A first course in partial differential equations with complex variables and transform methods / H. F. Weinberger.

Por: Weinberger, Hans FSeries A Blaisdell book in pure and applied mathematicsEditor: New York : Blaisdell, c1965Edición: [1st ed.]Descripción: ix, 446 p. ; 26 cmTema(s): Differential equations, PartialOtra clasificación: 35-01
Contenidos:
I. The one-dimensional wave equation
1. A physical problem and its mathematical models: the vibrating string [1]
2. The one-dimensional wave equation [8]
3. Discussion of the solution: characteristics [18]
4. Reflection and the free boundary problem [21]
5. The nonhomogeneous wave equation [24]
II. Linear second-order partial differential equations in two variables
6. Linearity and superposition [29]
7. Uniqueness for the vibrating string problem [36]
8. Classification of second-order equations with constant coefficients [41]
9. Classification of general second-order operators [44]
III. Some properties of elliptic and parabolic equations
10. Laplace’s equation [48]
11. Green’s theorem and uniqueness for the Laplace’s equation [52]
^12. The maximum principle [55]
*'13. The heat equation [58]
IV. Separation of variables and Fourier series
14. The method of separation of variables [63]
15. Orthogonality and least square approximation [70]
16. Completeness and the Parseval equation [73]
17. The Riemann-Lebesgue lemma [76]
8. Convergence of the trigonometric Fourier series [77]
19. Uniform convergence, Schwarz’s inequality, and completeness [81]
20. Sine and cosine series [87]
21. Change of scale [88]
22. The heat equation [92]
23. Laplace’s equation in a rectangle [95]
24. Laplace’s equation in a circle [100]
25. An extension of the validity of these solutions [105]
26. The damped wave equation [112]
V. Nonhomogeneous problems
27. Initial value problems for ordinary differential equations [117]
28. Boundary value problems and Green’s function for ordinary differential equations [120]
29. Nonhomogeneous problems and the finite Fourier transform [126]
30. Green’s function [132]
VI. Problems in higher dimensions and multiple Fourier series
31. Multiple Fourier series [141]
32. Laplace’s equation in a cube [146]
33. Laplace’s equation in a cylinder [149]
34. The three-dimensional wave equation in a cube [152]
35. Poisson’s equation in a cube [155]
VII. Sturm-Liouville theory and general Fourier expansions
36. Eigenfunction expansions for regular second-order ordinary differential equations [160]
37. Vibration of a variable string [169]
38. Some properties of eigenvalues and eigenfunctions [171]
39. Equations with singular endpoints [176]
40. Some properties of Bessel functions [179]
41. Vibration of a circular membrane [182]
42. Forced vibration of a circular membrane: natural frequencies and resonance [185]
43. The Legendre polynomials and associated Legendre functions [188]
44. Laplace’s equation in the sphere [194]
45. Poisson’s equation and Green’s function for the sphere [197]
VIII. Analytic functions of a complex variable
46. Complex numbers [201]
47. Complex power series and harmonic functions [207]
48. Analytic functions [213]
49. Contour integrals and Cauchy’s theorem [218]
50. Composition of analytic functions [225]
51. Taylor series of composite functions [231]
52. Conformal mapping and Laplace’s equation [236]
53. The bilinear transformation [246]
54. Laplace’s equation on unbounded domains [253]
55. Some special conformal mappings [257]
56. The Cauchy integral representation and Liouville’s theorem [261]
IX. Evaluation of integrals by complex variable methods
57. Singularities of analytic functions [269]
58. The calculus of residues [271]
59. Laurent series [278]
60. Infinite integrals [282]
61. Infinite series of residues [289]
62. Integrals along branch cuts [293]
X. The Fourier transform
63. The Fourier transform [298]
64. Jordan’s lemma [302]
65. Schwarz’s inequality and the triangle inequality for infinite integrals [305]
66. Fourier transforms of square integrable functions: the Parseval equation [310]
67. Fourier inversion theorems [313]
68. Sine and cosine transforms [320]
69. Some operational formulas [324]
70. The convolution product [326]
71. Multiple Fourier transforms: the heat equation in three dimensions [329]
72. The three-dimensional wave equation [333]
73. The Fourier transform with complex argument [337]
XI. The Laplace transform
74. The Laplace transform [346]
75. Initial value problems for ordinary differential equations [351]
76. Initial value problems for the one-dimensional heat equation [355]
77. A diffraction problem [362]
78. The Stokes rule and Duhamel’s principle [370]
XII. Approximation methods
79. “Exact” and approximate solutions [374]
80. The method of finite differences for initial-boundary value problems [375]
81. The finite difference method for Laplace’s equation [380]
82. The method of successive approximations [384]
83. The Rayleigh-Ritz method [392]
SOLUTIONS TO THE EXERCISES [443]
INDEX [399]
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Libros ordenados por tema 35 W423 (Browse shelf) Available A-2659

ECUACIONES DIFERENCIALES


Incluye referencias bibliográficas.

I. The one-dimensional wave equation --
1. A physical problem and its mathematical models: the vibrating string [1] --
2. The one-dimensional wave equation [8] --
3. Discussion of the solution: characteristics [18] --
4. Reflection and the free boundary problem [21] --
5. The nonhomogeneous wave equation [24] --
II. Linear second-order partial differential equations in two variables --
6. Linearity and superposition [29] --
7. Uniqueness for the vibrating string problem [36] --
8. Classification of second-order equations with constant coefficients [41] --
9. Classification of general second-order operators [44] --
III. Some properties of elliptic and parabolic equations --
10. Laplace’s equation [48] --
11. Green’s theorem and uniqueness for the Laplace’s equation [52] --
^12. The maximum principle [55] --
*'13. The heat equation [58] --
IV. Separation of variables and Fourier series --
14. The method of separation of variables [63] --
15. Orthogonality and least square approximation [70] --
16. Completeness and the Parseval equation [73] --
17. The Riemann-Lebesgue lemma [76] --
8. Convergence of the trigonometric Fourier series [77] --
19. Uniform convergence, Schwarz’s inequality, and completeness [81] --
20. Sine and cosine series [87] --
21. Change of scale [88] --
22. The heat equation [92] --
23. Laplace’s equation in a rectangle [95] --
24. Laplace’s equation in a circle [100] --
25. An extension of the validity of these solutions [105] --
26. The damped wave equation [112] --
V. Nonhomogeneous problems --
27. Initial value problems for ordinary differential equations [117] --
28. Boundary value problems and Green’s function for ordinary differential equations [120] --
29. Nonhomogeneous problems and the finite Fourier transform [126] --
30. Green’s function [132] --
VI. Problems in higher dimensions and multiple Fourier series --
31. Multiple Fourier series [141] --
32. Laplace’s equation in a cube [146] --
33. Laplace’s equation in a cylinder [149] --
34. The three-dimensional wave equation in a cube [152] --
35. Poisson’s equation in a cube [155] --
VII. Sturm-Liouville theory and general Fourier expansions --
36. Eigenfunction expansions for regular second-order ordinary differential equations [160] --
37. Vibration of a variable string [169] --
38. Some properties of eigenvalues and eigenfunctions [171] --
39. Equations with singular endpoints [176] --
40. Some properties of Bessel functions [179] --
41. Vibration of a circular membrane [182] --
42. Forced vibration of a circular membrane: natural frequencies and resonance [185] --
43. The Legendre polynomials and associated Legendre functions [188] --
44. Laplace’s equation in the sphere [194] --
45. Poisson’s equation and Green’s function for the sphere [197] --
VIII. Analytic functions of a complex variable --
46. Complex numbers [201] --
47. Complex power series and harmonic functions [207] --
48. Analytic functions [213] --
49. Contour integrals and Cauchy’s theorem [218] --
50. Composition of analytic functions [225] --
51. Taylor series of composite functions [231] --
52. Conformal mapping and Laplace’s equation [236] --
53. The bilinear transformation [246] --
54. Laplace’s equation on unbounded domains [253] --
55. Some special conformal mappings [257] --
56. The Cauchy integral representation and Liouville’s theorem [261] --
IX. Evaluation of integrals by complex variable methods --
57. Singularities of analytic functions [269] --
58. The calculus of residues [271] --
59. Laurent series [278] --
60. Infinite integrals [282] --
61. Infinite series of residues [289] --
62. Integrals along branch cuts [293] --
X. The Fourier transform --
63. The Fourier transform [298] --
64. Jordan’s lemma [302] --
65. Schwarz’s inequality and the triangle inequality for infinite integrals [305] --
66. Fourier transforms of square integrable functions: the Parseval equation [310] --
67. Fourier inversion theorems [313] --
68. Sine and cosine transforms [320] --
69. Some operational formulas [324] --
70. The convolution product [326] --
71. Multiple Fourier transforms: the heat equation in three dimensions [329] --
72. The three-dimensional wave equation [333] --
73. The Fourier transform with complex argument [337] --
XI. The Laplace transform --
74. The Laplace transform [346] --
75. Initial value problems for ordinary differential equations [351] --
76. Initial value problems for the one-dimensional heat equation [355] --
77. A diffraction problem [362] --
78. The Stokes rule and Duhamel’s principle [370] --
XII. Approximation methods --
79. “Exact” and approximate solutions [374] --
80. The method of finite differences for initial-boundary value problems [375] --
81. The finite difference method for Laplace’s equation [380] --
82. The method of successive approximations [384] --
83. The Rayleigh-Ritz method [392] --
SOLUTIONS TO THE EXERCISES [443] --
INDEX [399] --

MR, 31 #4969

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