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Summary
Summary
This book is based on an undergraduate course taught at the IAS/Park City Mathematics Institute (Utah) on linear and nonlinear waves. The first part of the text overviews the concept of a wave, describes one-dimensional waves using functions of two variables, provides an introduction to partial differential equations, and discusses computer-aided visualization techniques. The second part of the book discusses traveling waves, leading to a description of solitary waves and soliton solutions of the Klein-Gordon and Korteweg-deVries equations. The wave equation is derived to model the small vibrations of a taut string, and solutions are constructed via d'Alembert's formula and Fourier series.The last part of the book discusses waves arising from conservation laws. After deriving and discussing the scalar conservation law, its solution is described using the method of characteristics, leading to the formation of shock and rarefaction waves. Applications of these concepts are then given for models of traffic flow. The intent of this book is to create a text suitable for independent study by undergraduate students in mathematics, engineering, and science. The content of the book is meant to be self-contained, requiring no special reference material. Access to computer software such as MathematicaR, MATLABR, or MapleR is recommended, but not necessary. Scripts for MATLAB applications will be available via the Web. Exercises are given within the text to allow further practice with selected topics.
Author Notes
Robert Wooster, an associate professor of history at Texas A&M University-Corpus Christi, is the author of Nelson A. Miles and the Twilight of the Frontier Army (Nebraska 1993).
Reviews 1
Choice Review
Though the study of ordinary differential equations (ODEs) forms a standard part of the undergraduate curriculum in mathematics and engineering, a serious penetration of partial differential equations (PDEs) involves an altogether higher order of sophistication. Interesting PDEs rarely admit closed solutions, so formulating suitable questions requires care; issues of existence and uniqueness of solutions of PDEs require advanced tools from analysis and geometry; numerical simulation of PDEs (not addressed in this book) consumes large amounts of time, memory, and often requires sophisticated data structures. These notes form a stepping-stone from courses on ODEs to books like J.D. Logan's An Introduction to Nonlinear Partial Differential Equations (1994) or G.B. Whitham's Linear and Nonlinear Waves (1974). By concentrating only on the motion of one-dimensional waves, Knobel can present the issues and techniques characteristic of the subject in fewer than 200 pages. Though the mathematical theory mostly addresses the solutions of a given equation, undergraduates must learn to formulate the right equation to solve, based on given physical assumptions. This book's greatest strength is its careful evocation of the patterns of thought required for such modeling. A Web site offers additional supplementary material. Recommended for college libraries. Undergraduates through faculty. D. V. Feldman; University of New Hampshire
Table of Contents
| Introduction |
| Introduction to waves |
| A mathematical representation of waves |
| Partial differential equation |
| Traveling and standing waves: Traveling waves |
| The Korteweg-de Vries equation |
| The Sine-Gordon equation |
| The wave equation D'Alembert's solution of the wave equation |
| Vibrations of a semi-infinite string |
| Characteristic lines of the wave equation |
| Standing wave solutions of the wave equation |
| Standing waves of a nonhomogeneous string |
| Superposition of standing waves |
| Fourier series and the wave equation |
| Waves in conservation laws: Conservation laws |
| Examples of conservation laws |
| The method of characteristics |
| Gradient catastrophes and breaking times |
| Shock waves Shock wave example: Traffic at a red light |
| Shock waves and the viscosity method |
| Rarefaction waves An example with rarefaction and shock waves |
| Nonunique solutions and the entropy condition |
| Weak solutions of conservation laws |
| Bibliography |
| Index |
