Fourier Series and Boundary Value Problems

✍ Scribed by James Ward Brown, Ruel V. Churchill

• Publisher: Mcgraw-Hill College
• Year: 1993
• Tongue: English
• Leaves: 366
• Series: International Series in Pure and Applied Mathematics
• Edition: 5th
• Category: Library

✦ Synopsis

An introductory treatment of Fourier series and their applications to boundary value problems in partial equations that arise in engineering and physics. This revision incorporates up-to-date mathematics. Many sections have been rewritten to improve the motivation of the theory, and numerous illustrations and exercises have been added throughout the book. The new emphasis on solving boundary value problems with non-homogenous differential equations should benefit students who are faced with a wider range of physics and engineering problems than in the past. The introduction to the Sturm-Louisvill problems is motivated by several applications to boundary value problems.

✦ Table of Contents

Cover......Page 1
S Title......Page 2
List of Publications......Page 4
Picture of Joseph Fourier......Page 6
FOURIER SERIES AND BOUNDARY VALUE ROBLEMS, Fifth Edition......Page 8
0A404.B76 1993 515'.2433—dc2O......Page 9
ABOUTTHE AUTHORS......Page 10
Dedications......Page 11
JOSEPH FOURIER......Page 12
CONTENTS......Page 13
PREFACE......Page 15
CHAPTER 1 PARTIALD IFFERENTIAL EQUATIONS OF PHYSICS......Page 19
1. LINEAR BOUNDARY VALUE PROBLEMS......Page 20
2. CONDUCTION OF HEAT......Page 22
3. HIGHER DIMENSIONS AND BOUNDARY CONDITIONS......Page 24
4. THE LAPLACIAN IN CYLINDRICAL AND SPHERICAL COORDINATES......Page 27
PROBLEMS......Page 31
5. A VIBRATING STRING......Page 34
9 VIBRATIONS OF BARS AND MEMBRANES......Page 38
PROBLEMS......Page 40
7. TYPES OF EQUATIONS AND BOUNDARY CONDITIONS......Page 42
8. METHODS OF SOLUTION......Page 44
9. ON THE SUPERPOSITION OF SEPARATED SOLUTIONS......Page 47
PROBLEMS......Page 50
10. PIECEWISE CONTINUOUS FUNCTIONS......Page 57
11. INNER PRODUCTS AND ORTHONORMAL SETS......Page 60
PROBLEMS......Page 63
12. GENERALIZED FOURIER SERIES......Page 65
13. FOURIER COSINE SERIES......Page 68
14. FOURIER SINE SERIES......Page 70
PROBLEMS......Page 73
15. FOURIER SERIES......Page 75
16. BEST APPROXIMATION IN THE MEAN......Page 79
PROBLEMS......Page 83
17. ONE-SIDED DERIVATIVES......Page 85
18. TWO LEMMAS......Page 89
19. A FOURIER THEOREM......Page 91
20. DISCUSSION OF THE THEOREM AND ITS COROLLARY......Page 94
PROBLEMS......Page 97
21. FOURIER SERIES ON OTHER INTERVALS......Page 100
PROBLEMS......Page 103
22. UNIFORM CONVERGENCE OF FOURIER SERIES......Page 107
23. DIFFERENTIATION AND INTEGRATION OF FOURIER SERIES......Page 112
24. CONVERGENCE IN THE MEAN......Page 115
PROBLEMS......Page 119
25. LINEAR OPERATORS......Page 123
26. PRINCIPLE OF SUPERPOSITION......Page 125
27. A TEMPERATURE PROBLEM......Page 128
PROBLEMS......Page 136
29. A VIBRATING STRING PROBLEM......Page 137
30. VERIFICATION OF SOLUTION......Page 141
PROBLEMS......Page 144
31. HISTORICAL DEVELOPMENT......Page 145
CHAPTER 4 BOUNDARY VALUE PROBLEMS......Page 147
32. A SLAB WITH VARIOUS BOUNDARY CONDITIONS......Page 148
PROBLEMS......Page 153
33. THE SLAB WITH INTERNALLY GENERATED HEAT......Page 155
PROBLEMS......Page 158
34. DIRICHLET PROBLEMS......Page 161
35. OTHER TYPES OF BOUNDARY CONDITIONS......Page 164
PROBLEMS......Page 166
36. A STRING WITH PRESCRIBED INITIAL VELOCITY......Page 169
37. AN ELASTIC BAR......Page 171
38. RESONANCE......Page 173
PROBLEMS......Page 175
39. FOURIER SERIES IN TWO VARIABLES......Page 179
40. PERIODIC BOUNDARY CONDITIONS......Page 181
PROBLEMS......Page 183
41. REGULAR STURM-LIOUVILLE PROBLEMS......Page 186
42. MODIFICATIONS......Page 189
43. ORTHOGONALITY OF EIGEN FUNCTIONS......Page 190
PROBLEMS......Page 194
44. UNIQUENESS OF EIGENFUNCTIONS......Page 196
45. METHODS OF SOLUTION......Page 199
PROBLEMS......Page 202
46. EXAMPLES OF EIGENFUNCTION EXPANSIONS......Page 205
PROBLEMS......Page 209
47. SURFACE HEAT TRANSFER......Page 211
48. POLAR COORDINATES......Page 216
PROBLEMS......Page 217
49. MODIFICATIONS OF THE METHOD......Page 220
PROBLEMS......Page 224
50. A VERTICALLY HUNG ELASTIC BAR......Page 227
PROBLEMS......Page 230
51. THE FOURIER INTEGRAL FORMULA......Page 235
52. AN INTEGRATION FORMULA......Page 237
53. TWO LEMMAS......Page 238
54. A FOURIER INTEGRAL THEOREM......Page 242
PROBLEMS......Page 244
55. THE COSINE AND SINE INTEGRALS......Page 246
PROBLEMS......Page 248
56. MORE ON SUPERPOSITION OF SOLUTIONS......Page 250
57. TEMPERATURES IN A SEMI-INFINITE SOLID......Page 252
58. TEMPERATURES IN AN UNLIMITED MEDIUM......Page 255
PROBLEMS......Page 256
CHAPTER 7 BESSEL FUNCTIONS AND APPLICATIONS......Page 260
59. BESSEL FUNCTIONS J,,......Page 261
60. GENERAL SOLUTIONS OF BESSEL'S EQUATION......Page 264
61. RECURRENCE RELATIONS......Page 267
PROBLEMS......Page 269
62. BESSEL'S INTEGRAL FORM OF Jn(x)......Page 271
63. CONSEQUENCES OF THE INTEGRAL REPRESENTATIONS......Page 274
PROBLEMS......Page 276
64. THE ZEROS OF Jo(x)......Page 277
65. ZEROS OF RELATED FUNCTIONS......Page 280
66. ORTHOGONAL SETS OF BESSEL FUNCTIONS......Page 281
PROBLEMS......Page 286
67. THE ORTHONORMAL FUNCTIONS......Page 288
68. FOURIER-BESSEL SERIES......Page 289
PROBLEMS......Page 293
69. TEMPERATURES IN A LONG CYLINDER......Page 296
70. HEAT TRANSFER AT THE SURFACE OF THE CYLINDER......Page 300
PROBLEMS......Page 301
71. VIBRATION OF A CIRCULAR MEMBRANE......Page 305
PROBLEMS......Page 307
72. SOLUTIONS OF LEGENDRE'S EQUATION......Page 311
73. LEGENDRE POLYNOMIALS......Page 313
74. ORTHOGONALITY OF LEGENDRE POLYNOMIALS......Page 317
75. RODRIGUES' FORMULA AND NORMS......Page 319
PROBLEMS......Page 322
76. LEGENDRE SERIES......Page 325
PROBLEMS......Page 328
77. DIRICHLET PROBLEMS IN SPHERICAL REGIONS......Page 330
78. STEADY TEMPERATURES IN A HEMISPHERE......Page 333
PROBLEMS......Page 335
79. ABEL'S TEST FOR UNIFORM CONVERGENCE......Page 339
80. UNIQUENESS OF SOLUTIONS OF THE HEAT EQUATION......Page 342
81. SOLUTIONS OF LAPLACE'S OR POISSON'S EQUATION......Page 346
82. SOLUTIONS OF A WAVE EQUATION......Page 351
PROBLEMS......Page 352
BIBLIOGRAPHY......Page 354
INDEX......Page 359