I.
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The one-dimensional wave equation |
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1. |
A physical problem and its mathematical models: the vibrating string |
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2. |
The one-dimensional wave equation |
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3. |
Discussion of the solution: characteristics |
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4. |
Reflection and the free boundary problem |
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5. |
The nonhomogeneous wave equation |
II. |
Linear second-order partial differential equations in two variables |
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6. |
Linearity and superposition |
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7. |
Uniqueness for the vibrating string problem |
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8. |
Classification of second-order equations with constant coefficients |
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9. |
Classification of general second-order operators |
III. |
Some properties of elliptic and parabolic equations |
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10. |
Laplace's equation |
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11. |
Green's theorem and uniqueness for the Laplace's equation |
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12. |
The maximum principle |
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13. |
The heat equation |
IV. |
Separation of variables and Fourier series |
|
14. |
The method of separation of variables |
|
15. |
Orthogonality and least square approximation |
|
16. |
Completeness and the Parseval equation |
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17. |
The Riemann-Lebesgue lemma |
|
18. |
Convergence of the trigonometric Fourier series |
|
19. |
"Uniform convergence, Schwarz's inequality, and completeness" |
|
20. |
Sine and cosine series |
|
21. |
Change of scale |
|
22. |
The heat equation |
|
23. |
Laplace's equation in a rectangle |
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24. |
Laplace's equation in a circle |
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25. |
An extension of the validity of these solutions |
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26. |
The damped wave equation |
V. |
Nonhomogeneous problems |
|
27. |
Initial value problems for ordinary differential equations |
|
28. |
Boundary value problems and Green's function for ordinary differential equations |
|
29. |
Nonhomogeneous problems and the finite Fourier transform |
|
30. |
Green's function |
VI. |
Problems in higher dimensions and multiple Fourier series |
|
31. |
Multiple Fourier series |
|
32. |
Laplace's equation in a cube |
|
33. |
Laplace's equation in a cylinder |
|
34. |
The three-dimensional wave equation in a cube |
|
35. |
Poisson's equation in a cube |
VII. |
Sturm-Liouville theory and general Fourier expansions |
|
36. |
Eigenfunction expansions for regular second-order ordinary differential equations |
|
37. |
Vibration of a variable string |
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38. |
Some properties of eigenvalues and eigenfunctions |
|
39. |
Equations with singular endpoints |
|
40. |
Some properties of Bessel functions |
|
41. |
Vibration of a circular membrane |
|
42. |
Forced vibration of a circular membrane: natural frequencies and resonance |
|
43. |
The Legendre polynomials and associated Legendre functions |
|
44. |
Laplace's equation in the sphere |
|
45. |
Poisson's equation and Green's function for the sphere |
VIII. |
Analytic functions of a complex variable |
|
46. |
Complex numbers |
|
47. |
Complex power series and harmonic functions |
|
48. |
Analytic functions |
|
49. |
Contour integrals and Cauchy's theorem |
|
50. |
Composition of analytic functions |
|
51. |
Taylor series of composite functions |
|
52. |
Conformal mapping and Laplace's equation |
|
53. |
The bilinear transformation |
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54. |
Laplace's equation on unbounded domains |
|
55. |
Some special conformal mappings |
|
56. |
The Cauchy integral representation and Liouville's theorem |
IX. |
Evaluation of integrals by complex variable methods |
|
57. |
Singularities of analytic functions |
|
58. |
The calculus of residues |
|
59. |
Laurent series |
|
60. |
Infinite integrals |
|
61. |
Infinite series of residues |
|
62. |
Integrals along branch cuts |
X. |
The Fourier transform |
|
63. |
The Fourier transform |
|
64. |
Jordan's lemma |
|
65. |
Schwarz's inequality and the triangle inequality for infinite integrals |
|
66. |
Fourier transforms of square integrable functions: the Parseval equation |
|
67. |
Fourier inversion theorems |
|
68. |
Sine and cosine transforms |
|
69. |
Some operational formulas |
|
70. |
The convolution product |
|
71. |
Multiple Fourier transforms: the heat equation in three dimensions |
|
72. |
The three-dimensional wave equation |
|
73. |
The Fourier transform with complex argument |
XI. |
The Laplace transform |
|
74. |
The Laplace transform |
|
75. |
Initial value problems for ordinary differential equations |
|
76. |
Initial value problems for the one-dimensional heat equation |
|
77. |
A diffraction problem |
|
78. |
The Stokes rule and Duhamel's principle |
XII. |
Approximation methods |
|
79. |
"Exact" and approximate solutions" |
|
80. |
The method of finite differences for initial-boundary value problems |
|
81. |
The finite difference method for Laplace's equation |
|
82. |
The method of successive approximations |
|
83. |
The Raleigh-Ritz method |
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SOLUTIONS TO THE EXERCISES |
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INDEX |
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