Introduction to Perturbation Methods

by

Mark H. Holmes

Department of Mathematical Sciences

Rensselaer Polytechnic Institute

Troy, NY 12180

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Preface

First, let me say hello and welcome to the subject of perturbation methods. For those who may be unfamiliar with the topic, the title can be confusing. The first time I became aware of this was during a family reunion when someone asked what I did as a mathematician. This is not an easy question to answer but I started by describing how a certain segment of the applied mathematics community was interested in problems that arise from physical problems. Examples such as water waves, sound propagation, and the aerodynamics of airplanes were discussed. The difficulty of solving such problems was also described in exaggerated detail. Next came the part about how one generally ends up using a computer to actually find the solution. At this point I editorialized on the limitations of computer solutions and why it is important to derive, if at all possible, accurate approximations of the solution. This lead naturally to the mentioning of asymptotics and perturbation methods. These terms ended the conservation because I was unprepared for their reactions. They were not sure exactly what asymptotics meant and they were quite perplexed about perturbation methods. I tried, unsuccessfully, to explain what it means but it was not until sometime later that I realized the difficulty. For them, as in Webster's Collegiate Dictionary, the first two meanings for the word perturb are "to disturb greatly in mind (disquiet); to throw into confusion (disorder)." Although a cynic might suggest this is indeed appropriate for the subject, the intent is exactly the opposite. For a related comment, see Exercise 3.2.1(d).

In a nutshell, this book serves as an introduction into how to systematically construct an approximation of the solution of a problem that is otherwise untractable. The methods all rely on there being a parameter in the problem that is relatively small. Such a situation is relatively common in applications and this is one of the reasons that perturbation methods are a cornerstone of applied mathematics. One of the other cornerstones is scientific computing and it is interesting that the two subjects have grown up together. However, this is not unexpected given their respective capabilities. When using a computer one is capable of solving problems that are nonlinear, inhomogeneous, and multi-dimensional. Moreover, it is possible to achieve very high accuracy. The drawbacks are that computer solutions do not provide much insight into the physics of the problem (particularly for those who do not have access to the appropriate software or computer), and there is always the question as to whether or not the computed solution is correct. On the other hand, perturbation methods are also capable of dealing with nonlinear, inhomogeneous, and multi-dimensional problems (although not to the same extent as computer generated solutions). The principle objective when using perturbation methods, at least as far as the author is concerned, is to provide a reasonably accurate expression for the solution. By doing this one is able to derive an understanding of the physics of the problem. Also, one can use the result, in conjunction with the original problem, to obtain more efficient numerical procedures for computing the solution.

The methods covered in the text vary widely in their applicability. The first chapter introduces the fundamental ideas underlying asymptotic approximations. This includes their use in constructing approximate solutions of transcendental equations as well as differential equations. In the second chapter, matched asymptotic expansions are used to analyze problems with layers. Chapter Three describes a method for dealing with problems with more than one time scale. In Chapter Four, the WKB method for analyzing linear singular perturbation problems is developed, while in Chapter Five a method for dealing with materials containing disparate spatial scales (e.g., microscopic vs. macroscopic) is discussed. The last chapter examines the topics of multiple solutions and stability.

The mathematical prerequisites for this text include a basic background in differential equations and advanced calculus. In terms of difficulty, the chapters are written so the first sections are either elementary or intermediate, while the later sections are somewhat more advanced. Also, the ideas developed in each chapter are applied to a spectrum of problems, including ordinary differential equations, partial differential equations, and difference equations. Scattered through the exercises are applications to integral equations, integro-differential equations, differential-difference equations, and delay equations. What will not be found is an in-depth discussion of the theory underlying the methods. This aspect of the subject is important and references to the more theoretical work in the area are given in each chapter.

The exercises in each section vary in their complexity. In addition to the more standard textbook problems, an attempt has been made to include problems from the research literature. The latter are intended to provide a window into the wide range of areas which use perturbation methods. Solutions to some of the exercises are available and they can be obtained, at no charge, from ftp.rpi.edu using anonymous ftp. Also included, in the same file, is an errata sheet. Those who may want to make a contribution to this file, or have suggestions about the text, can reach the author at holmes@rpi.edu .

I would like to express my gratitude to the many students who took my course in perturbation methods at Rensselaer. They helped me immeasurably in understanding the subject and provided much needed encouragement to write this book. It is a pleasure to acknowledge the suggestions of Jon Bell, Ash Kapila, and Bob O'Malley, who read early versions of the manuscript. I would also like to thank Julian Cole, who first introduced me to perturbation methods and is still, to this day, showing me what the subject is about.

Mark H. Holmes

Troy, New York

August, 1994

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TABLE OF CONTENTS

Chapter 1: Introduction to Asymptotic Approximations

1.1 Introduction

1.2 Taylor's Theorem and l'Hospital's Rule

1.3 Order Symbols

1.4 Asymptotic Approximations

```	1.4.1  Asymptotic Expansions
1.4.2  Accuracy vs Convergence of an Asymptotic Series
1.4.3  Manipulating Asymptotic Expansions
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1.5 Asymptotic Solution of Algebraic and Transcendental Equations

1.6 Introduction to the Asymptotic Solution of Differential Equations

```	Example: The Projectile Problem
Example: A Nonlinear Potential Problem
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1.7 Uniformity

1.8 Symbolic Computing

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Chapter 2: Matched Asymptotic Expansions

2.1 Introduction

2.2 Introductory Example

2.3 Examples With Multiple Boundary Layers

2.4 Interior Layers

2.5 Corner Layers

2.6 Partial Differential Equations

```	Example 1.  Elliptic Problem
Outer Expansion
Boundary Layer Expansion
Composite Expansion
Parabolic Boundary Layer
Example 2.  Parabolic Problem
Outer Expansion
Inner Expansion
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2.7 Difference Equations

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Chapter 3: Multiple-Scales

3.1 Introduction

3.2 Introductory Example

```	Regular Expansion
Multiple-Scale Expansion
Three Times Scales
Discussion and Observations
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3.3 Slowly Varying Coefficients

3.4 Forced Motion Near Resonance

3.5 Boundary Layers

3.6 Introduction to Partial Differential Equations

3.7 Linear Wave Propagation

3.8 Nonlinear Waves

```	Example 1: Nonlinear Wave Equation
Example 2: Nonlinear Diffusion
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3.9 Difference Equations

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Chapter 4: The WKB and Related Methods

4.1 Introduction

4.2 Introductory Example

```	Examples
Second Term of the Expansion
General Discussion
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4.3 Turning Points

```	Solution in Transition Layer
Matching
Matching for  x > x[sub(t)]
Matching for  x < x[sub(t)]
Summary
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4.4 Wave Propagation and Energy Methods

```	Energy Methods
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4.5 Wave Propagation and Slender Body Approximations

```	Solution in Transition Region
Matching
Example
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4.6 Ray Methods

4.7 Parabolic Approximations

4.8 Discrete WKB Method

```	Turning Points
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Chapter 5: The Method of Homogenization

5.1 Introduction

5.2 Introductory Example

```	Properties of the Averaging Procedure
Summary
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5.3 Multidimensional Problem

```	Periodic Substructure
Implications of Periodicity
Homogenization Procedure
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5.4 Porous Flow

```	Reduction Using Homogenization
Averaging
Homogenized Problem
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Chapter 6: Introduction to Bifurcation and Stability

6.1 Introduction

6.2 Introductory Example

6.3 Analysis of a Bifurcation Point

6.4 Linearized Stability

6.5 Relaxation Dynamics

```	Outer Expansion
Initial Layer Expansion
Corner Layer Expansion
Interior Layer Expansion
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6.6 An Example Involving A Nonlinear Partial Differential Equation

```	Steady State Solutions
Linearized Stability Analysis
Stability of Zero Solution
Stability of the Branches that Bifurcate from the Zero Solution
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6.7 Bifurcation of Periodic Solutions

6.8 Systems of Ordinary Differential Equations

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Appendix 1: Solution and Properties of Transition Layer Equations

```	A.1.1  Airy Functions
A.1.2  Confluent Hypergeometric Functions
A.1.3  Higher Order Turning Points
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Appendix 2: Asymptotic Approximations of Integrals

Appendix 3: Numerical Solution of Nonlinear BVPs

References

Index