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## Über dieses Buch

This book is a revised and updated version, including a substantial portion of new material, of J. D. Cole's text Perturbation Methods in Applied Mathe­ matics, Ginn-Blaisdell, 1968. We present the material at a level which assumes some familiarity with the basics of ordinary and partial differential equations. Some of the more advanced ideas are reviewed as needed; therefore this book can serve as a text in either an advanced undergraduate course or a graduate level course on the subject. The applied mathematician, attempting to understand or solve a physical problem, very often uses a perturbation procedure. In doing this, he usually draws on a backlog of experience gained from the solution of similar examples rather than on some general theory of perturbations. The aim of this book is to survey these perturbation methods, especially in connection with differ­ ential equations, in order to illustrate certain general features common to many examples. The basic ideas, however, are also applicable to integral equations, integrodifferential equations, and even to_difference equations. In essence, a perturbation procedure consists of constructing the solution for a problem involving a small parameter B, either in the differential equation or the boundary conditions or both, when the solution for the limiting case B = 0 is known. The main mathematical tool used is asymptotic expansion with respect to a suitable asymptotic sequence of functions of B.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
We will use the conventional order symbols as a mathematical measure of the relative order of magnitude of various quantities. Although generalizations are straightforward, we need only be concerned with scalar functions of real variables. In the definitions which follow φ, ψ, etc. are scalar functions of the variable x (which may be a vector) and the scalar parameter ε. The variable x ranges over some domain D and ε belongs to some interval I.
J. Kevorkian, J. D. Cole

### Chapter 2. Limit Process Expansions Applied to Ordinary Differential Equations

Abstract
In this chapter a series of simple examples are considered, some model and some physical, in order to demonstrate the application of various techniques concerning limit process expansions. In general we expect analytic dependence of the exact solution on the small parameter ε, but one of the main tasks in the various problems is to discover the nature of this dependence by working with suitable approximate differential equations. Another problem is to systematize as much as possible the procedures for discovering these expansions.
J. Kevorkian, J. D. Cole

### Chapter 3. Multiple-Variable Expansion Procedures

Abstract
Various physical problems are characterized by the presence of a small disturbance which, because of being active over a long time, has a non-negligible cumulative effect. For example, the effect of a small damping force over many periods of oscillation is to produce a decay in the amplitude of a linear oscillator. A fancier example having the same physical and mathematical features is the motion of a satellite around the earth, where the dominant force is a spherically symmetric gravitational field. If this were the only force acting on the satellite the motion would (for sufficiently low energies) be periodic. The presence of a thin atmosphere, a slightly non-spherical earth, a small moon, a distant sun, etc. all produce small but cumulative effects which after a sufficient number of orbits drastically alter the nature of the motion.
J. Kevorkian, J. D. Cole

### Chapter 4. Applications to Partial Differential Equations

Abstract
In this chapter, the methods developed previously are applied to partial differential equations. The plan is the same as for the cases of ordinary differential equations discussed earlier. First, the very simplest case is discussed, in which a singular perturbation problem arises. This is a second-order equation which becomes a first-order one in the limit ε → 0. Following this, various more complicated physical examples of singular perturbations and boundary-layer theory are discussed. Next, the ideas of matching and inner and outer expansions are applied in some problems that are analogous to the singular boundary problems of Section 2.7. The final section deals with multiple variable expansions for partial differential equations, and several applications dealing with different aspects of the procedure are discussed.
J. Kevorkian, J. D. Cole

### Chapter 5. Examples from Fluid Mechanics

Abstract
In this chapter we consider some examples from fluid mechanics with the aim of illustrating the derivation of approximate equations, the role of several small parameters, and the calculation of uniformly valid results. Strictly speaking, there is no difference with the material discussed in Chapter 4. The emphasis here is on applications, so that we will not be dealing with model equations, but rather with nonlinear systems which reduce in the first approximation to some well known equation such as the transonic equation or the Korteweg-de Vries equation, etc. Typical to many problems is the occurrence of several small parameters. We will show how one must specify certain relations between these small parameters to account systematically for various competing perturbation terms and to derive meaningful approximations.
J. Kevorkian, J. D. Cole

### Backmatter

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