Author | John, Fritz. author |
---|---|
Title | Partial Differential Equations [electronic resource] / by Fritz John |
Imprint | New York, NY : Springer US, 1978 |
Edition | Third Edition |
Connect to | http://dx.doi.org/10.1007/978-1-4684-0059-5 |
Descript | online resource |
1 The single first-order equation -- 1. Introduction -- 2. Examples -- 3. Analytic Solution and Approximation Methods in a Simple Example -- 4. Quasi-linear Equations -- 5. The Cauchy Problem for the Quasi-linear Equation -- 6. Examples -- 7. The General First-order Equation for a Function of Two Variables -- 8. The Cauchy Problem -- 9. Solutions Generated as Envelopes -- 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 -- 4. The One-Dimensional Wave Equation -- 5. Systems of First-order Equations (Courant-Lax Theory) -- 6. A Quasi-linear System and Simple Waves -- 3 Characteristic manifolds and the Cauchy problem -- 1. Notation of Laurent Schwartz -- 2. The Cauchy Problem -- 3. Cauchy-Kowalewski Theorem -- 4. The Lagrange-Green Identity -- 5. The Uniqueness Theorem of Holmgren -- 6. Distribution Solutions -- 4 The Laplace equation -- 1. Greenโs Identity, Fundamental Solutions -- 2. The Maximum Principle -- 3. The Dirichlet Problem, Greenโs Function, and Poissonโs Formula -- 4. Proof of Existence of Solutions for the Dirichlet Problem Using Subharmonic Functions (โPerronโs Methodโ) -- 5. Solution of the Dirichlet Problem by Hilbert-Space Methods -- 5 Hyperbolic equations in higher dimensions -- 1. The Wave Equation in n-dimensional Space -- 2. Higher-order Hyperbolic Equations with Constant Coefficients -- 3. Symmetric Hyperbolic Systems -- 6 Higher-order elliptic equations with constant coefficients -- 1. The Fundamental Solution for Odd n -- 2. The Dirichlet Problem -- 7 Parabolic equations -- 1. The Heat Equation -- 2. The Initial-value Problem for General Second-order Linear Parabolic Equations