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 ProblemsChapter 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 ProblemChapter 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 ProblemsChapter 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") Problems 5.Solution of the Dirichlet Problem by Hilbert-Space Methods ProblemsChapter 5 Hyperbolic Equations in Higher Dimensions 1.The Wave Equation in n-Dimensional Space (a) The method of spherical means Problems (b) Hadamard's method of descent Problems (c)Duhamel's principle and the general Cauchy problem Problem (d)Initial-boundary-value problems ("Mixed" problems) Problems2.Higher-Order Hyperbolic Equations with Constant Coefficients (a)Standard form of the initial-value problem Problem (b)Solution by Fourier transformation Problems (c)Solution of a mixed problem by Fourier transformation (d)The method of plane waves Problems3.Symmetric Hyperbolic Systems (a)The basic energy inequality Problems (b)Existence of solutions by the method of finite differences Problems (c)Existence of solutions by the method of approximation by analytic functions (Method of Schauder)Chapter 6 Higher-Order Elliptic Equations with Constant Coefficients 1.The Fundamental Solution for Odd n Problems 2.The Dirichlet Problem Problems 3.More on the Hilbert Space H~ and the Assumption of Boundary Values in the Dirichlet Problem ProblemsChapter 7Parabolic Equations 1.The Heat Equation (a)The initial-value problem Problems (b)Maximum principle, uniqueness, and regularity Problem (c)A mixed problem Problems (d)Non-negative solutions Problems 2.The Initial-Value Problem for General Second-Order Linear Parabolic Equations (a)The method of finite differences and the maximum principle (b)Existence of solutions of the initial-value problem ProblemsH. Lewy's Example of a Linear Equation without Solutions ProblemsBibliographyGlossaryIndex
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