cause substantial delays to a driver trying to move quickly from one place to another on the standard two-lane, two-way highway encountered in so much of the United States.
Donald A. Drew
Chapter 14. Traffic Flow Theory
Abstract
Let us derive the basic conservation equation for traffic flow. We consider the flow of vehicles on a long road where the features of the flow we wish to calculate, such as bottlenecks, etc., are long compared with the average distances between vehicles. Let n(x, x + Δx,t) denote the number of vehicles between point x and point x + Δx on the road at time t (see Figure 14.1). We shall assume that k(x, t) exists such that for any x, Δx, and t,
$$ n(x,x + \Delta x,t) = \int_x^{{x + \Delta }} k (\hat{x},t)d\hat{x} $$
(1)
We note that, by the fundamental theorem of calculus,
$$ k(x,\,t) = \mathop{{\lim }}\limits_{{\Delta x \to 0}} \frac{{n(x,x + \Delta x,t)}}{{\Delta x}} $$
if k is continuous. We shall assume that we can adequately model the situations of interest with the assumption that k is continuous.
Donald A. Drew

Interacting Species: Steady States of Nonlinear Systems

Frontmatter
Chapter 15. Why the Percentage of Sharks Caught in the Mediterranean Sea Rose Dramatically during World War I
Abstract
In the mid-1920’s the Italian biologist Umberto D’Ancona was studying variations in the population of various species of fish that interact with each other. In the course of his research, he came across some data on percentages-of-total-catch of several species of fish that were brought into different Mediterranean ports in the years that spanned World War I. In particular, the data gave the percentage-of-total-catch of selachians (sharks, skates, rays, etc.) which are not very desirable as food fish.
Martin Braun
Chapter 16. Quadratic Population Models: Almost Never Any Cycles
Abstract
The populations of lynx and hare in the Canadian North Woods wax and wane together in a mysterious, 10-year cycle. The numbers of sharks and of food fish in the Adriatic jointly oscillate in a curious way.1 Our aim is to construct simple mathematical models for the interaction of any pair of species and then to look for cyclical steady states in these models. Our results appear to be contradictory. As we shall see, most simple systems have no cycles at all, but the famous system of D’Ancona-Volterra (or Lotka-Volterra) has nothing but cycles [5]. The attempt to construct a synthesis going beyond this contradiction leads to an inclusive view of modeling. Several distinct models of a phenomenon give us more understanding of the phenomenon than one model can ever impart.
Courtney S. Coleman
Chapter 17. The Principle of Competitive Exclusion in Population Biology
Abstract
It is often observed in nature that the struggle for existence between two similar species competing for the same limited food supply and living space nearly always ends in the complete extinction of one of the species. This phenomenon is known as the “principle of competitive exclusion,” and was first enunciated, in a slightly different form, by Darwin in 1859. In his paper, “The origin of species by natural selection,” he writes:
“As the species of the same genus usually have, though by no means invariably, much similarity in habits and constitutions and always in structure, the struggle will generally be more severe between them, if they come into competition with each other, than between the species of distinct genera.”
Martin Braun
Chapter 18. Biological Cycles and the Fivefold Way
Abstract
One of the more curious natural phenomena is the regular periodic variation in the populations of certain interacting species, a variation which does not correlate with known periodic external forces such as the cycles of darkness and light, the seasons, or weather cycles. Most commonly, these species are involved in a predator-prey relationship. The phenomenon has been observed in the microcosm of a laboratory culture (paramecium aurelia (predator) and saccharomyces exiguus (prey)—see D’Ancona [11]) and in the macrocosm of the Canadian coniferous forests (Canadian lynx (predator) and snowshoe hare (prey)—see Keith [17]). Other examples include the budworm-larch tree cycle in the Swiss Alps [2] and a lemming-vegetation cycle in Scandinavia [19]. We shall discuss only the lynx-hare cycle, but the mathematical models we shall develop are in principle applicable to any predator-prey interaction. In the course of our analysis we shall study the equilibria of coupled differential systems and the possible limit behavior of bounded orbits of such systems. This will all then be applied to the question of the existence of cycles in a predator-prey system.
Courtney S. Coleman
Chapter 19. Hilbert’s 16th Problem: How Many Cycles?
Abstract
In the above quotation Balthazar van der Pol has grouped the most varied phenomena into a single category of systems which exhibit periodic oscillations, or cycles, even though no “external periodic forces” cause such oscillations. (One wonders, however, about some of van der Pol’s systems; e.g., the sleeping of flowers is surely due to the diurnal cycle of darkness and light.) According to van der Pol, most of these systems can be modeled more or less accurately by a pair of differential equations of the form,
$$ \left\{ \begin{gathered} \frac{{dx}}{{dt}} = X(x,y) \hfill \\ \frac{{dy}}{{dt}} = Y(x,y) \hfill \\ \end{gathered} \right. $$
(1)
(1)
Courtney S. Coleman

Models Leading to Partial Differential Equations

Frontmatter
Chapter 20. Surge Tank Analysis
Abstract
Hydrodynamic generation of electrical power requires the efficient transfer of water under pressure from an elevated storage area, or reservoir, to the generating plant, often several hundred meters away. The obvious solution of connecting a large pipe to both ends has a serious shortcoming. A large surge in the power supplied to the electrical customers requires an abrupt increase in the amount of water flowing into the turbine. A falling-off in power usage causes the water suddenly to flow more slowly. In both cases, a large mass of water in the pipe must suddenly change its velocity. The slight compressibility of the water, combined with the slight elasticity of the pipe, causes a wave of high pressure to propagate up and down the pipe, resulting in “water hammer” and often rupturing the pipes.
Donald A. Drew
Chapter 21. Shaking a Piece of String to Rest
Abstract
Periodic disturbances play an important role in many diverse areas within science and engineering. Examples of periodic disturbances are the motion of a pendulum in a gravitational field, the motion of celestial bodies, and oscillations in an electrical circuit. When periodic disturbances travel in space, they are described collectively as “wave motion.” Some familiar examples of wave motion are the motion of ripples on the surface of a pond, sound waves, and transverse vibrations on a taut, flexible string.
Robert L. Borrelli
Chapter 22. Heat Transfer in Frozen Soil
Abstract
Between the Arctic Ocean and the Brooks Range of Alaska lies the North Slope. In this cold and barren tundra oil has been found and considerable oil exploration and production activity is anticipated for the coming years. A good part of this activity will require the erection of engineering structures such as drill rigs, pipelines, work camps, and, of course, roads and airfields.
Gunter H. Meyer
Chapter 23. Network Analysis of Steam Generator Flow
Abstract
In this first section we want to examine the role of the steam generator in the overall operation of a nuclear power plant. In this way we hope to put into perspective the mathematical problem which will eventually evolve and, at the same time, to emphasize its importance.
T. A. Porsching
Metadaten
Titel
Differential Equation Models
herausgegeben von
Martin Braun
Courtney S. Coleman
Donald A. Drew
Copyright-Jahr
1983
Verlag
Springer New York
Electronic ISBN
978-1-4612-5427-0
Print ISBN
978-1-4612-5429-4
DOI
https://doi.org/10.1007/978-1-4612-5427-0