nach oben

Erschienen in:

2016 | OriginalPaper | Buchkapitel

1. Introduction

verfasst von : David G. Schaeffer, John W. Cain

Erschienen in: Ordinary Differential Equations: Basics and Beyond

Verlag: Springer New York

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

This chapter summarizes various ideas that an introductory course in ordinary differential equations (ODEs) usually covers. Because of this wide focus, the exposition may seem somewhat diffuse—a small price to be paid for our not requiring previous training in ODEs.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Nächstes Kapitel Linear Systems with Constant Coefficients

Well, x′ = f(t) is simpler, but this equation belongs to calculus, not ODEs.

We resist the temptation to dazzle you with interesting higher-order ODEs drawn from numerous fields. Truth to tell, higher-order equations play a smaller role in the theory than systems of several coupled ODEs, which we introduce in Section 1.5.

More formally, we may say that equation (1.9) is linear if the function f is linear in its first n arguments; no restriction on the t-dependence is implied.

Equation (1.1) is also separable, but usually this term is reserved for nonlinear equations.

Observe that x ≡ 0 also satisfies (1.2). Thus, in the derivation, d x∕x is likewise meaningless for this solution, but the final answer nevertheless captures the solution. This behavior reminds us that solutions obtained using separability always need to be examined carefully.

You may groan that we use the same letter b for the constant of proportionality as for the initial conditions. Please trust us—in the larger scheme of things, this conflict of notation will not cause confusion.

This formula is more typical of the drag from a viscous fluid at low to moderate velocities—see Section 5.4 in [72] or look online. Reflecting this situation, in Figure 1.2 friction is represented by a “dashpot”: i.e., a piston sliding through a viscous fluid.

In fact, the more general equation (1.24) can be reduced to (1.14) by scaling the variables appropriately. The uses of scaling will be developed systematically in Chapter 5

The mass also experiences a radial acceleration, − ℓ(d x∕d t)², from which the tension in the arm may be calculated. However, the motion of the pendulum is determined by the tangential equation (1.26), without consideration of radial forces.

For now, you may regard (1.28) as a particular case of the force law (1.27), but in fact, the general case may be reduced to (1.28) through appropriate scaling (see Chapter 5).

You may wonder why we have used different letters—β in (1.28) and b in (1.24)—for analogous coefficients. As we will explore in Chapter 5, in ODEs arising in applications, most parameters have units (such as length, mass, inverse time) associated with them. Although we are not completely consistent, we try to use Latin letters for parameters with nontrivial dimensions and reserve Greek letters for dimensionless parameters. The latter usually are derived as composites of several dimensional parameters.

Although we introduced forces in connection with spring–mass systems, we want to consider more general force laws than can reasonably be associated with any spring. For that reason, we adopt the physicists’ phrase, “a particle in a force field.” If you are interested in mechanics, you should probably learn about the Lagrangian approach to this subject—see Chapter 7 in [88]. With this formalism, it is easy to derive equations of motion when there are constraints, e.g., a particle sliding along a curve in the plane.

One is reminded of the aphorism, “A simple lie may be more useful than a complicated truth” (adapted from de Tocqueville).

Reference [90] is one of the original papers. Modern derivations of van der Pol’s equation are available online.

Equation (1.34) may be derived from Kirchhoff’s laws—see [72] or look online. Incidentally, a circuit with the elements in series, rather than in parallel, is probably more familiar to most readers. We consider the parallel circuit because it relates more directly to van der Pol’s equation.

There is an unfortunate conflict between different fields in the use of the word system. At its first occurrence in this sentence, “system” is used in its biological sense “a group of interacting, interrelated, or interdependent elements forming a complex whole.” At its second occurrence, “system” is used in its more restricted mathematical sense, “several simultaneous equations.”

Teleost is the biologists’ term for what one normally thinks of as “fish.” Teleosts have a bony skeleton, in contrast to sharks, whose skeleton is made of cartilage. Teleosts appeared later in evolution, so they are sometimes called “modern” fish or “bony” fish. For purposes of the Lotka–Volterra model, teleost means “fish that are good to eat”—the model was developed to understand perplexing changes in fish harvests during World War I. (See Exercise 22.)

For greater realism, the underlying process should be modeled probabilistically. An ODE model provides a useful approximation for the evolution of average populations, provided the populations are large. The rate term proportional to x y may be derived from the probability that members of the two species encounter one another. For more detail, see Section 10.2 and the references therein. In chemical kinetics, the corresponding approximation is called the law of mass action.

This figure is our first instance of a phase plane plot: i.e., graphs of a few well-chosen trajectories for a two-dimensional equation that indicate the behavior of the general solution.

Approximate solutions, obtained either from numerical computations or asymptotic analysis, sometimes provide an adequate substitute. We will touch briefly on both kinds of approximate solutions, but they are not the main focus in this book.

We show in Chapter 5 that by scaling the variables, (1.35) can be reduced to (1.39). In the meantime, you may simply regard (1.39) as a special case of (1.35).

Food for thought: How do solutions on the boundary of the first quadrant, {x ≡ 0} or {y ≡ 0}, behave?

You may wonder why we should not choose K equal to unity, as in (1.2). We could in fact do this, but only at the expense of either losing some generality or making the y-equation in the system less transparent. The mystery surrounding this and other applications of scaling arguments should be dispelled by Chapter 5

A growth rate that depends on the population size is called the Allee effect.

We want ɛ < K, so that the carrying capacity exceeds the threshold for extinction, and when K > 1, we want ɛ < 1, so that the prey population at the coexistence equilibrium (1.44) exceeds the threshold for extinction.

Note that there are multiple equilibria—we shall see that this is typical for nonlinear systems.

Perhaps, for a computer-savvy generation, we should be giving pencil and paper a plug.

See also [82] for a discussion of a less dramatic resonance-induced bridge episode.

You can check this statement using the Lagrangian formulation of mechanics (cf. footnote number 11) to derive the equations for such constrained motion.

[9]

G. Birkhoff and G.-C. Rota, Ordinary differential equations, 4th edition, Wiley, New York, 1989.

[10]

M. Braun, Differential equations and their applications: An introduction to applied mathematics, 4th edition, Springer, New York, 1993.CrossRefMATH

[19]

L. Edelstein-Keshet, Mathematical models in biology, Society for Industrial and Applied Mathematics, Philadelphia, 2005.CrossRefMATH

[72]

R. A. Serway and J. W. Jewett, Jr. Principles of physics: A calculus-based text, 5th ed., Cengage, 2012.

[82]

S. H. Strogatz, D. M. Abrams, A. McRobie, B. Eckhardt, and E. Ott, Crowd synchrony on the Millennium Bridge, Nature 438 (2005), 43–44.CrossRef

[84]

J. Szarski, Differential inequalities, Polish Scientific Publishers, Warsaw, 1965.MATH

[88]

S. T. Thornton and J. B. Marion, Classical dynamics of particles and systems, Thomson Brooks/Cole, 2004.

[90]

B. van der Pol, On “relaxation-oscillations”, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Series 7 2 (1926), 978–992.

Titel: Introduction
verfasst von: David G. Schaeffer
John W. Cain
Verlag: Springer New York
Buch: Ordinary Differential Equations: Basics and Beyond
Print ISBN: 978-1-4939-6387-4

Electronic ISBN: 978-1-4939-6389-8

Copyright-Jahr: 2016
DOI: https://doi.org/10.1007/978-1-4939-6389-8_1

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"