Partial differential equations / Fritz John ; [editors, Fritz John ... et al.].

Por: John, Fritz, 1910-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)
Contenidos:
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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