Chapter 1 The Single First-Order Equation
1.Introduction
2.Examples
3.Analytic Solution and Approximation Methods in a Simple Example
Problems
4.Quasi-linear Equations
5.The Cauchy Problem for the Quasi-linear Equation
6.Examples
Problems
7.The General First-Order Equation for a Function of Two Variables
8.The Cauchy Problem
9.Solutions Generated as Envelopes
Problems
Chapter 2 Second-Order Equations: Hyperbolic Equations for Functions of Two Independent Variables
1.Characteristics for Linear and Quasi-linear Second-order Equations
2.Propagation of Singularities
3.The Linear Second-Order Equation
Problems
4.The One-Dimensional Wave Equation
Problems
5.Systems of First-Order Equations
6.A Quasi-linear System and Simple Waves
Problem
Chapter 3 Characteristic Manifolds and the Cauchy Problem
1.Notation of Laurent Schwartz
Problems
2.The Cauchy Problem
Problems
3.Real Analytic Functions and the Cauchy-Kowalevski Theorem
(a) Multiple infinite series
Problems
(b) Real analytic functions
Problems
(c) Analytic and real analytic functions
Problems
(d) The proof of the Cauchy-Kowalevski theorem
Problems
4.The Lagrange-Green Identity
5. The Uniqueness Theorem of Holmgren
Problems
6.Distribution Solutions
Problems
Chapter 4 The Laplace Equation
1.Green's Identity, Fundamental Solutions, and Poisson's Equation
Problems
2.The Maximum Principle
Problems
3.The Dirichlet Problem, Green's Function, and Poisson's Formula
Problems
4.Proof of Existence of Solutions for the Dirichlet Problem Using Subharmonic Functions ("Perron's Method")
