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1998 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Ordinary Differential Equations

verfasst von: Wolfgang Walter

Verlag: Springer New York

Buchreihe : Graduate Texts in Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

Develops the theory of initial-, boundary-, and eigenvalue problems, real and complex linear systems, asymptotic behavior and stability. Using novel approaches to many subjects, the book emphasizes differential inequalities and treats more advanced topics such as Caratheodory theory, nonlinear boundary value problems and radially symmetric elliptic problems. New proofs are given which use concepts and methods from functional analysis. Applications from mechanics, physics, and biology are included, and exercises, which range from routine to demanding, are dispersed throughout the text. Solutions for selected exercises are included at the end of the book. All required material from functional analysis is developed in the book and is accessible to students with a sound knowledge of calculus and familiarity with notions from linear algebra. This text would be an excellent choice for a course for beginning graduate or advanced undergraduate students.

Inhaltsverzeichnis

Frontmatter
Introduction
Abstract
A differential equation is an equation containing independent variables, functions, and derivatives of functions. The equation
$$ y' + 2xy = 0 $$
(1)
is a differential equation. Here x is the independent variable and y is the unknown function. A solution is a function y = Ø(x) that satisfies (1) identically in.x,that is, Ø′(x)+ 2x ⋅ Ø(x) ≡0. It is easy to check that the function \( y = e^{ - x^2 } \) is a solution of (1):
$$ \frac{d} {{dx}}(e^{ - x^2 } ) + 2xe^{ - x^2 } \equiv 0 for - \infty < x < \infty $$
.
Wolfgang Walter
Chapter I. First Order Equations: Some Integrable Cases
Abstract
We consider the explicit first order differential equation
$$ y' = f\left( {x,y} \right). $$
(1)
The right-hand side f (x, y) of the equation is assumed to be defined as a real-valued function on a set D in the xy-plane.
Wolfgang Walter
Chapter II. Theory of First Order Differential Equations
Abstract
Many questions in the theory of differential equations can be answered in a particularly elegant manner using general concepts of functional analysis. In this and later chapters, functional-analytic methods will be used to derive theorems on existence, uniqueness, and dependence of solutions on parameters. We begin by introducing the concept of a Banach space.
Wolfgang Walter
Chapter III. First Order Systems. Equations of Higher Order
Abstract
Systems of Differential Equations.Direction Fields. By a first order system of differential equations (in explicit form) we mean a set of simultaneous equations of the form
$$ \begin{array}{*{20}{c}} {{{y'}_1}}& = &{{f_1}\left( {x,{y_1}, \ldots ,{y_n}} \right)} \\ \vdots &{}& \vdots \\ {{{y'}_n}}& = &{{f_n}\left( {x,{y_1}, \ldots ,{y_n}} \right)} \end{array} $$
(1)
.
Wolfgang Walter
Chapter IV. Linear Differential Equations
Abstract
We denote ri x n matrices by uppercase italic letters,
$$ A = \left( {\begin{array}{*{20}{c}} {{a_{11}} \ldots {a_{1n}}} \\ { \vdots \ddots \vdots } \\ {{a_{n1}} \cdots {a_{nn}}} \end{array}} \right) = ({a_{ij}}), $$
where aij E R or C. With the usual definitions of addition and scalar multiplication of matrices,
$$ A + B = ({a_{ij}} + {b_{ij}}),{\text{ }}\lambda A = (\lambda {a_{ij}}), $$
the set of all n x n matrices forms a real or complex vector space. One can inter-pret this space as Rn2 (or, for complex aij, bij, A, as Cn2). Matrix multiplication is defined by
$$ AB = C \Leftrightarrow {C_{ij}} = \sum\limits_{k = 1}^n {{a_{ik}}{b_{kj}}} $$
Wolfgang Walter
Chapter V. Complex Linear Systems
Abstract
The subject of this chapter is the homogeneous linear system
$$ w'(z) = A(z)w(z), $$
(1)
where w(z) = (w1(z),... wn(z))T is a complex-valued vector function and A(z)=(aij (z)) is a complex-valued n x n matrix. We also investigate homogeneous linear differential equations of higher order. Let G C C be open and denote by H(G) the complex linear space of functions that are single-valued and holomorphic on \( w(z) \in H(G)orA(z) \in H(G) \) if every component Wi (z) or aij (z) belongs to H(G). Compatible norms for complex column vectors and n x n matrices will be denoted by single vertical bars, and the properties (14.2-3),
$$ \left| {AB} \right| \leq \left| A \right|\left| B \right|{\text{ }}and{\text{ }}\left| {Aw} \right| \leq \left| A \right|\left| W \right| $$
are taken for granted. Throughout this chapter, matrices are understood to be complex n x n matrices.
Wolfgang Walter
Chapter VI. Boundary Value and Eigenvalue Problems
Abstract
In a boundary value problem for an nth order differential equation
$$ {u^{(n)}} = f(x,u,...,{u^{(n - 1)}}), $$
the rt additional conditions that (we expect to) define a solution uniquely are not prescribed at a single point a, as in the case of the initial value problem, but at two points a and b that are the endpoints of the interval a < x < b where the solution is considered. Boundary value problems for (real) linear second order equations
$$ u'' + {a_1}\left( x \right)u' + {a_0}\left( x \right)u = g\left( x \right){\text{ }}for{\text{ }}a{\text{ }} \leq x \leq b $$
(1)
are particularly important because of numerous applications in science and technology.
Wolfgang Walter
Chapter VII. Stability and Asymptotic Behavior
Abstract
We resume the problems treated in §12. In contrast to the case investigated there, we now consider solutions defined on infinite intervals. In this setting, continuous dependence on initial conditions and on the right side of the differential equation is a significantly more complicated matter than in §12, where general results were obtained under restricted assumptions. Even in the simplest examples, new phenomena emerge when the interval is infinite.
Wolfgang Walter
Backmatter
Metadaten
Titel
Ordinary Differential Equations
verfasst von
Wolfgang Walter
Copyright-Jahr
1998
Verlag
Springer New York
Electronic ISBN
978-1-4612-0601-9
Print ISBN
978-1-4612-6834-5
DOI
https://doi.org/10.1007/978-1-4612-0601-9