[图书简介]
One of the most fundamental and active areas in mathematics, the theory of partial differential equations (PDEs) is essential in the modeling of natural phenomena. PDEs have a wide range of interesting and important applications in every branch of applied mathematics, physics, and engineering, including fluid dynamics, elasticity, and optics.
This significantly expanded fourth edition is designed as an introduction to the theory and applications of linear PDEs. The authors provide fundamental concepts, underlying principles, a wide range of applications, and various methods of solutions to PDEs. In addition to essential standard material on the subject, the book contains new material that is not usually covered in similar texts and reference books, including conservation laws, the spherical wave equation, the cylindrical wave equation, higher-dimensional boundary-value problems, the finite element method, fractional partial differential equations, and nonlinear partial differential equations with applications.
Key features include:
* Applications to a wide variety of physical problems in numerous interdisciplinary areas
* Over 900 worked examples and exercises dealing with problems in fluid mechanics, gas dynamics, optics, plasma physics, elasticity, biology, and chemistry
* Historical comments on partial differential equations
* Solutions and hints to selected exercises
* A comprehensive bibliography?comprised of many standard texts and reference books, as well as a set of selected classic and recent papers?for readers interested in learning more about the modern treatment of the subject
Linear Partial Differential Equations for Scientists and Engineers, Fourth Edition will primarily serve as a textbook for the first two courses in PDEs, or in a course on advanced engineering mathematics. The book may also be used as a reference for graduate students, researchers, and professionals in modern applied mathematics, mathematical physics, and engineering. Readers will gain a solid mathematical background in PDEs, sufficient to start interdisciplinary collaborative research in a variety of fields.
[Table Of Contents]

Preface to the Fourth Edition

Preface to the Third Edition

       1 Introduction

             1.1 Brief Historical Comments

             1.2 Basic Concepts and Definitions

             1.3 Mathematical Problems

             1.4 Linear Operators

             1.5 Superposition Principle

             1.6 Exercises

       2 First-Order, Quasi-Linear Equations and Method of Characteristics

             2.1 Introduction

             2.2 Classification of First-Order Equations

             2.3 Construction of a First-Order Equation

             2.4 Geometrical Interpretation of a First-Order Equation

             2.5 Method of Characteristics and General Solutions

             2.6 Canonical Forms of First-Order Linear Equations

             2.7 Method of Separation of Variables

             2.8 Exercises

       3 Mathematical Models

             3.1 Classical Equations

             3.2 The Vibrating String

             3.3 The Vibrating Membrane

             3.4 Waves in an Elastic Medium

             3.5 Conduction of Heat in Solids

             3.6 The Gravitational Potential

             3.7 Conservation Laws and The Burgers Equation

             3.8 The SchrOdinger and the Korteweg?e Vries Equations

             3.9 Exercises

       4 Classification of Second-Order Linear Equations

             4.1 Second-Order Equations in Two Independent Variables

             4.2 Canonical Forms

             4.3 Equations with Constant Coefficients

             4.4 General Solutions

             4.5 Summary and Further Simplification

             4.6 Exercises

       5 The Cauchy Problem and Wave Equations

             5.1 The Cauchy Problem

             5.2 The Cauchy-Kowalewskaya Theorem

             5.3 Homogeneous Wave Equations

             5.4 Initial Boundary-Value Problems

             5.5 Equations with Nonhomogeneous Boundary Conditions

             5.6 Vibration of Finite String with Fixed Ends

             5.7 Nonhomogeneous Wave Equations

             5.8 The Riemann Method

             5.9 Solution of the Goursat Problem

             5.10 Spherical Wave Equation

             5.11 Cylindrical Wave Equation

             5.12 Exercises

       6 Fourier Series and Integrals with Applications

             6.1 Introduction

             6.2 Piecewise Continuous Functions and Periodic Functions

             6.3 Systems of Orthogonal Functions

             6.4 Fourier Series

             6.5 Convergence of Fourier Series

             6.6 Examples and Applications of Fourier Series

             6.7 Examples and Applications of Cosine and Sine Fourier Series

             6.8 Complex Fourier Series

             6.9 Fourier Series on an Arbitrary Interval

             6.10 The Riemann-Lebesgue Lemma and Pointwise Convergence Theorem

             6.11 Uniform Convergence, Differentiation, and Integration

             6.12 Double Fourier Series

             6.13 Fourier Integrals

             6.14 Exercises

       7 Method of Separation of Variables

             7.1 Introduction

             7.2 Separation of Variables

             7.3 The Vibrating String Problem

             7.4 Existence and Uniqueness of Solution of the Vibrating String Problem

             7.5 The Heat Conduction Problem

             7.6 Existence and Uniqueness of Solution of the Heat Conduction Problem

             7.7 The Laplace and Beam Equations

             7.8 Nonhomogeneous Problems

             7.9 Exercises

       8 Eigenvalue Problems and Special Functions

             8.1 Sturm- Liouville Systems

             8.2 Eigenvalues and Eigenfunctions

             8.3 Eigenfunction Expansions

             8.4 Convergence in the Mean

             8.5 Completeness and Parseval's Equality

             8.6 Bessel's Equation and Bessel's Function

             8.7 Adjoint Forms and Lagrange Identity

             8.8 Singular Sturm-Liouville Systems

             8.9 Legendre's Equation and Legendre's Function

             8.10 Boundary-Value Problems Involving Ordinary Differential Equations

             8.11 Green's Functions for Ordinary Differential Equations

             8.12 Construction of Green's Functions

             8.13 The Schr?inger Equation and Linear Harmonic Oscillator

             8.14 Exercises

       9 Boundary-Value Problems and Applications

             9.1 Boundary-Value Problems

             9.2 Maximum and Minimum Principles

             9.3 Uniqueness and Continuity Theorems

             9.4 Dirichlet Problem for a Circle

             9.5 Dirichlet Problem for a Circular Annulus

             9.6 Neumann Problem for a Circle

             9.7 Dirichlet Problem for a Rectangle

             9.8 Dirichlet Problem Involving the Poisson Equation

             9.9 The Neumann Problem for a Rectangle

             9.10 Exercises

10 Higher-Dimensional Boundary-Value Problems

             10.1 Introduction

             10.2 Dirichlet Problem for a Cube

             10.3 Dirichlet Problem for a Cylinder

             10.4 Dirichlet Problem for a Sphere

             10.5 Three-Dimensional Wave and Heat Equations

             10.6 Vibrating Membrane

             10.7 Heat Flow in a Rectangular Plate

             10.8 Waves in Three Dimensions

             10.9 Heat Conduction in a Rectangular Volume

             10.10 The Schriidinger Equation and the Hydrogen Atom

             10.11 Method of Eigenfunctions and Vibration of Membrane

             10.12 Time-Dependent Boundary-Value Problems

             10.13 Exercises

11 Green's Functions and Boundary-Value Problems

             11.1 Introduction

             11.2 The Dirac Delta Function

             11.3 Properties of Green's Functions

             11.4 Method of Green's Functions

             11.5 Dirichlet's Problem for the Laplace Operator

             11.6 Dirichlet's Problem for the Helmholtz Operator

             11.7 Method of Images

             11.8 Method of Eigenfunctions

             11.9 Higher-Dimensional Problems

             11.10 Neumann Problem

             11.11 Exercises

12 Integral Transform Methods with Applications

             12.1 Introduction

             12.2 Fourier Transforms

             12.3 Properties of Fourier Transforms

             12.4 Convolution Theorem of the Fourier Transform

             12.5 The Fourier Transforms of Step and Impulse Functions

             12.6 Fourier Sine and Cosine Transforms

             12.7 Asymptotic Approximation of Integrals by Stationary Phase Method

             12.8 Laplace Transforms

             12.9 Properties of Laplace Transforms

             12.10 Convolution Theorem of the Laplace Transform

             12.11 Laplace Transforms of the Heaviside and Dirac Delta Functions

             12.12 Hankel Transforms

             12.13 Properties of Hankel Transforms and Applications

             12.14 Mellin Transforms and their Operational Properties

             12.15 Finite Fourier Transforms and Applications

             12.16 Finite Hankel Transforms and Applications

             12.17 Solution of Fractional Partial Differential Equations

             12.18 Exercises

13 Nonlinear Partial Differential Equations with Applications

             13.1 Introduction

             13.2 One-Dimensional Wave Equation and Method of Characteristics

             13.3 Linear Dispersive Waves

             13.4 Nonlinear Dispersive Waves and Whitham's Equations

             13.5 Nonlinear Instability

             13.6 The Traffic Flow Model

             13.7 Flood Waves in Rivers

             13.8 Riemann's Simple Waves of Finite Amplitude

             13.9 Discontinuous Solutions and Shock Waves

             13.10 Structure of Shock Waves and Burgers' Equation

             13.11 The Korteweg-de Vries Equation and Solitons

             13.12 The Nonlinear Schr?inger Equation and Solitary Waves

             13.13 The Lax Pair and the Zakharov and Shabat Scheme

             13.14 Exercises

14 Numerical and Approximation Methods

             14.1 Introduction

             14.2 Finite Difference Approximations, Convergence, and Stability

             14.3 Lax-Wendroff Explicit Method

             14.4 Explicit Finite Difference Methods

             14.5 Implicit Finite Difference Methods

             14.6 Variational Methods and the Euler-Lagrange Equations

             14.7 The Rayleigh-Ritz Approximation Method

             14.8 The Galerkin Approximation Method

             14.9 The Kantorovich Method

             14.10 The Finite Element Method

             14.11 Exercises

15 Tables of Integral Transforms

             15.1 Fourier Transforms

             15.2 Fourier Sine Transforms

             15.3 Fourier Cosine Transforms

             15.4 Laplace Transforms

             15.5 Hankel Transforms

             15.6 Finite Hankel Transforms

Answers and Hints to Selected Exercises

                           1.6 Exercises

                           2.8 Exercises

                           3.9 Exercises

                           4.6 Exercises

                    5.12 Exercises

                    6.14 Exercises

                           7.9 Exercises

                    8.14 Exercises

                    9.10 Exercises

             10.13 Exercises

             11.11 Exercises

             12.18 Exercises

             14.11 Exercises

Appendix: Some Special Functions and Their Properties

             A-1 Gamma, Beta, Error. and Airy Functions

             A-2 Hermite Polynomials and Weber-Hermite Functions

Bibliography

Index