Handbook of linear partial differential equations for engineers and scientists [missing CH 1-3]

Table of Contents......Page 0
HANDBOOK OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS for ENGINEERS and SCIENTISTS......Page 1
FOREWORD......Page 3
Special Functions (See Also Supplement A)......Page 4
AUTHOR......Page 6
CONTENTS......Page 7
0.1.1-2. Types of equations. Characteristic equations.......Page 16
0.1.1-4. Canonical form of hyperbolic equations (case b2 - ac > 0).......Page 17
0.1.2. Equations with Many Independent Variables......Page 18
0.2.1. Initial and Boundary Conditions. Cauchy Problem. Boundary Value Problems......Page 19
0.2.1-2. Hyperbolic equations. Initial and boundary conditions.......Page 20
0.2.2. First, Second, Third, and Mixed Boundary Value Problems......Page 21
0.3.1-2. Usage of particular solutions for the construction of other particular solutions.......Page 22
0.3.1-3. Separable solutions.......Page 24
0.3.2-1. Simplest properties of nonhomogeneous linear equations.......Page 25
0.4.1-1. Scheme of solving linear boundary value problems by separation of variables.......Page 26
0.4.1-2. Search for particular solutions. Derivation of equations and boundary conditions.......Page 27
0.4.1-3. Solution of eigenvalue problems. Orthogonality of eigenfunctions.......Page 29
0.4.2-2. Solution of boundary value problems for hyperbolic equations.......Page 30
0.4.2-4. Linear nonhomogeneous equations with nonhomogeneous boundary conditions.......Page 31
0.5.1. Main Integral Transforms......Page 32
0.5.2-1. The Laplace transform. The inverse Laplace transform.......Page 33
0.5.2-2. Main properties of the Laplace transform.......Page 34
0.5.2-3. Solving linear problems of mathematical physics by the Laplace transform.......Page 35
0.5.3-1. The Fourier transform and its properties.......Page 36
0.5.3-2. Solving linear problems of mathematical physics by the Fourier transform.......Page 37
0.6.1-1. General formula for the solution of the Cauchy problem.......Page 38
0.6.2. Cauchy Problem for Hyperbolic Equations......Page 39
0.7.1-2. Representation of the problem solution in terms of the Green’s function.......Page 40
0.7.2-2. Representation of the problem solution in terms of the Green’s function.......Page 42
0.8.1-2. Representation of the problem solution in terms of the Green’s function.......Page 43
0.8.2-1. Statement of the problem.......Page 44
0.8.3-1. Statement of the problem.......Page 45
0.8.4. Comparison of the Solution Structures for Boundary Value Problems for Equations of Various Types......Page 46
0.9.1-1. Expressions of the Green’s function in terms of infinite series.......Page 47
0.9.2-1. Boundary value problems for rectangular domains.......Page 48
0.9.2-2. Boundary value problems for a cylindrical domain with arbitrary cross-section.......Page 49
0.9.3-2. Domain:  Boundary value problems for elliptic equations.......Page 50
0.9.3-3. Domain: Boundary value problems for elliptic equations.......Page 51
0.9.3-4. Boundary value problems for parabolic equations.......Page 52
0.10.1-2. Hyperbolic equations with two independent variables.......Page 53
0.10.2-1. Parabolic equations.......Page 54
0.11.1. Transformations That Lead to Homogeneous Boundary Conditions......Page 55
0.11.2. Transformations That Lead to Homogeneous Initial and Boundary Conditions......Page 56
2.1.1-2. Formulas to construct particular solutions. Remarks on the Green’s functions.......Page 58
2.1.1-4. Domain:  Cauchy problem.......Page 59
2.1.1-5. Domain:  First boundary value problem.......Page 61
2.1.1-8. Domain:  First boundary value problem.......Page 62
2.1.1-10. Domain:  Third boundary value problem.......Page 63
2.1.1-11. Domain:  Mixed boundary value problems.......Page 64
2.1.1-13. Domain:  Second boundary value problem.......Page 65
2.1.1-15. Domain:  Mixed boundary value problems.......Page 66
2.1.1-16. Domain:  First boundary value problem.......Page 67
2.1.1-18. Domain:  Third boundary value problem.......Page 68
2.1.1-19. Domain: Mixed boundary value problems.......Page 69
2.1.2-2. Domain:  First boundary value problem.......Page 70
2.1.2-4. Domain:  Third boundary value problem.......Page 71
2.1.2-6. Domain:  Second boundary value problem.......Page 72
2.1.2-8. Domain: First boundary value problem.......Page 73
2.1.2-10. Domain: First boundary value problem.......Page 74
2.1.2-12. Domain: Mixed boundary value problem.......Page 75
2.1.3-1. Particular solutions. Remarks on the Green’s functions.......Page 76
2.1.3-3. Domain:  Second boundary value problem.......Page 77
2.1.3-4. Domain:  Third boundary value problem.......Page 78
2.1.3-5. Domain:  Mixed boundary value problems.......Page 79
2.1.3-6. Domain:   First boundary value problem.......Page 80
2.1.3-7. Domain:  Second boundary value problem.......Page 81
2.1.3-9. Domain:  Mixed boundary value problems.......Page 82
2.1.3-10. Domain:  First boundary value problem.......Page 83
2.1.3-11. Domain:  Second boundary value problem.......Page 84
2.2.1-2. Domain: First boundary value problem.......Page 85
2.2.1-5. Domain:  First boundary value problem.......Page 86
2.2.1-7. Domain:  Third boundary value problem.......Page 87
2.2.1-10. Domain:  Second boundary value problem.......Page 88
2.2.1-13. Domain:  First boundary value problem.......Page 89
2.2.1-15. Domain:  Third boundary value problem.......Page 90
2.2.2-2. Domain:  Different boundary value problems.......Page 91
2.2.3-1. Domain:  Different boundary value problems.......Page 92
2.2.3-4. Domain:  Third boundary value problem.......Page 93
2.2.3-5. Domain:  Mixed boundary value problem.......Page 94
2.3.1. Equations Containing Arbitrary Parameters......Page 95
2.3.2. Equations Containing Arbitrary Functions......Page 97
3.1.1-2. Formulas to construct particular solutions. Remarks on the Green’s functions.......Page 101
3.1.1-3. Domain: Cauchy problem.......Page 102
3.1.1-6. Domain:  Third boundary value problem.......Page 103
 3.1.1-8. Domain:  Second boundary value problem.......Page 104
3.1.1-10. Domain:  Mixed boundary value problem.......Page 105
3.1.1-12. Domain:  Second boundary value problem.......Page 106
3.1.1-14. Domain:  Mixed boundary value problems.......Page 107
3.1.1-15. Domain: 0 £ < , 0 £ < , 0 £ < . First boundary value problem.......Page 108
3.1.1-17. Domain: 0 £ < , 0 £ < , 0 £ < . Third boundary value problem.......Page 109
3.1.1-18. Domain: 0 £ < , 0 £ < , 0 £ < . Mixed boundary value problems.......Page 110
3.1.1-20. Domain: 0 £ £ 1, 0 £ £ 2, - < < . Second boundary value problem.......Page 111
3.1.1-21. Domain: 0 £ £ 1, 0 £ £ 2, - < < . Third boundary value problem.......Page 112
3.1.1-23. Domain: 0 £ £ 0 £ £ 0 £ < . First boundary value problem.......Page 113
3.1.1-24. Domain: 0 £ £ 0 £ £ 0 £ < . Second boundary value problem.......Page 114
3.1.1-26. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ < . Mixed boundary value problems.......Page 115
3.1.1-27. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. First boundary value problem.......Page 117
3.1.1-29. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. Third boundary value problem.......Page 118
3.1.1-30. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. Mixed boundary value problems.......Page 119
3.1.2-2. Domain: 0 £ £ , 0 £ £ 2 , - < < . First boundary value problem.......Page 121
3.1.2-4. Domain: 0 £ £ , 0 £ £ 2 , - < < . Third boundary value problem.......Page 122
3.1.2-6. Domain: 0 £ £ , 0 £ £ 2 , 0 £ < . Second boundary value problem.......Page 123
3.1.2-8. Domain: 0 £ £ , 0 £ £ 2 , 0 £ < . Mixed boundary value problems.......Page 124
3.1.2-9. Domain:  First boundary value problem.......Page 125
3.1.2-11. Domain:  Third boundary value problem.......Page 126
3.1.2-12. Domain: 0 £ £ , 0 £ £ 2 , 0 £ £ . Mixed boundary value problems.......Page 127
3.1.2-14. Domain: 1 £ £ 2, 0 £ £ 2 , - < < . Second boundary value problem.......Page 128
3.1.2-15. Domain: 1 £ £ 2, 0 £ £ 2 , - < < . Third boundary value problem.......Page 129
3.1.2-17. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ < . Second boundary value problem.......Page 130
3.1.2-19. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ < . Mixed boundary value problems.......Page 131
3.1.2-21. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . Second boundary value problem.......Page 133
3.1.2-22. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . Third boundary value problem.......Page 134
3.1.2-23. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . Mixed boundary value problems.......Page 135
3.1.2-24. Domain: 0 £ < , 0 £ £ 0, - < < . First boundary value problem.......Page 136
3.1.2-26. Domain: 0 £ < , 0 £ £ 0, 0 £ < . First boundary value problem.......Page 137
3.1.2-28. Domain: 0 £ < , 0 £ £ 0, 0 £ < . Mixed boundary value problems.......Page 138
3.1.2-30. Domain: 0 £ < , 0 £ £ 0, 0 £ £ . Second boundary value problem.......Page 140
3.1.2-31. Domain: 0 £ < , 0 £ £ 0, 0 £ £ . Mixed boundary value problems.......Page 141
3.1.2-32. Domain: 0 £ £ , 0 £ £ 0, - < < . First boundary value problem.......Page 142
3.1.2-33. Domain: 0 £ £ , 0 £ £ 0 £ < . First boundary value problem.......Page 143
3.1.2-35. Domain: 0 £ £ , 0 £ £ 0, 0 £ £ . First boundary value problem.......Page 144
3.1.2-36. Domain: 0 £ £ , 0 £ £ 0 £ £ . Mixed boundary value problem.......Page 145
3.1.3-1. Domain: 0 £ £ , 0 £ £ , 0 £ £ 2 . First boundary value problem.......Page 146
3.1.3-3. Domain: 0 £ £ , 0 £ £ , 0 £ £ 2 . Third boundary value problem.......Page 147
3.1.3-5. Domain: 1 £ £ 2, 0 £ £ , 0 £ £ 2 . Second boundary value problem.......Page 148
3.1.3-6. Domain: 1 £ £ 2, 0 £ £ , 0 £ £ 2 . Third boundary value problem.......Page 149
3.2.1-1. Domain: - < < , - < < , - < < . Cauchy problem.......Page 150
3.2.1-4. Domain: - < < , 0 £ < , 0 £ £ . Different boundary value problems.......Page 151
3.2.1-7. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ < . Different boundary value problems.......Page 152
3.2.2-2. Domain: 0 £ £ , 0 £ £ 2 , 0 £ < . Different boundary value problems.......Page 153
3.2.2-5. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ < . Different boundary value problems.......Page 154
3.2.2-9. Domain: 0 £ < , 0 £ £ 0, 0 £ £ . Different boundary value problems.......Page 155
3.2.3-1. Domain: 0 £ £ , 0 £ £ , 0 £ £ 2 . Different boundary value problems.......Page 156
3.3.1. Equations Containing Arbitrary Parameters......Page 157
3.3.2. Equations Containing Arbitrary Functions......Page 159
3.3.3-1. First boundary value problem.......Page 162
3.3.3-3. Third boundary value problem.......Page 163
3.4.1-1. Homogeneous equation......Page 164
3.4.1-4. Domain: = {0 £ £ ; = 1, , }. Second boundary value problem.......Page 165
3.4.2. Other Equations Containing Arbitrary Parameters......Page 166
3.4.3. Equations Containing Arbitrary Functions......Page 167
4.1.1-1. General solution. Some formulas.......Page 174
4.1.1-3. Domain: 0 £ < . First boundary value problem.......Page 175
4.1.1-5. Domain: 0 £ £ . First boundary value problem.......Page 176
4.1.1-7. Domain: 0 £ £ . Third boundary value problem.......Page 177
4.1.1-8. Domain: 0 £ £ . Mixed boundary value problem.......Page 178
4.1.2-2. Domain: 0 £ < . First boundary value problem.......Page 179
4.1.2-4. Domain: 0 £ £ . First boundary value problem.......Page 180
4.1.2-7. Domain: 0 £ £ . Mixed boundary value problem.......Page 181
4.1.3-2. Some formulas and transformations of the homogeneous equation.......Page 182
4.1.3-3. Domain: - < < . Cauchy problem.......Page 183
4.1.3-5. Domain: 0 £ £ . Second boundary value problem.......Page 184
4.1.3-7. Domain: 0 £ £ . Mixed boundary value problem.......Page 185
4.1.4-3. Domain: 0 £ £ . First boundary value problem.......Page 186
4.1.4-5. Domain: 0 £ £ . Third boundary value problem.......Page 187
4.1.5-3. Domain: 0 £ £ . First boundary value problem.......Page 188
4.1.5-4. Domain: 0 £ £ . Second boundary value problem.......Page 189
4.2.1-1. Domain: 0 £ £ . First boundary value problem.......Page 190
4.2.1-4. Domain: 1 £ £ 2. First boundary value problem.......Page 191
4.2.1-6. Domain: 1 £ £ 2. Third boundary value problem.......Page 192
4.2.3. Equation of the Form......Page 193
4.2.3-5. Domain: 0 £ £ . Second boundary value problem.......Page 194
4.2.3-7. Domain: 1 £ £ 2. First boundary value problem.......Page 195
4.2.4-1. Reduction to a nonhomogeneous constant coefficient equation.......Page 196
4.2.5. Equation of the Form......Page 197
4.2.5-3. Domain: 0 £ £ . Third boundary value problem.......Page 198
4.2.5-5. Domain: 1 £ £ 2. Second boundary value problem.......Page 199
4.2.6. Equation of the Form......Page 200
4.2.6-3. Domain: 0 £ £ . Third boundary value problem.......Page 201
4.2.6-5. Domain: 1 £ £ 2. Second boundary value problem.......Page 202
4.3.1. Equations of the Form......Page 203
4.3.2. Equations of the Form......Page 207
4.3.3. Other Equations......Page 209
4.4.1. Equations of the Form......Page 215
4.4.2. Equations of the Form......Page 221
4.4.3. Other Equations......Page 226
4.5.1-1. General relations to solve linear nonhomogeneous boundary value problems.......Page 228
4.5.1-3. Second boundary value problem (case a1 = a2 = 1, b1 = b2 = 0).......Page 229
4.5.2-1. General relations to solve linear nonhomogeneous boundary value problems.......Page 230
4.5.2-2. First,  second,  third,  and mixed boundary value problems.......Page 231
4.5.3. Other Equations......Page 232
5.1.1-1. Particular solutions and some relations.......Page 236
5.1.1-3. Domain: 0 £ £ 1, 0 £ £ 2. First boundary value problem.......Page 238
5.1.1-5. Domain: 0 £ £ 1, 0 £ £ 2. Third boundary value problem.......Page 239
5.1.1-6. Domain: 0 £ £ 1, 0 £ £ 2. Mixed boundary value problems.......Page 240
5.1.2-1. Domain: 0 £ £ , 0 £ £ 2 . First boundary value problem.......Page 241
5.1.2-4. Domain: 1 £ £ 2, 0 £ £ 2 . First boundary value problem.......Page 242
5.1.2-5. Domain: 1 £ £ 2, 0 £ £ 2 . Second boundary value problem.......Page 243
5.1.2-7. Domain: 0 £ £ , 0 £ £ 0. First boundary value problem.......Page 244
5.1.2-9. Domain: 0 £ £ , 0 £ £ 0. Mixed boundary value problem.......Page 245
5.1.3-2. Domain: 0 £ £ , 0 £ £ . Second boundary value problem.......Page 246
5.1.3-3. Domain: 0 £ £ , 0 £ £ . Third boundary value problem.......Page 247
5.1.3-4. Domain: 0 £ £ , 0 £ £ . Mixed boundary value problems.......Page 248
5.1.3-6. Domain: 1 £ £ 2, 0 £ £ . Second boundary value problem.......Page 249
5.2.1-1. Domain: - < < , - < < . Cauchy problem.......Page 250
5.2.1-4. Domain: 0 £ £ 1, 0 £ £ 2. Third boundary value problem.......Page 251
5.2.2-1. Domain: 0 £ £ , 0 £ £ 2 . First boundary value problem.......Page 252
5.2.2-5. Domain: 1 £ £ 2, 0 £ £ 2 . Second boundary value problem.......Page 253
5.2.2-9. Domain: 0 £ £ , 0 £ £ Mixed boundary value problem.......Page 254
5.2.3-3. Domain: 0 £ £ , 0 £ £ . Third boundary value problem.......Page 255
5.2.3-6. Domain: 1 £ £ 2, 0 £ £ . Second boundary value problem.......Page 256
5.3.1-2. Domain: - < < , - < < . Cauchy problem.......Page 257
5.3.1-4. Domain: 0 £ £ 1, 0 £ £ 2. Second boundary value problem.......Page 258
5.3.1-5. Domain: 0 £ £ 1, 0 £ £ 2. Third boundary value problem.......Page 259
5.3.1-6. Domain: 0 £ £ 1, 0 £ £ 2. Mixed boundary value problems.......Page 260
5.3.2-1. Domain: 0 £ £ , 0 £ £ 2 . First boundary value problem.......Page 261
5.3.2-3. Domain: 0 £ £ , 0 £ £ 2 . Third boundary value problem.......Page 262
5.3.2-5. Domain: 1 £ £ 2, 0 £ £ 2 . Second boundary value problem.......Page 263
5.3.2-7. Domain: 0 £ £ , 0 £ £ 0. First boundary value problem.......Page 264
5.3.2-8. Domain: 0 £ £ , 0 £ £ 0. Second boundary value problem.......Page 265
5.3.3-1. Domain: 0 £ £ , 0 £ £ . First boundary value problem.......Page 266
5.3.3-2. Domain: 0 £ £ , 0 £ £ . Second boundary value problem.......Page 267
5.3.3-4. Domain: 0 £ £ , 0 £ £ . Mixed boundary value problems.......Page 268
5.3.3-6. Domain: 1 £ £ 2, 0 £ £ . Second boundary value problem.......Page 270
5.4.1-1. Reduction to the two-dimensional Klein–Gordon equation.......Page 271
5.4.1-4. Domain: 0 £ £ 1, 0 £ £ 2. First boundary value problem.......Page 272
5.4.1-5. Domain: 0 £ £ 0 £ £ Second boundary value problem.......Page 273
5.4.1-7. Domain: 0 £ £ 1, 0 £ £ 2. Mixed boundary value problems.......Page 274
5.4.2-2. Domain: 0 £ £ , 0 £ £ 2 . Second boundary value problem.......Page 276
5.4.2-4. Domain: 1 £ £ 2, 0 £ £ 2 . First boundary value problem.......Page 277
5.4.2-5. Domain: 1 £ £ 2, 0 £ £ 2 . Second boundary value problem.......Page 278
5.4.2-7. Domain: 0 £ £ , 0 £ £ 0. First boundary value problem.......Page 279
5.4.2-8. Domain: 0 £ £ , 0 £ £ Second boundary value problem.......Page 280
5.4.3. Axisymmetric Problems......Page 281
5.4.3-2. Domain: 0 £ £ , 0 £ £ . Second boundary value problem.......Page 282
5.4.3-3. Domain: 0 £ £ , 0 £ £ . Third boundary value problem.......Page 283
5.4.3-4. Domain: 0 £ £ , 0 £ £ . Mixed boundary value problems.......Page 284
5.4.3-5. Domain: 1 £ £ 2, 0 £ £ . First boundary value problem.......Page 285
5.5. Other Equations with Two Space Variables......Page 286
6.1.1-1. Particular solutions and their properties.......Page 287
6.1.1-3. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. First boundary value problem.......Page 288
6.1.1-4. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. Second boundary value problem.......Page 289
6.1.1-5. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. Third boundary value problem.......Page 290
6.1.1-6. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. Mixed boundary value problems.......Page 291
6.1.2-2. Domain: 0 £ £ , 0 £ £ 2 , 0 £ £ . Second boundary value problem.......Page 293
6.1.2-3. Domain: 0 £ £ , 0 £ £ 2 , 0 £ £ . Third boundary value problem.......Page 294
6.1.2-4. Domain: 0 £ £ , 0 £ £ 2 , 0 £ £ . Mixed boundary value problems.......Page 295
6.1.2-5. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . First boundary value problem.......Page 296
6.1.2-6. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . Second boundary value problem.......Page 297
6.1.2-8. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . Mixed boundary value problems.......Page 298
6.1.2-9. Domain: 0 £ £ , 0 £ £ 0, 0 £ £ . First boundary value problem.......Page 300
6.1.2-10. Domain: 0 £ £ , 0 £ £ 0 £ £ . Mixed boundary value problem.......Page 301
6.1.3-2. Domain: 0 £ £ , 0 £ £ , 0 £ £ 2 . Second boundary value problem.......Page 302
6.1.3-4. Domain: 1 £ £ 2, 0 £ £ , 0 £ £ 2 . First boundary value problem.......Page 303
6.1.3-5. Domain: 1 £ £ 2, 0 £ £ , 0 £ £ 2 . Second boundary value problem.......Page 304
6.1.3-6. Domain: 1 £ £ 2, 0 £ £ , 0 £ £ 2 . Third boundary value problem.......Page 305
6.2.2. Problems in Cylindrical Coordinates......Page 306
6.2.3. Problems in Spherical Coordinates......Page 307
6.3.1-1. Fundamental solutions.......Page 308
6.3.1-3. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. First boundary value problem.......Page 309
6.3.1-4. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. Second boundary value problem.......Page 310
6.3.1-5. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. Third boundary value problem.......Page 311
6.3.1-6. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. Mixed boundary value problems.......Page 312
6.3.2-1. Domain: 0 £ £ , 0 £ £ 2 , 0 £ £ . First boundary value problem.......Page 314
6.3.2-3. Domain: 0 £ £ , 0 £ £ 2 , 0 £ £ . Third boundary value problem.......Page 315
6.3.2-4. Domain: 0 £ £ , 0 £ £ 2 , 0 £ £ . Mixed boundary value problems.......Page 316
6.3.2-5. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . First boundary value problem.......Page 317
6.3.2-6. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . Second boundary value problem.......Page 318
6.3.2-7. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . Third boundary value problem.......Page 319
6.3.2-8. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . Mixed boundary value problems.......Page 320
6.3.2-9. Domain: 0 £ £ , 0 £ £ 0, 0 £ £ . First boundary value problem.......Page 322
6.3.3. Problems in Spherical Coordinates......Page 323
6.3.3-2. Domain: 0 £ £ , 0 £ £ , 0 £ £ 2 . Second boundary value problem.......Page 324
6.3.3-4. Domain: 1 £ £ 2, 0 £ £ , 0 £ £ 2 . First boundary value problem.......Page 325
6.3.3-5. Domain: 1 £ £ 2, 0 £ £ , 0 £ £ 2 . Second boundary value problem.......Page 326
6.3.3-6. Domain: 1 £ £ 2, 0 £ £ , 0 £ £ 2 . Third boundary value problem.......Page 327
6.4.1-2. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. First boundary value problem.......Page 328
6.4.1-3. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. Second boundary value problem.......Page 329
6.4.1-4. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. Third boundary value problem.......Page 330
6.4.1-5. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. Mixed boundary value problems.......Page 331
6.4.2. Problems in Cylindrical Coordinates......Page 332
6.4.2-2. Domain: 0 £ £ , 0 £ £ 2 , 0 £ £ . Second boundary value problem.......Page 333
6.4.2-3. Domain: 0 £ £ , 0 £ £ 2 , 0 £ £ . Third boundary value problem.......Page 334
6.4.2-4. Domain: 0 £ £ , 0 £ £ 2 , 0 £ £ . Mixed boundary value problems.......Page 335
6.4.2-5. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . First boundary value problem.......Page 336
6.4.2-6. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . Second boundary value problem.......Page 337
6.4.2-7. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . Third boundary value problem.......Page 338
6.4.2-8. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . Mixed boundary value problems.......Page 339
6.4.2-9. Domain: 0 £ £ , 0 £ £ 0, 0 £ £ . First boundary value problem.......Page 341
6.4.3. Problems in Spherical Coordinates......Page 342
6.4.3-2. Domain:   Second boundary value problem.......Page 343
6.4.3-4. Domain: 1 £ £ 2, 0 £ £ , 0 £ £ 2 . First boundary value problem.......Page 344
6.4.3-5. Domain: £ £ 0 £ £ , 0 £ £ 2 . Second boundary value problem.......Page 345
6.4.3-6. Domain: 1 £ £ 2, 0 £ £ , 0 £ £ 2 . Third boundary value problem.......Page 346
6.5.2-1. First boundary value problem.......Page 347
6.6. Equations with n Space Variables......Page 349
6.6.1-3. Domain: - < < ; = 1, , . Cauchy problem.......Page 350
6.6.2-2. Domain: ={0 £ £ ; = 1, , }. First boundary value problem.......Page 351
6.6.2-3. Domain:  Second boundary value problem.......Page 352
6.6.2-5. Domain: ={0 £ £ ; = 1, , }. Mixed boundary value problem.......Page 353
6.6.3-2. Domain: ={0 £ £ ; = 1, , }. First boundary value problem.......Page 354
6.6.3-3. Domain: ={0 £ £ ; = 1, , }. Second boundary value problem.......Page 355
6.6.3-5. Domain: ={0 £ £ ; = 1, , }. Mixed boundary value problem.......Page 356
6.6.4. Equations Containing the First Time Derivative......Page 357
7.1.1-1. Particular solutions and a method for their construction.......Page 361
7.1.1-2. Specific features of stating boundary value problems for the Laplace equation.......Page 362
7.1.1-7. Domain:  Second boundary value problem.......Page 363
7.1.1-9. Domain:  First boundary value problem.......Page 364
7.1.1-12. Domain:  Mixed boundary value problems.......Page 365
7.1.2-1. Particular solutions:......Page 366
7.1.2-3. Domain:  Second boundary value problem.......Page 367
7.1.2-4. Domain:  Third boundary value problem.......Page 368
7.1.2-6. Domain:  Second boundary value problem.......Page 369
7.1.3-1. Parabolic,  elliptic,  and bipolar coordinate systems.......Page 370
7.1.3-3. Reduction of the two-dimensional Neumann problem to the Dirichlet problem.......Page 371
7.2.1-1. First boundary value problem.......Page 372
7.2.1-3. Third boundary value problem.......Page 373
7.2.2-4. Domain:  Second boundary value problem.......Page 374
7.2.2-7. Domain:  Third boundary value problem.......Page 375
7.2.2-10. Domain:  Third boundary value problem.......Page 376
7.2.2-12. Domain:  First boundary value problem.......Page 377
7.2.2-14. Domain:  Third boundary value problem.......Page 378
7.2.3-1. Domain:  First boundary value problem.......Page 379
7.2.3-3. Domain: First boundary value problem.......Page 380
7.2.3-6. Domain:  First boundary value problem.......Page 381
7.2.3-8. Domain:  First boundary value problem.......Page 382
7.2.4-2. General formula for the Green’s function. Example boundary value problems.......Page 383
7.3.1-1. Some definitions.......Page 384
7.3.1-2. Properties of eigenvalues and eigenfunctions.......Page 385
7.3.1-3. Nonhomogeneous Helmholtz equation with homogeneous boundary conditions.......Page 386
7.3.1-5. Boundary conditions at infinity in the case of an infinite domain.......Page 387
7.3.2-2. Domain:......Page 388
7.3.2-4. Domain:  Second boundary value problem.......Page 389
7.3.2-6. Domain:  Second boundary value problem.......Page 390
7.3.2-8. Domain:  Second boundary value problem.......Page 391
7.3.2-11. Domain:  First boundary value problem.......Page 392
7.3.2-14. Domain:  Mixed boundary value problems.......Page 393
7.3.2-15. Domain:  First boundary value problem.......Page 394
7.3.2-16. Domain:  Second boundary value problem.......Page 395
7.3.2-18. Domain:  Mixed boundary value problems.......Page 396
7.3.3-1. Particular solutions of the homogeneous equation:......Page 397
7.3.3-3. Domain:  Second boundary value problem.......Page 398
7.3.3-6. Domain:  Second boundary value problem.......Page 399
7.3.3-7. Domain:  Third boundary value problem.......Page 400
7.3.3-10. Domain:  Third boundary value problem.......Page 401
7.3.4-1. Parabolic coordinate system.......Page 402
7.3.4-3. Domain:  First boundary value problem.......Page 403
7.4.1. Stationary Schròdinger Equation......Page 404
7.4.2. Convective Heat and Mass Transfer Equations......Page 406
7.4.3. Equations of Heat and Mass Transfer in Anisotropic Media......Page 412
7.4.4. Other Equations Arising in Applications......Page 420
7.4.5-1. Statements of boundary value problems. Relations for the Green’s function.......Page 423
7.4.5-2. Representation of solutions to boundary value problems using the Green’s function.......Page 425
8.1.1-1. Particular solutions and some relations.......Page 427
8.1.1-4. Domain:  First boundary value problem.......Page 428
8.1.2-2. Domain:  First boundary value problem.......Page 429
8.1.2-3. Domain:  First boundary value problem.......Page 430
8.1.3-2. Domain:  First boundary value problem.......Page 431
8.1.3-3. Domain:  Second boundary value problem.......Page 432
8.2.1. Preliminary Remarks. Solution Structure......Page 433
8.2.1-1. First boundary value problem.......Page 435
8.2.1-2. Second boundary value problem.......Page 436
8.2.1-3. Third boundary value problem.......Page 437
8.2.2-3. Domain:  Third boundary value problem.......Page 438
8.2.2-5. Domain:  First boundary value problem.......Page 439
8.2.2-7. Domain:  Mixed boundary value problem.......Page 440
8.2.2-9. Domain:  First boundary value problem.......Page 441
8.2.2-11. Domain:  Mixed boundary value problems.......Page 442
8.2.2-12. Domain:  First boundary value problem.......Page 443
8.2.2-13. Domain:  First boundary value problem.......Page 444
8.2.2-15. Domain:  Mixed boundary value problems.......Page 445
8.2.2-16. Domain:  First boundary value problem.......Page 446
8.2.2-18. Domain:  Mixed boundary value problem.......Page 447
8.2.3-1. Domain:  First boundary value problem.......Page 448
8.2.3-3. Domain:  First boundary value problem.......Page 449
8.2.3-5. Domain:  Mixed boundary value problem.......Page 450
8.2.3-8. Domain:  Mixed boundary value problem.......Page 451
8.2.4-1. Domain:  First boundary value problem.......Page 452
8.2.4-3. Domain:  Third boundary value problem.......Page 453
8.2.4-7. Domain:  First boundary value problem.......Page 454
8.3.1-1. Some definitions.......Page 455
8.3.1-3. Nonhomogeneous Helmholtz equation with homogeneous boundary conditions.......Page 456
8.3.1-4. Solution of nonhomogeneous boundary value problems of general form.......Page 457
8.3.1-6. Green’s function for an infinite cylindrical domain of arbitrary cross-section.......Page 458
8.3.1-7. Green’s function for a semiinfinite cylindrical domain.......Page 459
8.3.1-8. Green’s function for a cylindrical domain of finite dimensions.......Page 460
8.3.2. Problems in Cartesian Coordinates......Page 461
8.3.2-3. Domain:  First boundary value problem.......Page 462
8.3.2-6. Domain:  Second boundary value problem.......Page 463
8.3.2-9. Domain:  First boundary value problem.......Page 464
8.3.2-11. Domain:  Third boundary value problem.......Page 465
8.3.2-12. Domain:  Mixed boundary value problems.......Page 466
8.3.2-13. Domain:  First boundary value problem.......Page 467
8.3.2-15. Domain:  Third boundary value problem.......Page 468
8.3.2-16. Domain:  Mixed boundary value problems.......Page 469
8.3.2-17. Domain:  First boundary value problem.......Page 470
8.3.2-19. Domain:  Third boundary value problem.......Page 471
8.3.2-20. Domain:  Mixed boundary value problems.......Page 472
8.3.2-23. Domain:  Mixed boundary value problems.......Page 473
8.3.3-3. Domain:  Second boundary value problem.......Page 474
8.3.3-5. Domain:  First boundary value problem.......Page 475
8.3.3-7. Domain:  Third boundary value problem.......Page 476
8.3.3-8. Domain:  Mixed boundary value problem.......Page 477
8.3.3-10. Domain:  Second boundary value problem.......Page 478
8.3.3-12. Domain:  First boundary value problem.......Page 479
8.3.3-14. Domain:  Mixed boundary value problems.......Page 480
8.3.3-16. Domain:  Mixed boundary value problem.......Page 481
8.3.4-2. Domain:  First boundary value problem.......Page 482
8.3.4-3. Domain:  Second boundary value problem.......Page 483
8.3.4-6. Domain:  First boundary value problem.......Page 484
8.3.5. Other Orthogonal Curvilinear Coordinates......Page 485
8.4.1. Equations Containing Arbitrary Functions......Page 487
8.4.2-1. First boundary value problem.......Page 489
8.4.2-3. Third boundary value problem.......Page 490
8.5.1-2. Domain:......Page 491
8.5.2. Other Equations......Page 492
9.1. Third-Order Partial Differential Equations......Page 495
9.2.1-1. Particular solutions of the homogeneous equation:......Page 496
9.2.1-4. The function and its second derivative are prescribed at the boundaries:......Page 497
9.2.1-8. Mixed conditions are prescribed at the boundaries (case 2):......Page 498
9.2.2-3. Domain:  Free vibration of a semiinfinite rod.......Page 499
9.2.3-2. Both ends of the rod are clamped.......Page 500
9.2.3-5. One end of the rod is clamped and the other is hinged.......Page 501
9.2.4-1. Particular solutions of the homogeneous equation:......Page 502
9.2.4-5. The first and third derivatives are prescribed at the ends:......Page 503
9.2.4-9. Mixed boundary conditions are prescribed at the ends (case 3):......Page 504
9.2.5-1. Equations containing the first derivative with respect to t.......Page 505
9.2.5-2. Equations containing the second derivative with respect to t.......Page 506
9.3.1-2. The function and its first derivatives are prescribed at the sides of a rectangle:......Page 507
9.3.1-5. The second and third derivatives are prescribed at the sides of a rectangle:......Page 508
9.3.2-2. Domain:  Cauchy problem.......Page 509
9.3.2-6. Domain:  Mixed boundary conditions are set at the sides:......Page 510
9.3.3-1. Three-dimensional case. Cauchy problem.......Page 511
9.3.3-4. n-dimensional case. Boundary value problem.......Page 512
9.3.4-3. The first and third derivatives are prescribed at the sides of a rectangle:......Page 513
9.3.5-2. The function and its first derivatives are prescribed at the sides of a rectangle:......Page 514
9.4.1-1. Two-dimensional equation. Particular solutions.......Page 515
9.4.1-3. Two-dimensional boundary value problems for the upper half-plane.......Page 516
9.4.1-5. Three-dimensional equation.......Page 517
9.4.1-6. n-dimensional equation.......Page 518
9.4.2-2. Domain:  Boundary value problem.......Page 519
9.4.3-1. Homogeneous equation.......Page 520
9.4.3-4. Domain:  Eigenvalue problem with.......Page 521
9.4.4-1. Homogeneous equation.......Page 522
9.4.5-1. Particular solutions of the homogeneous equation:......Page 523
9.4.6-1. Stokes equation for the stream function in the spherical coordinate system.......Page 524
9.4.6-2. Stokes equation in the bipolar coordinate system.......Page 526
9.5.1-1. Domain:......Page 527
9.5.1-3. Solution of the Cauchy problem for general initial conditions.......Page 528
9.5.2-2. Elliptic differential operator of general form.......Page 529
9.5.2-4. Fundamental solution of a general elliptic equation.......Page 530
9.5.4-1. Equations with two independent variables.......Page 531
9.5.4-2. Equations with many independent variables.......Page 533
9.5.5. Some Special-Type Equations......Page 534
9.6.1-2. The case of general homogeneous boundary conditions. The Green’s function.......Page 538
9.6.1-3. The case of nonhomogeneous boundary conditions. Preliminary transformations.......Page 539
9.6.1-4. The case of special nonhomogeneous boundary conditions.......Page 540
9.6.1-5. The case of general nonhomogeneous boundary conditions.......Page 541
9.6.2-2. The case of nonhomogeneous initial and boundary conditions.......Page 542
9.6.3-1. Equations with the first-order partial derivative with respect to t.......Page 543
9.6.3-2. Equations with the second-order partial derivative with respect to t.......Page 544
9.6.4. Some Special-Type Equations......Page 545
A.1.2. Binomial Coefficients......Page 548
A.2.2. Exponential Integral......Page 549
A.3.1. Sine Integral......Page 550
A.3.3. Fresnel Integrals......Page 551
A.4.1-2. Some formulas.......Page 552
A.5.1. Incomplete Gamma Function......Page 553
A.6.1-1. The Bessel functions of the first and the second kinds.......Page 554
A.6.1-4. The Bessel functions for v  = n, where n = 0, 1, 2, .........Page 555
A.6.2-2. Integrals with Bessel functions:......Page 556
A.6.3-2. Orthogonality properties of Bessel functions.......Page 557
A.6.4. Hankel Functions (Bessel Functions of the Third Kind)......Page 558
A.7.1-4. Modified Bessel functions for v = n, where n = 0, 1, 2, .........Page 559
A.7.2-3. Asymptotic expansions as :......Page 560
A.8.2-2. Asymptotic expansions as.......Page 561
A.9.1-1. The degenerate hypergeometric functions and.......Page 562
A.9.1-4. Degenerate hypergeometric functions for n = 0,  1,  2, ...:......Page 563
A.9.2-3. Asymptotic expansion as |x | :......Page 564
A.11. Whittaker Functions......Page 565
A.12.3. Associated Legendre Functions......Page 567
A.14.1-1. Mathieu equation and Mathieu functions.......Page 568
A.14.1-2. Properties of the Mathieu functions.......Page 569
A.16. Orthogonal Polynomials......Page 570
A.16.1-2. Generalized Laguerre polynomials.......Page 571
A.16.3. Hermite Polynomial......Page 572
A.16.4. Jacobi Polynomials......Page 573
B.1.1. Preliminary Remarks......Page 574
B.1.2. Simple Cases of Variable Separation in Nonlinear Equations......Page 575
B.1.3. Examples of Nontrivial Variable Separation in Nonlinear Equations......Page 576
B.2.1-2. General form of functional differential equations.......Page 578
B.2.2-2. Examples of constructing exact generalized separable solutions.......Page 579
B.2.3-1. Preliminary remarks. Description of the method.......Page 581
B.2.3-2. Solutions of simple functional equations and their application.......Page 582
B.2.4-1. Description of the simplified scheme.......Page 584
B.2.4-2. Examples of constructing exact solutions of higher-order equations.......Page 585
B.3.2. Special Functional Separable Solutions......Page 586
B.3.2-1. Solutions of the form (1) with z linear in one of the independent variables.......Page 587
B.3.2-2. Solution by reduction to equations with quadratic nonlinearities.......Page 588
B.3.3-2. Examples of constructing functional separable solutions.......Page 589
B.3.4-1. Splitting method. Reduction to a standard functional equation.......Page 593
B.3.5-2. The functional equation where.......Page 594
B.3.5-3. The functional equation  where.......Page 597
B.3.5-4. Equation.......Page 598
B.4.2. Individual Equations......Page 599
B.5.1-1. Equations of the form......Page 602
B.5.1-2. Equations of the form......Page 603
B.5.1-3. Equations of the form......Page 607
B.5.1-4. Equations of the form......Page 609
B.5.1-5. Equations of the form......Page 611
B.5.1-7. Equations with three independent variables.......Page 613
B.5.2-1. Equations of the form......Page 614
B.5.2-2. Equations of the form......Page 618
B.5.3-1. Equations of the form......Page 620
B.5.3-2. Equations of the form......Page 622
B.5.3-3. Other equations with two independent variables.......Page 623
B.5.3-4. Equations with three independent variables.......Page 625
B.5.4-1. Monge–Ampère equations.......Page 626
B.5.4-2. Other equations with quadratic nonlinearities.......Page 628
B.5.5-1. Equations of the form......Page 629
B.5.5-2. Equations of the form......Page 631
B.6.1. Stationary Hydrodynamic Boundary Layer Equations......Page 632
B.6.2. Nonstationary Hydrodynamic Boundary Layer Equations......Page 634
B.7.1. Stationary Hydrodynamic Equations (Navier–Stokes Equations)......Page 642
B.7.2. Nonstationary Hydrodynamic Equations......Page 645
B.8.1. Equations of the Form......Page 650
B.8.2. Equations of the Form......Page 654
B.8.3. Other Equations......Page 658
REFERENCES......Page 661