Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben

1993 | Buch | 4. Auflage

First-order differential equations Kapitel lesen Erstes Kapitel lesen

Differential Equations and Their Applications

An Introduction to Applied Mathematics

verfasst von: Martin Braun

Verlag: Springer New York

Buchreihe : Texts in Applied Mathematics

Enthalten in: Professional Book Archive

Einloggen, um Zugang zu erhalten
insite
SUCHEN

Über dieses Buch

There are two major changes in the Fourth Edition of Differential Equations and Their Applications. The first concerns the computer programs in this text. In keeping with recent trends in computer science, we have replaced all the APL programs with Pascal and C programs. The Pascal programs appear in the text in place of the APL programs, where they are followed by the Fortran programs, while the C programs appear in Appendix C.

Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the classical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics (TAM).
The development of new courses is a natural consequence of a high Ievel of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mathematical Sciences (AMS) series, which will focus on advanced textbooks and research Ievel monographs.

Inhaltsverzeichnis

Frontmatter
1. First-order differential equations
Abstract
This book is a study of differential equations and their applications. A differential equation is a relationship between a function of time and its derivatives.
Martin Braun
2. Second-order linear differential equations
Abstract
A second-order differential equation is an equation of the form
$$\frac{{{d^2}y}}{{d{t^2}}} = f\left( {t,y,\frac{{dy}}{{dt}}} \right)$$
(1)
.
Martin Braun
3. Systems of differential equations
Abstract
In this chapter we will consider simultaneous first-order differential equations in several variables, that is, equations of the form
$$\begin{gathered} \frac{{d{x_1}}}{{dt}} = {f_1}\left( {t,{x_1},...,{x_n}} \right), \hfill \\ \frac{{d{x_2}}}{{dt}} = {f_2}\left( {t,{x_1},...,{x_n}} \right), \hfill \\ \vdots \hfill \\ \frac{{d{x_n}}}{{dt}} = {f_n}\left( {t,{x_1},...,{x_n}} \right). \hfill \\ \end{gathered} $$
(1)
A solution of (1) is n functions x 1(t),..., x n(t) such that dx j(t)/dt = f j(t, x 1(t),..., x n(t)), j = 1,2,..., n. For example, x 1(t) = t and x 2(t) = t 2 is a solution of the simultaneous first-order differential equations
$$\frac{{d{x_1}}}{{dt}} = 1{\kern 1pt} and{\kern 1pt} \frac{{d{x_2}}}{{dt}} = 2{x_1}$$
since dx 1(t)/dt = 1 and dx 2(t)/dt = 2t = 2x 1(t).
Martin Braun
4. Qualitative theory of differential equations
Abstract
In this chapter we consider the differential equation
$$\dot x = f\left( {t,x} \right)$$
(1)
where
$$ x = \left[ {\begin{array}{*{20}{c}} {{x_1}\left( t \right)} \\ \vdots\\ {{x_n}\left( t \right)} \end{array}} \right], $$
and
$$ f\left( {t,x} \right) = \left[ {\begin{array}{*{20}{c}} {{f_1}\left( {t,{x_1}, \ldots ,{x_n}} \right)} \\ \vdots\\ {{f_n}\left( {t,{x_1}, \ldots ,{x_n}} \right)} \end{array}} \right] $$
is a nonlinear function of x 1, ..., x n. Unfortunately, there are no known methods of solving Equation. This, of course, is very disappointing. However, it is not necessary, in most applications, to find the solutions of explicitly. For example, let x 1(t) and x 2(t) denote the populations, at time t, of two species competing amongst themselves for the limited food and living space in their microcosm. Suppose, moreover, that the rates of growth of x 1(t) and x 2(t) are governed by the differential equation. In this case, we are not really interested in the values of x 1(t) and x 2(t) at every time t. Rather, we are interested in the qualitative properties of x 1(t) and x 2(t). Specically, we wish to answer the following questions.
Martin Braun
5. Separation of variables and Fourier series
Abstract
In the applications which we will study in this chapter, we will be confronted with the following problem.
Martin Braun
6. Sturm-Liouville boundary value problems
Abstract
In Section 5.5 we described the remarkable result that an arbitrary piecewise differentiable function f(x) could be expanded in either a pure sine series of the form
$$f\left( x \right) = \sum\limits_{n = 1}^\infty{{b_n}\sin \frac{{n\pi x}}{l}} $$
(1)
or a pure cosine series of the form
$$f\left( x \right) = \frac{{{a^0}}}{2}\sum\limits_{n = 1}^\infty{{a_n}\cos \frac{{n\pi x}}{l}} $$
(2)
on the interval 0 < x < l. We were led to the trigonometric functions appearing in the series (1) and (2) by considering the 2 point boundary value problems
$$y'' + \lambda y = 0,y\left( 0 \right) = 0,y\left( l \right) = 0,$$
(3)
and
$$y'' + \lambda y = 0,y'\left( 0 \right) = 0,y'\left( l \right) = 0.$$
(4)
Recall that Equations (3) and (4) have nontrivial solutions
$${y_n}\left( x \right) = c\sin \frac{{n\pi x}}{l}and{\kern 1pt} {y_n}\left( x \right) = c\cos \frac{{n\pi x}}{l},$$
respectively, only if λ = λ n = n 2 π 2/l 2. These special values of λ were called eigenvalues, and the corresponding solutions were called eigenfunctions.
Martin Braun
Backmatter
Metadaten
Titel
Differential Equations and Their Applications
verfasst von
Martin Braun
Copyright-Jahr
1993
Verlag
Springer New York
Electronic ISBN
978-1-4612-4360-1
Print ISBN
978-0-387-94330-5
DOI
https://doi.org/10.1007/978-1-4612-4360-1