A First Look at Perturbation Theory

James G. Simmonds, James E. Mann, James E. Mann

• Publisher: Dover Publications
• Year: 2013
• Print ISBN: 9780486675510
• eBook ISBN: 9780486315584
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Undergraduates in engineering and the physical sciences receive a thorough introduction to perturbation theory in this useful and accessible text. Students discover methods for obtaining an approximate solution of a mathematical problem by exploiting the presence of a small, dimensionless parameter — the smaller the parameter, the more accurate the approximate solution. Knowledge of perturbation theory offers a twofold benefit: approximate solutions often reveal the exact solution's essential dependence on specified parameters; also, some problems resistant to numerical solutions may yield to perturbation methods. In fact, numerical and perturbation methods can be combined in a complementary way.
The text opens with a well-defined treatment of finding the roots of polynomials whose coefficients contain a small parameter. Proceeding to differential equations, the authors explain many techniques for handling perturbations that reorder the equations or involve an unbounded independent variable. Two disparate practical problems that can be solved efficiently with perturbation methods conclude the volume.
Written in an informal style that moves from specific examples to general principles, this elementary text emphasizes the "why" along with the "how"; prerequisites include a knowledge of one-variable calculus and ordinary differential equations. This newly revised second edition features an additional appendix concerning the approximate evaluation of integrals.

Chapter 1

Introduction and Overview

Perturbation theory is the study of the effects of small disturbances. If the effects are small, the disturbances or perturbations are said to be regular; otherwise, they are said to be singular. The basic idea in perturbation theory is to obtain an approximate solution of a mathematical problem by exploiting the presence of a small dimensionless parameter—the smaller the parameter, the more accurate the approximate solution.

Regular perturbations are assumed nearly every time we construct a mathematical model of a real world phenomenon. Our choice of language reflects this: the flow is almost steady, the density varies essentially with altitude only, the conductivity is virtually independent of temperature, the spring is nearly linear, the friction is practically negligible.

Singular perturbations are probably less familiar. Fig. 1.1 illustrates two examples. The top in Fig. 1.1a is set spinning rapidly about a vertical axis. During one revolution, the effects of aerodynamic drag and the friction at the tip are small (regular perturbations). Eventually, though, the top falls and comes to rest in a position far from its initial one. Thus, over a long period of time, the perturbations are singular. Models such as this are characterized by what may be called a singularity in the domain. For the top, this means that we are interested in w...

Table of contents

• Cover
• Title Page
• Copyright
• Dedication
• Contents
• Preface
• 1 Introduction and Overview
• 2 Roots of Polynomials
• 3 Singular Perturbations in Ordinary Differential Equations
• 4 Periodic Solutions of the Simplest Nonlinear Differential Equations. Poincaré’s Method
• 5 Introduction to the Two-Scale Method
• 6 The WKB Approximation
• 7 Transition Point Problems and Langer’s Method of Uniform Approximation
• 8 Introduction to Boundary Layer Theory
• 9 Cables and Cells: Ancient and Modern Problems
• Bibliography
• A Roots of Tε(z) and T0(z)
• B Proof that RN+1 = O (βN+1)
• C Approximate Evaluation of Integrals
• Index