Partial differential equations / Fritz John ; [editors, Fritz John ... et al.].
Series Applied mathematical sciences (Springer-Verlag New York Inc.): v. 1.Editor: New York : Springer-Verlag, c1978Edición: 3rd edDescripción: ix, 198 p. : il. ; 25 cmISBN: 0387903275Tema(s): Differential equations, PartialOtra clasificación: 35-01 (35-02)Chapter [1] The single first-order equation [1] 1. Introduction [1] 2. Examples [2] 3. Analytic Solution and Approximation Methods in a Simple Example [4] Problems [8] 4. Quasi-linear Equations [8] 5. The Cauchy Problem for the Quasi-linear Equation [10] 6. Examples [14] Problems [18] 7. The General First-order Equation for a Function of Two Variables [19] 8. The Cauchy Problem [23] 9. Solutions Generated as Envelopes [28] Problems [30] Chapter [2] Second-order equations: hyperbolic equations for functions of two independent variables [31] 1. Characteristics for Linear and Quasi-linear Second-order Equations [31] 2. Propagation of Singularities [33] 3. The Linear Second-order Equation [35] Problems [37] 4. The One-Dimensional Wave Equation [38] Problems [43] 5. Systems of First-order Equations (Courant-Lax Theory) [44] 6. A Quasi-linear System and Simple Waves [50] Problem [51] Chapter [3] Characteristic manifolds and the Cauchy problem [52] 1. Notation of Laurent Schwartz [52] Problems [53] 2. The Cauchy Problem [54] Problems [59] 3. Cauchy-Kowalewski Theorem [59] Problems [63] 4. The Lagrange-Green Identity [64] 5. The Uniqueness Theorem of Holmgren [65] 6. Distribution Solutions [67] Problems [70] Chapter [4] The Laplace equation [72] 1. Green’s Identity, Fundamental Solutions [72] Problems [79] 2. The Maximum Principle [81] Problems [83] 3. The Dirichlet Problem, Green’s Function, and Poisson’s Formula [84] Problems [88] 4. Proof of Existence of Solutions for the Dirichlet Problem Using Subharmonic Functions (“Perron’s Method”) [89] Problem [94] 5. Solution of the Dirichlet Problem by Hilbert-Space Methods [94] Problems [101] Chapter [5] Hyperbolic equations in higher dimensions [103] 1. The Wave Equation in n-dimensional Space [103] (a) The method of spherical means [103] Problems [109] (b) Hadamard’s method of descent [110] Problems [111] (c) Duhamel’s principle and the general Cauchy problem [112] Problem [116] (d) Initial-boundary-value problems (“Mixed” problems) [116] Problems [119] 2. Higher-order Hyperbolic Equations with Constant Coefficients [120] (a) Standard form of the initial-value problem [120] Problem [122] (b) Solution by Fourier transformation [122] Problems [132] (c) Solution of a mixed problem by Fourier transformation [133] (d) The method of plane waves [135] Problems [138] 3. Symmetric Hyperbolic Systems [139] (a) The basic energy inequality [139] Problems [145] (b) Existence of solutions by the method of finite differences [146] Problems [155] Chapter [6] Higher-order elliptic equations with constant coefficients [156] 1. The Fundamental Solution for Odd n [157] Problems [159] 2. The Dirichlet Problem [160] Problems [164] Chapter [7] Parabolic equations [166] 1. The Heat Equation [166] (a) The initial-value problem [166] Problems [173] (b) Maximum principle, uniqueness, and regularity [174] Problems [179] (c) A mixed problem [179] Problems [181] 2. The Initial-value Problem for General Second-order Linear Parabolic Equations [181] (a) The method of finite differences and the maximum principle [181] (b) Existence of solutions of the initial-value problem [185] Problems [188] Bibliography [191] Glossary [193] Index [195]
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Incluye índice.
Bibliografía: p. 191-192.
Chapter [1] --
The single first-order equation [1] --
1. Introduction [1] --
2. Examples [2] --
3. Analytic Solution and Approximation Methods in a Simple --
Example [4] --
Problems [8] --
4. Quasi-linear Equations [8] --
5. The Cauchy Problem for the Quasi-linear Equation [10] --
6. Examples [14] --
Problems [18] --
7. The General First-order Equation for a Function of Two Variables [19] --
8. The Cauchy Problem [23] --
9. Solutions Generated as Envelopes [28] --
Problems [30] --
Chapter [2] --
Second-order equations: hyperbolic equations for functions of two independent variables [31] --
1. Characteristics for Linear and Quasi-linear Second-order --
Equations [31] --
2. Propagation of Singularities [33] --
3. The Linear Second-order Equation [35] --
Problems [37] --
4. The One-Dimensional Wave Equation [38] --
Problems [43] --
5. Systems of First-order Equations --
(Courant-Lax Theory) [44] --
6. A Quasi-linear System and Simple Waves [50] --
Problem [51] --
Chapter [3] --
Characteristic manifolds and the Cauchy problem [52] --
1. Notation of Laurent Schwartz [52] --
Problems [53] --
2. The Cauchy Problem [54] --
Problems [59] --
3. Cauchy-Kowalewski Theorem [59] --
Problems [63] --
4. The Lagrange-Green Identity [64] --
5. The Uniqueness Theorem of Holmgren [65] --
6. Distribution Solutions [67] --
Problems [70] --
Chapter [4] --
The Laplace equation [72] --
1. Green’s Identity, Fundamental Solutions [72] --
Problems [79] --
2. The Maximum Principle [81] --
Problems [83] --
3. The Dirichlet Problem, Green’s Function, and Poisson’s Formula [84] --
Problems [88] --
4. Proof of Existence of Solutions for the Dirichlet Problem Using Subharmonic Functions (“Perron’s Method”) [89] --
Problem [94] --
5. Solution of the Dirichlet Problem by Hilbert-Space Methods [94] --
Problems [101] --
Chapter [5] --
Hyperbolic equations in higher dimensions [103] --
1. The Wave Equation in n-dimensional Space [103] --
(a) The method of spherical means [103] --
Problems [109] --
(b) Hadamard’s method of descent [110] --
Problems [111] --
(c) Duhamel’s principle and the general Cauchy problem [112] --
Problem [116] --
(d) Initial-boundary-value problems (“Mixed” problems) [116] --
Problems [119] --
2. Higher-order Hyperbolic Equations with Constant Coefficients [120] --
(a) Standard form of the initial-value problem [120] --
Problem [122] --
(b) Solution by Fourier transformation [122] --
Problems [132] --
(c) Solution of a mixed problem by Fourier transformation [133] --
(d) The method of plane waves [135] --
Problems [138] --
3. Symmetric Hyperbolic Systems [139] --
(a) The basic energy inequality [139] --
Problems [145] --
(b) Existence of solutions by the method of finite differences [146] --
Problems [155] --
Chapter [6] --
Higher-order elliptic equations with constant coefficients [156] --
1. The Fundamental Solution for Odd n [157] --
Problems [159] --
2. The Dirichlet Problem [160] --
Problems [164] --
Chapter [7] --
Parabolic equations [166] --
1. The Heat Equation [166] --
(a) The initial-value problem [166] --
Problems [173] --
(b) Maximum principle, uniqueness, and regularity [174] --
Problems [179] --
(c) A mixed problem [179] --
Problems [181] --
2. The Initial-value Problem for General Second-order Linear --
Parabolic Equations [181] --
(a) The method of finite differences and the maximum principle [181] --
(b) Existence of solutions of the initial-value problem [185] --
Problems [188] --
Bibliography [191] --
Glossary [193] --
Index [195] --
MR, 80f:35001
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