Knobel R. An Introduction to the Mathematical Theory of Waves
размером 1,94 МБ
Providence: American Mathematical Society, 1999. —212 p.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.Contents:
Foreword Preface Introduction Introduction to Waves Wave phenomena Examples of waves A Mathematical Representation of Waves Representation of one-dimensional waves Methods for visualizing functions of two variables ial Differential Equations Introduction and examples An intuitive view Terminology Traveling and Standing Waves Traveling Waves Traveling waves Wave fronts and pulses Wave trains and dispersion The Korteweg-de Vries Equation The KdV equation Solitary wave solutions The Sine-Gordon Equation A mechanical transmission line The Sine-Gordon equation Traveling wave solutions The Wave Equation Vibrating strings A derivation of the wave equation Solutions of the wave equation D'Alembert's Solution of the Wave Equation General solution of the wave equation The d'Alembert form of a solution Vibrations of a Semi-infinite String A semi-infinite string with fixed end A semi-infinite string with free end Characteristic Lines of the Wave Equation Domain of dependence and range of influence Characteristics and solutions of the wave equation Solutions of the semi-infinite problem Standing Wave Solutions of the Wave Equation Standing waves Standing wave solutions of the wave equation Standing waves of a finite string Modes of vibration Standing Waves of a Nonhomogeneous String The wave equation for a nonhomogeneous string Standing waves of a finite string Modes of vibration Numerical calculation of natural frequencies Superposition of Standing Waves Finite superposition Infinite superposition Fourier Series and the Wave Equation Fourier sine series Fourier series solution of the wave equation Waves in Conservation Laws Conservation Laws Derivation of a general scalar conservation law Constitutive equations Examples of Conservation Laws Plug flow chemical reactor Diffusion Traffic flow The Method of Characteristics Advection equation Nonhomogeneous advection equation General linear conservation laws Nonlinear conservation laws Gradient Catastrophes and Breaking Times Gradient catastrophe Breaking time Shock Waves Piecewise smooth solutions of a conservation law Shock wave solutions of a conservation law Shock Wave Example: Traffic at a Red Light An initial value problem Shock wave solution Shock Waves and the Viscosity Method Another model of traffic flow Traveling wave solutions of the new model Viscosity Rarefaction Waves An example of a rarefaction wave Stopped traffic at a green light An Example with Rarefaction and Shock Waves Nonunique Solutions and the Entropy Condition Nonuniqueness of piecewise smooth solutions The entropy condition Weak Solutions of Conservation Laws Classical solutions The weak form of a conservation law Bibliography Index
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