Partial Differential Equations An Introduction

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ISBN: 9780486438344 | 0486438341
Cover: Paperback
Copyright: 11/30/2004

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This text offers students in mathematics, engineering, and the applied sciences a solid foundation for advanced studies in mathematics. Classical topics presented in a modern context include coverage of integral equations and basic scattering theory. Includes examples of inverse problems arising from improperly posed applications as well as exercises, many with answers. 1988 edition.

Introduction

(60)

Physical Examples

(3)

First Order Linear Equations

(5)

Classification of Second Order Equations and Canonical Forms

(7)

Types of Second Order Equations

(1)

Reduction of Second Order Equations with Constant Coefficients to Canonical Form

(1)

Reduction of Second Order Equations in Two Independent Variables to Canonical Form

(5)

Fourier Series and Integrals

(11)

Analytic Functions

(20)

Power Series and Analytic Functions

(4)

Integration of Analytic Functions

(6)

Singularities of Analytic Functions and the Residue Theorem

(7)

A Linear Partial Differential Equation with No Solution

(3)

A Brief History of the Theory of Partial Differential Equations

(12)

The Wave Equation

(45)

The Wave Equation in Two Independent Variables

(3)

The Cauchy Problem for Hyperbolic Equations in Two Independent Variables

(5)

The Cauchy Problem for the Wave Equation in More Than Two Independent Variables

(8)

The Initial-Boundary Value Problem for the Wave Equation in Two Independent Variables

(5)

Fourier's Method for the Wave Equation in Three Independent Variables

(7)

The Equations of Gas Dynamics

(16)

The Heat Equation

106

(45)

The Weak Maximum Principle for Parabolic Equations

106

(6)

The Initial-Boundary Value Problem for the Heat Equation in a Rectangle

112

(3)

Cauchy's Problem for the Heat Equation

115

(8)

Regularity of Solutions to the Heat Equation

123

(3)

The Strong Maximum Principle for the Heat Equation

126

(4)

The Stefan Problem and Analytic Continuation

130

(4)

Hermite Polynomials and the Numerical Solution of the Heat Equation in a Rectangle

134

(2)

Nonlinear Problems in Heat Conduction

136

(15)

Laplace's Equation

151

(44)

Green's Formulas

152

(2)

Basic Properties of Harmonic Functions

154

(3)

Boundary Value Problems for Laplace's Equation

157

(2)

Separation of Variables in Polar and Spherical Coordinates

159

(7)

Green's Function and Poisson's Formula

166

(7)

Finite Difference Methods for Laplace's Equation

173

(2)

Poisson's Equation

175

(6)

Time Harmonic Wave Propagation in a Nonhomogeneous Medium

181

(14)

Potential Theory and Fredholm Integral Equations

195

(48)

Potential Theory

196

(10)

The Fredholm Alternative

206

(9)

Applications to the Dirichlet and Neumann Problems

215

(7)

The Interior Dirichlet Problem

216

(1)

The Interior Neumann Problem

217

(1)

The Exterior Neumann Problem

218

(2)

The Exterior Dirichlet Problem

220

(2)

Hilbert-Schmidt Theory and Eigenvalue Problems

222

(13)

The Numerical Solution of Fredholm Integral Equations of the Second Kind

235

(8)

Scattering Theory

243

(57)

The Gamma Function

244

(4)

Bessel Functions

248

(12)

Definitions

248

(1)

Wronskians

249

(1)

A Generating Function for Jn(z)

250

(1)

Integral Representations

251

(2)

Asymptotic Expansions for Positive z

253

(5)

Addition Formulas

258

(2)

The Scattering of Acoustic Waves

260

(4)

Maxwell's Equations

264

(3)

Scattering by a Cylinder of Arbitrary Cross Section

267

(10)

The Inverse Scattering Problem

277

(23)

Preliminaries

277

(3)

Herglotz Wave Functions

280

(5)

Optimal Solutions to the Inverse Scattering Problem

285

(15)

References

300

(5)

Index

305

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Partial Differential Equations An Introduction

Summary

Table of Contents

Supplemental Materials