Mathematical Methods for Wave Phenomena (Computer Science and Applied Mathematics)

✍ Scribed by Norman Bleistein

• Publisher: Academic Press
• Year: 1984
• Tongue: English
• Leaves: 357
• Edition: 1
• Category: Library

✦ Synopsis

Computer Science and Applied Mathematics: Mathematical Methods for Wave Phenomena focuses on the methods of applied mathematics, including equations, wave fronts, boundary value problems, and scattering problems. The publication initially ponders on first-order partial differential equations, Dirac delta function, Fourier transforms, asymptotics, and second-order partial differential equations. Discussions focus on prototype second-order equations, asymptotic expansions, asymptotic expansions of Fourier integrals with monotonic phase, method of stationary phase, propagation of wave fronts, and variable index of refraction. The text then examines wave equation in one space dimension, as well as initial boundary value problems, characteristics for the wave equation in one space dimension, and asymptotic solution of the Klein-Gordon equation. The manuscript offers information on wave equation in two and three dimensions and Helmholtz equation and other elliptic equations. Topics include energy integral, domain of dependence, and uniqueness, scattering problems, Green's functions, and problems in unbounded domains and the Sommerfeld radiation condition. The asymptotic techniques for direct scattering problems and the inverse methods for reflector imaging are also elaborated. The text is a dependable reference for computer science experts and mathematicians pursuing studies on the mathematical methods of wave phenomena.

✦ Table of Contents

Title page
Copyright page
CONTENTS
Preface
1 FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS
1.1 First-Order Quasi-Linear Differential Equations
1.2 An Illustrative Example
1.3 First-Order Nonlinear Differential Equations
1.4 Examples—The Eikonal Equation—and More Theory
1.5 Propagation of Wave Fronts
1.6 Variable Index of Refraction
1.7 Higher Dimensions
References
2 THE DIRAC DELTA FUNCTION, FOURIER TRANSFORMS, AND ASYMPTOTICS
2.1 The Dirac Delta Function and Related Distributions
2.2 Fourier Transforms
2.3 Fourier Transforms of Distributions
2.4 Multidimensional Fourier Transforms
2.5 Asymptotic Expansions
2.6 Asymptotic Expansions of Fourier Integrals with Monotonic Phase
2.7 The Method of Stationary Phase
2.8 Multidimensional Fourier Integrals
References
3 SECOND-ORDER PARTIAL DIFFERENTIAL EQUATIONS
3.1 Prototype Second-Order Equations
3.2 Some Simple Examples
References
4 THE WAVE EQUATION IN ONE SPACE DIMENSION
4.1 Characteristics for the Wave Equation in One Space Dimension
4.2 The Initial Boundary Value Problem
4.3 The Initial Boundary Value Problem Continued
4.4 The Adjoint Equation and the Riemann Function
4.5 The Green's Function
4.6 Asymptotic Solution of the Klein-Gordon Equation
4.7 More on Asymptotic Solutions
References
5 THE WAVE EQUATION IN TWO AND THREE DIMENSIONS
5.1 Characteristics and Ill-Posed Cauchy Problems
5.2 The Energy Integral, Domain of Dependence, and Uniqueness
5.3 The Green's Function
5.4 Scattering Problems
References
6 THE HELMHOLTZ EQUATION AND OTHER ELLIPTIC EQUATIONS
6.1 Green's Identities and Uniqueness Results
6.2 Some Special Features of Laplace's Equation
6.3 Green's Functions
6.4 Problems in Unbounded Domains and the Sommerfeld Radiation Condition
6.5 Some Exact Solutions
References
7 MORE ON ASYMPTOTICS
7.1 Watson's Lemma
7.2 The Method of Steepest Descents: Preliminary Results
7.3 Formulas for the Method of Steepest Descents
7.4 The Method of Steepest Descents: Implementation
References
8 ASYMPTOTIC TECHNIQUES FOR DIRECT SCATTERING PROBLEMS
8.1 Scattering by a Half-Space: Analysis by Steepest Descents
8.2 Introduction to Ray Methods
8.3 Determination of Ray Data
8.4 The Kirchhoff Approximation
References
9 INVERSE METHODS FOR REFLECTOR IMAGING
9.1 The Singular Function and the Characteristic Function
9.2 Physical Optics Far-Field Inverse Scattering (POFFIS)
9.3 The Seismic Inverse Problem
References
Index