A first course in partial differential equations with complex variables and transform methods / H. F. Weinberger.
Series 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: 35-01I. 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]
| Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 35 W423 (Browse shelf) | Checked out | 2024-04-22 | A-2659 |
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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