Partial differential equations 偏微分方程

Intended for a college senior or first-year graduate-level course in partial differential equations, this text fills the gap between elementary and more sophisticated books.Students of mathematics, engineering, and the applied sciences, with a background in advanced calculus and ordinary differential equations, will find this volume an excellent introduction to the beauty and power of partial differential equations. It is also the source of a solid foundation for advanced studies in mathematics.
Classical topics presented in a modern context include coverage of integral equa-tions and basic scattering theory. Complete and accessible treatment of the theory of partial differential equations includes a variety of examples of inverse problems arising from improperly posed applications, including a discussion of an inverse scattering problem in acoustic wave propagation. This treatment exposes stu-dents to an active area of research with striking philosophical implications.Exercises at the ends of chapters, many with answers, offer a clear progression in developing an understanding of this essential area of mathematics.

Chapter 1 INTRODUCTION
1.1 Physical Examples
1.2 First Order Linear Equations
1.3 Classification of Second Order Equations and Canonical Forms
Types of Second Order Equations
Reduction of Second Order Equations with Constant Coefficients to Canonical Form
Reduction of Second Order Equations in Two Independent Variables to Canonical Form
1.4 Fourier Series and Integrals
1.5 Analytic Functions
Power Series and Analytic Functions
Integration of Analytic Functions
Singularities of Analytic Functions and the Residue Theorem
A Linear Partial Differential Equation with No Solution
1.6 A Brief History of the Theory of Partial Differential Equations
Chapter 2 THE WAVE EQUATION
2.1 The Wave Equation in Two Independent Variables
2.2 The Cauchy Problem for Hyperbolic Equations in Two Independent Variables
2.3 The Cauchy Problem for the Wave Equation in More Than Two Independent Variables
2.4 The Initial-Boundary Value Problem for the Wave Equation in Two Independent Variables
2.5 Fourier's Method for the Wave Equation in Three Independent Variables
2.6 The Equations of Gas Dynamics
Chapter 3 THE HEAT EQUATION
3.1 The Weak Maximum Principle for Parabolic Equatioffs
3.2 The Initial-Boundary Value Problem for the Heat Equation in a Rectangle
3.3 Cauchy's Problem for the Heat Equation
3.4 Regularity of Solutions to the Heat Equation
*3.5 The Strong Maximum Principle for the Heat Equation
*3.6 The Stefan Problem and Analytic Continuation
3.7 Hermite Polynomials and the Numerical Solution of the Heat Equation in a Rectangle
3.8 Nonlinear Problems in Heat Conduction
Chapter 4 LAPLACE'S EQUATION
4.1 Green's Formulas
4.2 Basic Properties of Harmonic Functions
4.3 Boundary Value Problems for Laplace's Equation
4.4 Separation of Variables in Polar and Spherical Coordinates
4.5 Green's Function and Poisson's Formula
4.6 Finite Difference Methods for Laplace's Equation
4.7 Poisson's Equation
4.8 Time Harmonic Wave Propagation in a Nonhomogeneous Medium
Chapter 5 POTENTIAL THEORY AND FREDHOLM INTEGRAL EQUATIONS
5.1 Potential Theory
5.2 The Fredholm Alternative
5.3 Applications to the Dirichlet and Neumann Problems
The Interior Diriehlet Problem
The Interior Neumann Problem
The Exterior Neumann Problem
The Exterior Dirichlet Problem
5.4 Hilbert-Schmidt Theory and Eigenvalue Problems
5.5 The Numerical Solution of Fredholm Integral Equations of the Second Kind
Chapter 6 SCATTERING THEORY
REFERENCES
INDEX