Partial differential equations / Fritz John.

Includes index.

Ch.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 Variâbles -- 8. The Cauchy Problem -- 9. Solutions Generated as Envelopes Problems -- Ch. 2 Second-Order Equations: Hyperbolic Equations for Functions of Two Independent Variâbles -- 1. Characteristîcs for Linear and Quasi-linear Second-order Equation -- 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 Manİfolds 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) Mulliple infınite series Problems -- (b) Real analytic functions Problems -- (c) Analytic and real analytic functions Problems -- (d)The proof of the Cauchy-Kowalevski ıheorem Problems -- 4. The Lagrange-Green Identity -- 5. The Uniqueness Theorem of Holmgren Problems -- 6. Distribution Solutions Problems -- Ch. 4 The Laplace Equation -- 1. Green's İdentity, 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 Problems -- Ch. 5 Hyperbolic Equations in Higher Dimensions -- 1. The Wave Equation in /ı-Dimensional Space -- (a) The rnethod 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) Problems -- 2. Higher-Order Hyperbolic Equations with Constant Coefficients -- (a) Standard form of the initîal-value problem Problem -- (b) Solution by Fourier transformation Problems -- (c) Solution of a mixed problem by Fourier transformation -- (d) The method of plane waves Problems -- 3. 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) -- Ch. 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 and the Assumption of Boundary Values in the Dirichfet Problem Problems -- Ch. 7 Parabolic Equations -- 1.The Heat Equation -- (a) The üıitial-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 -- Ch. 8 H. Lewy's Example of a Linear Equation without Solutions Problems