**Introduction to Perturbation Methods**

**by**

**Mark H. Holmes**

**Department of Mathematical Sciences**

**Rensselaer Polytechnic Institute**

**Troy, NY 12180**

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

**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 Expansions1.5 Asymptotic Solution of Algebraic and Transcendental Equations1.4.2 Accuracy vs Convergence of an Asymptotic Series

1.4.3 Manipulating Asymptotic Expansions

1.6 Introduction to the Asymptotic Solution of Differential Equations

Example: The Projectile Problem1.7 UniformityExample: A Nonlinear Potential Problem

1.8 Symbolic Computing

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 Problem2.7 Difference EquationsOuter Expansion

Boundary Layer Expansion

Composite Expansion

Parabolic Boundary Layer

Example 2. Parabolic Problem

Outer Expansion

Inner Expansion

3.1 Introduction

3.2 Introductory Example

Regular Expansion3.3 Slowly Varying CoefficientsMultiple-Scale Expansion

Three Times Scales

Discussion and Observations

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 Equation3.9 Difference EquationsExample 2: Nonlinear Diffusion

4.1 Introduction

4.2 Introductory Example

Examples4.3 Turning PointsSecond Term of the Expansion

General Discussion

Solution in Transition Layer4.4 Wave Propagation and Energy MethodsMatching

Matching for x > x[sub(t)]

Matching for x < x[sub(t)]

Summary

Energy Methods4.5 Wave Propagation and Slender Body Approximations

Solution in Transition Region4.6 Ray MethodsMatching

Example

4.7 Parabolic Approximations

4.8 Discrete WKB Method

Turning Points

5.1 Introduction

5.2 Introductory Example

Properties of the Averaging Procedure5.3 Multidimensional ProblemSummary

Periodic Substructure5.4 Porous FlowImplications of Periodicity

Homogenization Procedure

Reduction Using HomogenizationAveraging

Homogenized Problem

6.1 Introduction

6.2 Introductory Example

6.3 Analysis of a Bifurcation Point

6.4 Linearized Stability

6.5 Relaxation Dynamics

Outer Expansion6.6 An Example Involving A Nonlinear Partial Differential EquationInitial Layer Expansion

Corner Layer Expansion

Interior Layer Expansion

Steady State Solutions6.7 Bifurcation of Periodic SolutionsLinearized Stability Analysis

Stability of Zero Solution

Stability of the Branches that Bifurcate from the Zero Solution

6.8 Systems of Ordinary Differential Equations

A.1.1 Airy FunctionsA.1.2 Confluent Hypergeometric Functions

A.1.3 Higher Order Turning Points

**Appendix 2: Asymptotic Approximations of Integrals**

**Appendix 3: Numerical Solution of Nonlinear BVPs**

**References**

**Index**