Handbook of linear partial differential equations for engineers and scientists

Andrei D. Polyanin1584882999, 9781584882992

Following in the footsteps of the authors’ bestselling Handbook of Integral Equations and Handbook of Exact Solutions for Ordinary Differential Equations, this handbook presents brief formulations and exact solutions for more than 2,200 equations and problems in science and engineering.”Parabolic, hyperbolic, and elliptic equations with constant and variable coefficients”New exact solutions to linear equations and boundary value problems”Equations and problems of general form that depend on arbitrary functions”Formulas for constructing solutions to nonhomogeneous boundary value problems”Second- and higher-order equations and boundary value problemsAn introductory section outlines the basic definitions, equations, problems, and methods of mathematical physics. It also provides useful formulas for expressing solutions to boundary value problems of general form in terms of the Green’s function. Two supplements at the end of the book furnish more tools and information: Supplement A lists the properties of common special functions, including the gamma, Bessel, degenerate hypergeometric, and Mathieu functions, and Supplement B describes the methods of generalized and functional separation of variables for nonlinear partial differential equations.

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Table of contents :
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
1.1.1-1. Particular solutions ( A, B, and are arbitrary constants)…….Page 16
Table of Contents……Page 0
1.1.1-3. Infinite series solutions…….Page 17
1.1.1-6. Domain: First boundary value problem…….Page 18
1.1.1-8. Domain: Third boundary value problem…….Page 19
1.1.1-9. Domain: First boundary value problem…….Page 20
1.1.1-12. Domain: Mixed boundary value problems…….Page 21
1.1.1-13. Problems without initial conditions…….Page 22
1.1.1-14. Conjugate heat and mass transfer problems…….Page 23
1.1.2-3. Domain: Second boundary value problem…….Page 24
1.1.2-5. Domain: First boundary value problem…….Page 25
1.1.2-8. Domain: Mixed boundary value problems…….Page 26
1.1.3. Equation of the Form……Page 27
1.1.3-2. Reduction to the heat equation. Remarks on the Green’s functions…….Page 28
1.1.3-6. Domain: Third boundary value problem…….Page 29
1.1.3-9. Domain: Third boundary value problem…….Page 30
1.1.4-1. Homogeneous equation…….Page 31
1.1.4-3. Domain: Cauchy problem…….Page 32
1.1.4-6. Domain: Third boundary value problem…….Page 33
1.1.4-8. Domain: Second boundary value problem…….Page 34
1.1.5-1. Homogeneous equation……Page 35
1.1.5-4. Domain: First boundary value problem…….Page 36
1.1.5-7. Domain: First boundary value problem…….Page 37
1.2.1. Equation of the Form……Page 38
1.2.1-2. Particular solutions in the form of an infinite series…….Page 39
1.2.1-4. Domain: Second boundary value problem…….Page 40
1.2.1-6. Domain: First boundary value problem…….Page 41
1.2.1-8. Domain: Third boundary value problem…….Page 42
1.2.2. Equation of the Form……Page 43
1.2.2-3. Domain: Third boundary value problem…….Page 44
1.2.2-5. Domain: Second boundary value problem…….Page 45
1.2.3. Equation of the Form……Page 46
1.2.3-3. Infinite series particular solutions…….Page 47
1.2.3-4. Domain: First boundary value problem…….Page 48
1.2.3-6. Domain: Third boundary value problem…….Page 49
1.2.3-8. Domain: Second boundary value problem…….Page 50
1.2.3-11. Domain: First boundary value problem…….Page 51
1.2.4-2. Domain: Second boundary value problem…….Page 52
1.2.4-5. Domain: Second boundary value problem…….Page 53
1.2.5. Equation of the Form……Page 54
1.2.5-2. Infinite series solutions…….Page 55
1.2.5-5. Domain: Second boundary value problem…….Page 56
1.2.6-2. Domain: Second boundary value problem…….Page 57
1.3.1-1. The function f depends on the space coordinate x alone…….Page 58
1.3.1-3. The function f depends on both x and t…….Page 61
1.3.2. Equations of the Form……Page 63
1.3.3. Equations of the Form……Page 66
1.3.4. Equations of the Form……Page 68
1.3.5. Equations of the Form……Page 71
1.3.6. Equations of the Form……Page 73
1.3.7. Equations of the Form……Page 77
1.3.8-2. Mass exchange between fluid film and gas…….Page 78
1.3.8-3. Dissolution of a plate by a laminar fluid film…….Page 79
1.3.9. Equations of the Form……Page 80
1.4.1. Equations of the Form……Page 81
1.4.2. Equations of the Form……Page 83
1.4.3. Equations of the Form……Page 86
1.4.5. Equations of the Form……Page 87
1.5.1. Equations Containing a Hyperbolic Cosine……Page 90
1.5.2. Equations Containing a Hyperbolic Sine……Page 91
1.5.3. Equations Containing a Hyperbolic Tangent……Page 92
1.5.4. Equations Containing a Hyperbolic Cotangent……Page 93
1.6.1. Equations of the Form……Page 94
1.6.2. Equations of the Form……Page 95
1.7.1. Equations Containing a Cosine……Page 96
1.7.2. Equations Containing a Sine……Page 97
1.7.3. Equations Containing a Tangent……Page 98
1.7.4. Equations Containing a Cotangent……Page 99
1.8.1. Equations of the Form……Page 100
1.8.2. Equations of the Form……Page 103
1.8.3. Equations of the Form……Page 106
1.8.4. Equations of the Form……Page 109
1.8.5. Equations of the Form……Page 110
1.8.6. Equations of the Form……Page 111
1.8.7. Equations of the Form……Page 119
1.8.8. Equations of the Form……Page 121
1.8.9-2. General properties of the Sturm–Liouville problem (5)…….Page 124
1.8.9-3. First boundary value problem: the case of a1 = a2 = 0 and b1 = b2 = 1…….Page 125
1.8.9-4. Second boundary value problem: the case of a1 = a2 = 1 and b1 = b2 = 0…….Page 126
1.8.9-7. Mixed boundary value problem: the case of a1 = b2 = 1 and a2 = b1 = 0…….Page 127
1.9.1. Equations of the Diffusion (Thermal) Boundary Layer……Page 128
1.9.2-1. Eigenvalue problem. Cauchy problem…….Page 130
1.9.2-4. Linear harmonic oscillator:……Page 131
1.9.2-7. Morse potential:……Page 132
1.9.2-9. Potential with a trigonometric function:……Page 133
2.1.1-2. Formulas to construct particular solutions. Remarks on the Green’s functions…….Page 134
2.1.1-4. Domain: Cauchy problem…….Page 135
2.1.1-5. Domain: First boundary value problem…….Page 137
2.1.1-8. Domain: First boundary value problem…….Page 138
2.1.1-10. Domain: Third boundary value problem…….Page 139
2.1.1-11. Domain: Mixed boundary value problems…….Page 140
2.1.1-13. Domain: Second boundary value problem…….Page 141
2.1.1-15. Domain: Mixed boundary value problems…….Page 142
2.1.1-16. Domain: First boundary value problem…….Page 143
2.1.1-18. Domain: Third boundary value problem…….Page 144
2.1.1-19. Domain: Mixed boundary value problems…….Page 145
2.1.2-2. Domain: First boundary value problem…….Page 146
2.1.2-4. Domain: Third boundary value problem…….Page 147
2.1.2-6. Domain: Second boundary value problem…….Page 148
2.1.2-8. Domain: First boundary value problem…….Page 149
2.1.2-10. Domain: First boundary value problem…….Page 150
2.1.2-12. Domain: Mixed boundary value problem…….Page 151
2.1.3-1. Particular solutions. Remarks on the Green’s functions…….Page 152
2.1.3-3. Domain: Second boundary value problem…….Page 153
2.1.3-4. Domain: Third boundary value problem…….Page 154
2.1.3-5. Domain: Mixed boundary value problems…….Page 155
2.1.3-6. Domain: First boundary value problem…….Page 156
2.1.3-7. Domain: Second boundary value problem…….Page 157
2.1.3-9. Domain: Mixed boundary value problems…….Page 158
2.1.3-10. Domain: First boundary value problem…….Page 159
2.1.3-11. Domain: Second boundary value problem…….Page 160
2.2.1-2. Domain: First boundary value problem…….Page 161
2.2.1-5. Domain: First boundary value problem…….Page 162
2.2.1-7. Domain: Third boundary value problem…….Page 163
2.2.1-10. Domain: Second boundary value problem…….Page 164
2.2.1-13. Domain: First boundary value problem…….Page 165
2.2.1-15. Domain: Third boundary value problem…….Page 166
2.2.2-2. Domain: Different boundary value problems…….Page 167
2.2.3-1. Domain: Different boundary value problems…….Page 168
2.2.3-4. Domain: Third boundary value problem…….Page 169
2.2.3-5. Domain: Mixed boundary value problem…….Page 170
2.3.1. Equations Containing Arbitrary Parameters……Page 171
2.3.2. Equations Containing Arbitrary Functions……Page 173
3.1.1-2. Formulas to construct particular solutions. Remarks on the Green’s functions…….Page 177
3.1.1-3. Domain: Cauchy problem…….Page 178
3.1.1-6. Domain: Third boundary value problem…….Page 179
3.1.1-8. Domain: Second boundary value problem…….Page 180
3.1.1-10. Domain: Mixed boundary value problem…….Page 181
3.1.1-12. Domain: Second boundary value problem…….Page 182
3.1.1-14. Domain: Mixed boundary value problems…….Page 183
3.1.1-15. Domain: 0 £ < , 0 £ < , 0 £ < . First boundary value problem…….Page 184
3.1.1-17. Domain: 0 £ < , 0 £ < , 0 £ < . Third boundary value problem…….Page 185
3.1.1-18. Domain: 0 £ < , 0 £ < , 0 £ < . Mixed boundary value problems…….Page 186
3.1.1-20. Domain: 0 £ £ 1, 0 £ £ 2, – < < . Second boundary value problem…….Page 187
3.1.1-21. Domain: 0 £ £ 1, 0 £ £ 2, – < < . Third boundary value problem…….Page 188
3.1.1-23. Domain: 0 £ £ 0 £ £ 0 £ < . First boundary value problem…….Page 189
3.1.1-24. Domain: 0 £ £ 0 £ £ 0 £ < . Second boundary value problem…….Page 190
3.1.1-26. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ < . Mixed boundary value problems…….Page 191
3.1.1-27. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. First boundary value problem…….Page 193
3.1.1-29. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. Third boundary value problem…….Page 194
3.1.1-30. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. Mixed boundary value problems…….Page 195
3.1.2-2. Domain: 0 £ £ , 0 £ £ 2 , – < < . First boundary value problem…….Page 197
3.1.2-4. Domain: 0 £ £ , 0 £ £ 2 , – < < . Third boundary value problem…….Page 198
3.1.2-6. Domain: 0 £ £ , 0 £ £ 2 , 0 £ < . Second boundary value problem…….Page 199
3.1.2-8. Domain: 0 £ £ , 0 £ £ 2 , 0 £ < . Mixed boundary value problems…….Page 200
3.1.2-9. Domain: First boundary value problem…….Page 201
3.1.2-11. Domain: Third boundary value problem…….Page 202
3.1.2-12. Domain: 0 £ £ , 0 £ £ 2 , 0 £ £ . Mixed boundary value problems…….Page 203
3.1.2-14. Domain: 1 £ £ 2, 0 £ £ 2 , – < < . Second boundary value problem…….Page 204
3.1.2-15. Domain: 1 £ £ 2, 0 £ £ 2 , – < < . Third boundary value problem…….Page 205
3.1.2-17. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ < . Second boundary value problem…….Page 206
3.1.2-19. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ < . Mixed boundary value problems…….Page 207
3.1.2-21. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . Second boundary value problem…….Page 209
3.1.2-22. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . Third boundary value problem…….Page 210
3.1.2-23. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . Mixed boundary value problems…….Page 211
3.1.2-24. Domain: 0 £ < , 0 £ £ 0, – < < . First boundary value problem…….Page 212
3.1.2-26. Domain: 0 £ < , 0 £ £ 0, 0 £ < . First boundary value problem…….Page 213
3.1.2-28. Domain: 0 £ < , 0 £ £ 0, 0 £ < . Mixed boundary value problems…….Page 214
3.1.2-30. Domain: 0 £ < , 0 £ £ 0, 0 £ £ . Second boundary value problem…….Page 216
3.1.2-31. Domain: 0 £ < , 0 £ £ 0, 0 £ £ . Mixed boundary value problems…….Page 217
3.1.2-32. Domain: 0 £ £ , 0 £ £ 0, – < < . First boundary value problem…….Page 218
3.1.2-33. Domain: 0 £ £ , 0 £ £ 0 £ < . First boundary value problem…….Page 219
3.1.2-35. Domain: 0 £ £ , 0 £ £ 0, 0 £ £ . First boundary value problem…….Page 220
3.1.2-36. Domain: 0 £ £ , 0 £ £ 0 £ £ . Mixed boundary value problem…….Page 221
3.1.3-1. Domain: 0 £ £ , 0 £ £ , 0 £ £ 2 . First boundary value problem…….Page 222
3.1.3-3. Domain: 0 £ £ , 0 £ £ , 0 £ £ 2 . Third boundary value problem…….Page 223
3.1.3-5. Domain: 1 £ £ 2, 0 £ £ , 0 £ £ 2 . Second boundary value problem…….Page 224
3.1.3-6. Domain: 1 £ £ 2, 0 £ £ , 0 £ £ 2 . Third boundary value problem…….Page 225
3.2.1-1. Domain: – < < , – < < , – < < . Cauchy problem…….Page 226
3.2.1-4. Domain: – < < , 0 £ < , 0 £ £ . Different boundary value problems…….Page 227
3.2.1-7. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ < . Different boundary value problems…….Page 228
3.2.2-2. Domain: 0 £ £ , 0 £ £ 2 , 0 £ < . Different boundary value problems…….Page 229
3.2.2-5. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ < . Different boundary value problems…….Page 230
3.2.2-9. Domain: 0 £ < , 0 £ £ 0, 0 £ £ . Different boundary value problems…….Page 231
3.2.3-1. Domain: 0 £ £ , 0 £ £ , 0 £ £ 2 . Different boundary value problems…….Page 232
3.3.1. Equations Containing Arbitrary Parameters……Page 233
3.3.2. Equations Containing Arbitrary Functions……Page 235
3.3.3-1. First boundary value problem…….Page 238
3.3.3-3. Third boundary value problem…….Page 239
3.4.1-1. Homogeneous equation……Page 240
3.4.1-4. Domain: = {0 £ £ ; = 1, , }. Second boundary value problem…….Page 241
3.4.2. Other Equations Containing Arbitrary Parameters……Page 242
3.4.3. Equations Containing Arbitrary Functions……Page 243
4.1.1-1. General solution. Some formulas…….Page 250
4.1.1-3. Domain: 0 £ < . First boundary value problem…….Page 251
4.1.1-5. Domain: 0 £ £ . First boundary value problem…….Page 252
4.1.1-7. Domain: 0 £ £ . Third boundary value problem…….Page 253
4.1.1-8. Domain: 0 £ £ . Mixed boundary value problem…….Page 254
4.1.2-2. Domain: 0 £ < . First boundary value problem…….Page 255
4.1.2-4. Domain: 0 £ £ . First boundary value problem…….Page 256
4.1.2-7. Domain: 0 £ £ . Mixed boundary value problem…….Page 257
4.1.3-2. Some formulas and transformations of the homogeneous equation…….Page 258
4.1.3-3. Domain: – < < . Cauchy problem…….Page 259
4.1.3-5. Domain: 0 £ £ . Second boundary value problem…….Page 260
4.1.3-7. Domain: 0 £ £ . Mixed boundary value problem…….Page 261
4.1.4-3. Domain: 0 £ £ . First boundary value problem…….Page 262
4.1.4-5. Domain: 0 £ £ . Third boundary value problem…….Page 263
4.1.5-3. Domain: 0 £ £ . First boundary value problem…….Page 264
4.1.5-4. Domain: 0 £ £ . Second boundary value problem…….Page 265
4.2.1-1. Domain: 0 £ £ . First boundary value problem…….Page 266
4.2.1-4. Domain: 1 £ £ 2. First boundary value problem…….Page 267
4.2.1-6. Domain: 1 £ £ 2. Third boundary value problem…….Page 268
4.2.3. Equation of the Form……Page 269
4.2.3-5. Domain: 0 £ £ . Second boundary value problem…….Page 270
4.2.3-7. Domain: 1 £ £ 2. First boundary value problem…….Page 271
4.2.4-1. Reduction to a nonhomogeneous constant coefficient equation…….Page 272
4.2.5. Equation of the Form……Page 273
4.2.5-3. Domain: 0 £ £ . Third boundary value problem…….Page 274
4.2.5-5. Domain: 1 £ £ 2. Second boundary value problem…….Page 275
4.2.6. Equation of the Form……Page 276
4.2.6-3. Domain: 0 £ £ . Third boundary value problem…….Page 277
4.2.6-5. Domain: 1 £ £ 2. Second boundary value problem…….Page 278
4.3.1. Equations of the Form……Page 279
4.3.2. Equations of the Form……Page 283
4.3.3. Other Equations……Page 285
4.4.1. Equations of the Form……Page 291
4.4.2. Equations of the Form……Page 297
4.4.3. Other Equations……Page 302
4.5.1-1. General relations to solve linear nonhomogeneous boundary value problems…….Page 304
4.5.1-3. Second boundary value problem (case a1 = a2 = 1, b1 = b2 = 0)…….Page 305
4.5.2-1. General relations to solve linear nonhomogeneous boundary value problems…….Page 306
4.5.2-2. First, second, third, and mixed boundary value problems…….Page 307
4.5.3. Other Equations……Page 308
5.1.1-1. Particular solutions and some relations…….Page 312
5.1.1-3. Domain: 0 £ £ 1, 0 £ £ 2. First boundary value problem…….Page 314
5.1.1-5. Domain: 0 £ £ 1, 0 £ £ 2. Third boundary value problem…….Page 315
5.1.1-6. Domain: 0 £ £ 1, 0 £ £ 2. Mixed boundary value problems…….Page 316
5.1.2-1. Domain: 0 £ £ , 0 £ £ 2 . First boundary value problem…….Page 317
5.1.2-4. Domain: 1 £ £ 2, 0 £ £ 2 . First boundary value problem…….Page 318
5.1.2-5. Domain: 1 £ £ 2, 0 £ £ 2 . Second boundary value problem…….Page 319
5.1.2-7. Domain: 0 £ £ , 0 £ £ 0. First boundary value problem…….Page 320
5.1.2-9. Domain: 0 £ £ , 0 £ £ 0. Mixed boundary value problem…….Page 321
5.1.3-2. Domain: 0 £ £ , 0 £ £ . Second boundary value problem…….Page 322
5.1.3-3. Domain: 0 £ £ , 0 £ £ . Third boundary value problem…….Page 323
5.1.3-4. Domain: 0 £ £ , 0 £ £ . Mixed boundary value problems…….Page 324
5.1.3-6. Domain: 1 £ £ 2, 0 £ £ . Second boundary value problem…….Page 325
5.2.1-1. Domain: – < < , – < < . Cauchy problem…….Page 326
5.2.1-4. Domain: 0 £ £ 1, 0 £ £ 2. Third boundary value problem…….Page 327
5.2.2-1. Domain: 0 £ £ , 0 £ £ 2 . First boundary value problem…….Page 328
5.2.2-5. Domain: 1 £ £ 2, 0 £ £ 2 . Second boundary value problem…….Page 329
5.2.2-9. Domain: 0 £ £ , 0 £ £ Mixed boundary value problem…….Page 330
5.2.3-3. Domain: 0 £ £ , 0 £ £ . Third boundary value problem…….Page 331
5.2.3-6. Domain: 1 £ £ 2, 0 £ £ . Second boundary value problem…….Page 332
5.3.1-2. Domain: – < < , – < < . Cauchy problem…….Page 333
5.3.1-4. Domain: 0 £ £ 1, 0 £ £ 2. Second boundary value problem…….Page 334
5.3.1-5. Domain: 0 £ £ 1, 0 £ £ 2. Third boundary value problem…….Page 335
5.3.1-6. Domain: 0 £ £ 1, 0 £ £ 2. Mixed boundary value problems…….Page 336
5.3.2-1. Domain: 0 £ £ , 0 £ £ 2 . First boundary value problem…….Page 337
5.3.2-3. Domain: 0 £ £ , 0 £ £ 2 . Third boundary value problem…….Page 338
5.3.2-5. Domain: 1 £ £ 2, 0 £ £ 2 . Second boundary value problem…….Page 339
5.3.2-7. Domain: 0 £ £ , 0 £ £ 0. First boundary value problem…….Page 340
5.3.2-8. Domain: 0 £ £ , 0 £ £ 0. Second boundary value problem…….Page 341
5.3.3-1. Domain: 0 £ £ , 0 £ £ . First boundary value problem…….Page 342
5.3.3-2. Domain: 0 £ £ , 0 £ £ . Second boundary value problem…….Page 343
5.3.3-4. Domain: 0 £ £ , 0 £ £ . Mixed boundary value problems…….Page 344
5.3.3-6. Domain: 1 £ £ 2, 0 £ £ . Second boundary value problem…….Page 346
5.4.1-1. Reduction to the two-dimensional Klein–Gordon equation…….Page 347
5.4.1-4. Domain: 0 £ £ 1, 0 £ £ 2. First boundary value problem…….Page 348
5.4.1-5. Domain: 0 £ £ 0 £ £ Second boundary value problem…….Page 349
5.4.1-7. Domain: 0 £ £ 1, 0 £ £ 2. Mixed boundary value problems…….Page 350
5.4.2-2. Domain: 0 £ £ , 0 £ £ 2 . Second boundary value problem…….Page 352
5.4.2-4. Domain: 1 £ £ 2, 0 £ £ 2 . First boundary value problem…….Page 353
5.4.2-5. Domain: 1 £ £ 2, 0 £ £ 2 . Second boundary value problem…….Page 354
5.4.2-7. Domain: 0 £ £ , 0 £ £ 0. First boundary value problem…….Page 355
5.4.2-8. Domain: 0 £ £ , 0 £ £ Second boundary value problem…….Page 356
5.4.3. Axisymmetric Problems……Page 357
5.4.3-2. Domain: 0 £ £ , 0 £ £ . Second boundary value problem…….Page 358
5.4.3-3. Domain: 0 £ £ , 0 £ £ . Third boundary value problem…….Page 359
5.4.3-4. Domain: 0 £ £ , 0 £ £ . Mixed boundary value problems…….Page 360
5.4.3-5. Domain: 1 £ £ 2, 0 £ £ . First boundary value problem…….Page 361
5.5. Other Equations with Two Space Variables……Page 362
6.1.1-1. Particular solutions and their properties…….Page 363
6.1.1-3. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. First boundary value problem…….Page 364
6.1.1-4. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. Second boundary value problem…….Page 365
6.1.1-5. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. Third boundary value problem…….Page 366
6.1.1-6. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. Mixed boundary value problems…….Page 367
6.1.2-2. Domain: 0 £ £ , 0 £ £ 2 , 0 £ £ . Second boundary value problem…….Page 369
6.1.2-3. Domain: 0 £ £ , 0 £ £ 2 , 0 £ £ . Third boundary value problem…….Page 370
6.1.2-4. Domain: 0 £ £ , 0 £ £ 2 , 0 £ £ . Mixed boundary value problems…….Page 371
6.1.2-5. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . First boundary value problem…….Page 372
6.1.2-6. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . Second boundary value problem…….Page 373
6.1.2-8. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . Mixed boundary value problems…….Page 374
6.1.2-9. Domain: 0 £ £ , 0 £ £ 0, 0 £ £ . First boundary value problem…….Page 376
6.1.2-10. Domain: 0 £ £ , 0 £ £ 0 £ £ . Mixed boundary value problem…….Page 377
6.1.3-2. Domain: 0 £ £ , 0 £ £ , 0 £ £ 2 . Second boundary value problem…….Page 378
6.1.3-4. Domain: 1 £ £ 2, 0 £ £ , 0 £ £ 2 . First boundary value problem…….Page 379
6.1.3-5. Domain: 1 £ £ 2, 0 £ £ , 0 £ £ 2 . Second boundary value problem…….Page 380
6.1.3-6. Domain: 1 £ £ 2, 0 £ £ , 0 £ £ 2 . Third boundary value problem…….Page 381
6.2.2. Problems in Cylindrical Coordinates……Page 382
6.2.3. Problems in Spherical Coordinates……Page 383
6.3.1-1. Fundamental solutions…….Page 384
6.3.1-3. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. First boundary value problem…….Page 385
6.3.1-4. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. Second boundary value problem…….Page 386
6.3.1-5. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. Third boundary value problem…….Page 387
6.3.1-6. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. Mixed boundary value problems…….Page 388
6.3.2-1. Domain: 0 £ £ , 0 £ £ 2 , 0 £ £ . First boundary value problem…….Page 390
6.3.2-3. Domain: 0 £ £ , 0 £ £ 2 , 0 £ £ . Third boundary value problem…….Page 391
6.3.2-4. Domain: 0 £ £ , 0 £ £ 2 , 0 £ £ . Mixed boundary value problems…….Page 392
6.3.2-5. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . First boundary value problem…….Page 393
6.3.2-6. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . Second boundary value problem…….Page 394
6.3.2-7. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . Third boundary value problem…….Page 395
6.3.2-8. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . Mixed boundary value problems…….Page 396
6.3.2-9. Domain: 0 £ £ , 0 £ £ 0, 0 £ £ . First boundary value problem…….Page 398
6.3.3. Problems in Spherical Coordinates……Page 399
6.3.3-2. Domain: 0 £ £ , 0 £ £ , 0 £ £ 2 . Second boundary value problem…….Page 400
6.3.3-4. Domain: 1 £ £ 2, 0 £ £ , 0 £ £ 2 . First boundary value problem…….Page 401
6.3.3-5. Domain: 1 £ £ 2, 0 £ £ , 0 £ £ 2 . Second boundary value problem…….Page 402
6.3.3-6. Domain: 1 £ £ 2, 0 £ £ , 0 £ £ 2 . Third boundary value problem…….Page 403
6.4.1-2. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. First boundary value problem…….Page 404
6.4.1-3. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. Second boundary value problem…….Page 405
6.4.1-4. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. Third boundary value problem…….Page 406
6.4.1-5. Domain: 0 £ £ 1, 0 £ £ 2, 0 £ £ 3. Mixed boundary value problems…….Page 407
6.4.2. Problems in Cylindrical Coordinates……Page 408
6.4.2-2. Domain: 0 £ £ , 0 £ £ 2 , 0 £ £ . Second boundary value problem…….Page 409
6.4.2-3. Domain: 0 £ £ , 0 £ £ 2 , 0 £ £ . Third boundary value problem…….Page 410
6.4.2-4. Domain: 0 £ £ , 0 £ £ 2 , 0 £ £ . Mixed boundary value problems…….Page 411
6.4.2-5. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . First boundary value problem…….Page 412
6.4.2-6. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . Second boundary value problem…….Page 413
6.4.2-7. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . Third boundary value problem…….Page 414
6.4.2-8. Domain: 1 £ £ 2, 0 £ £ 2 , 0 £ £ . Mixed boundary value problems…….Page 415
6.4.2-9. Domain: 0 £ £ , 0 £ £ 0, 0 £ £ . First boundary value problem…….Page 417
6.4.3. Problems in Spherical Coordinates……Page 418
6.4.3-2. Domain: Second boundary value problem…….Page 419
6.4.3-4. Domain: 1 £ £ 2, 0 £ £ , 0 £ £ 2 . First boundary value problem…….Page 420
6.4.3-5. Domain: £ £ 0 £ £ , 0 £ £ 2 . Second boundary value problem…….Page 421
6.4.3-6. Domain: 1 £ £ 2, 0 £ £ , 0 £ £ 2 . Third boundary value problem…….Page 422
6.5.2-1. First boundary value problem…….Page 423
6.6. Equations with n Space Variables……Page 425
6.6.1-3. Domain: – < < ; = 1, , . Cauchy problem…….Page 426
6.6.2-2. Domain: ={0 £ £ ; = 1, , }. First boundary value problem…….Page 427
6.6.2-3. Domain: Second boundary value problem…….Page 428
6.6.2-5. Domain: ={0 £ £ ; = 1, , }. Mixed boundary value problem…….Page 429
6.6.3-2. Domain: ={0 £ £ ; = 1, , }. First boundary value problem…….Page 430
6.6.3-3. Domain: ={0 £ £ ; = 1, , }. Second boundary value problem…….Page 431
6.6.3-5. Domain: ={0 £ £ ; = 1, , }. Mixed boundary value problem…….Page 432
6.6.4. Equations Containing the First Time Derivative……Page 433
7.1.1-1. Particular solutions and a method for their construction…….Page 437
7.1.1-2. Specific features of stating boundary value problems for the Laplace equation…….Page 438
7.1.1-7. Domain: Second boundary value problem…….Page 439
7.1.1-9. Domain: First boundary value problem…….Page 440
7.1.1-12. Domain: Mixed boundary value problems…….Page 441
7.1.2-1. Particular solutions:……Page 442
7.1.2-3. Domain: Second boundary value problem…….Page 443
7.1.2-4. Domain: Third boundary value problem…….Page 444
7.1.2-6. Domain: Second boundary value problem…….Page 445
7.1.3-1. Parabolic, elliptic, and bipolar coordinate systems…….Page 446
7.1.3-3. Reduction of the two-dimensional Neumann problem to the Dirichlet problem…….Page 447
7.2.1-1. First boundary value problem…….Page 448
7.2.1-3. Third boundary value problem…….Page 449
7.2.2-4. Domain: Second boundary value problem…….Page 450
7.2.2-7. Domain: Third boundary value problem…….Page 451
7.2.2-10. Domain: Third boundary value problem…….Page 452
7.2.2-12. Domain: First boundary value problem…….Page 453
7.2.2-14. Domain: Third boundary value problem…….Page 454
7.2.3-1. Domain: First boundary value problem…….Page 455
7.2.3-3. Domain: First boundary value problem…….Page 456
7.2.3-6. Domain: First boundary value problem…….Page 457
7.2.3-8. Domain: First boundary value problem…….Page 458
7.2.4-2. General formula for the Green’s function. Example boundary value problems…….Page 459
7.3.1-1. Some definitions…….Page 460
7.3.1-2. Properties of eigenvalues and eigenfunctions…….Page 461
7.3.1-3. Nonhomogeneous Helmholtz equation with homogeneous boundary conditions…….Page 462
7.3.1-5. Boundary conditions at infinity in the case of an infinite domain…….Page 463
7.3.2-2. Domain:……Page 464
7.3.2-4. Domain: Second boundary value problem…….Page 465
7.3.2-6. Domain: Second boundary value problem…….Page 466
7.3.2-8. Domain: Second boundary value problem…….Page 467
7.3.2-11. Domain: First boundary value problem…….Page 468
7.3.2-14. Domain: Mixed boundary value problems…….Page 469
7.3.2-15. Domain: First boundary value problem…….Page 470
7.3.2-16. Domain: Second boundary value problem…….Page 471
7.3.2-18. Domain: Mixed boundary value problems…….Page 472
7.3.3-1. Particular solutions of the homogeneous equation:……Page 473
7.3.3-3. Domain: Second boundary value problem…….Page 474
7.3.3-6. Domain: Second boundary value problem…….Page 475
7.3.3-7. Domain: Third boundary value problem…….Page 476
7.3.3-10. Domain: Third boundary value problem…….Page 477
7.3.4-1. Parabolic coordinate system…….Page 478
7.3.4-3. Domain: First boundary value problem…….Page 479
7.4.1. Stationary Schròdinger Equation……Page 480
7.4.2. Convective Heat and Mass Transfer Equations……Page 482
7.4.3. Equations of Heat and Mass Transfer in Anisotropic Media……Page 488
7.4.4. Other Equations Arising in Applications……Page 496
7.4.5-1. Statements of boundary value problems. Relations for the Green’s function…….Page 499
7.4.5-2. Representation of solutions to boundary value problems using the Green’s function…….Page 501
8.1.1-1. Particular solutions and some relations…….Page 503
8.1.1-4. Domain: First boundary value problem…….Page 504
8.1.2-2. Domain: First boundary value problem…….Page 505
8.1.2-3. Domain: First boundary value problem…….Page 506
8.1.3-2. Domain: First boundary value problem…….Page 507
8.1.3-3. Domain: Second boundary value problem…….Page 508
8.2.1. Preliminary Remarks. Solution Structure……Page 509
8.2.1-1. First boundary value problem…….Page 511
8.2.1-2. Second boundary value problem…….Page 512
8.2.1-3. Third boundary value problem…….Page 513
8.2.2-3. Domain: Third boundary value problem…….Page 514
8.2.2-5. Domain: First boundary value problem…….Page 515
8.2.2-7. Domain: Mixed boundary value problem…….Page 516
8.2.2-9. Domain: First boundary value problem…….Page 517
8.2.2-11. Domain: Mixed boundary value problems…….Page 518
8.2.2-12. Domain: First boundary value problem…….Page 519
8.2.2-13. Domain: First boundary value problem…….Page 520
8.2.2-15. Domain: Mixed boundary value problems…….Page 521
8.2.2-16. Domain: First boundary value problem…….Page 522
8.2.2-18. Domain: Mixed boundary value problem…….Page 523
8.2.3-1. Domain: First boundary value problem…….Page 524
8.2.3-3. Domain: First boundary value problem…….Page 525
8.2.3-5. Domain: Mixed boundary value problem…….Page 526
8.2.3-8. Domain: Mixed boundary value problem…….Page 527
8.2.4-1. Domain: First boundary value problem…….Page 528
8.2.4-3. Domain: Third boundary value problem…….Page 529
8.2.4-7. Domain: First boundary value problem…….Page 530
8.3.1-1. Some definitions…….Page 531
8.3.1-3. Nonhomogeneous Helmholtz equation with homogeneous boundary conditions…….Page 532
8.3.1-4. Solution of nonhomogeneous boundary value problems of general form…….Page 533
8.3.1-6. Green’s function for an infinite cylindrical domain of arbitrary cross-section…….Page 534
8.3.1-7. Green’s function for a semiinfinite cylindrical domain…….Page 535
8.3.1-8. Green’s function for a cylindrical domain of finite dimensions…….Page 536
8.3.2. Problems in Cartesian Coordinates……Page 537
8.3.2-3. Domain: First boundary value problem…….Page 538
8.3.2-6. Domain: Second boundary value problem…….Page 539
8.3.2-9. Domain: First boundary value problem…….Page 540
8.3.2-11. Domain: Third boundary value problem…….Page 541
8.3.2-12. Domain: Mixed boundary value problems…….Page 542
8.3.2-13. Domain: First boundary value problem…….Page 543
8.3.2-15. Domain: Third boundary value problem…….Page 544
8.3.2-16. Domain: Mixed boundary value problems…….Page 545
8.3.2-17. Domain: First boundary value problem…….Page 546
8.3.2-19. Domain: Third boundary value problem…….Page 547
8.3.2-20. Domain: Mixed boundary value problems…….Page 548
8.3.2-23. Domain: Mixed boundary value problems…….Page 549
8.3.3-3. Domain: Second boundary value problem…….Page 550
8.3.3-5. Domain: First boundary value problem…….Page 551
8.3.3-7. Domain: Third boundary value problem…….Page 552
8.3.3-8. Domain: Mixed boundary value problem…….Page 553
8.3.3-10. Domain: Second boundary value problem…….Page 554
8.3.3-12. Domain: First boundary value problem…….Page 555
8.3.3-14. Domain: Mixed boundary value problems…….Page 556
8.3.3-16. Domain: Mixed boundary value problem…….Page 557
8.3.4-2. Domain: First boundary value problem…….Page 558
8.3.4-3. Domain: Second boundary value problem…….Page 559
8.3.4-6. Domain: First boundary value problem…….Page 560
8.3.5. Other Orthogonal Curvilinear Coordinates……Page 561
8.4.1. Equations Containing Arbitrary Functions……Page 563
8.4.2-1. First boundary value problem…….Page 565
8.4.2-3. Third boundary value problem…….Page 566
8.5.1-2. Domain:……Page 567
8.5.2. Other Equations……Page 568
9.1. Third-Order Partial Differential Equations……Page 571
9.2.1-1. Particular solutions of the homogeneous equation:……Page 572
9.2.1-4. The function and its second derivative are prescribed at the boundaries:……Page 573
9.2.1-8. Mixed conditions are prescribed at the boundaries (case 2):……Page 574
9.2.2-3. Domain: Free vibration of a semiinfinite rod…….Page 575
9.2.3-2. Both ends of the rod are clamped…….Page 576
9.2.3-5. One end of the rod is clamped and the other is hinged…….Page 577
9.2.4-1. Particular solutions of the homogeneous equation:……Page 578
9.2.4-5. The first and third derivatives are prescribed at the ends:……Page 579
9.2.4-9. Mixed boundary conditions are prescribed at the ends (case 3):……Page 580
9.2.5-1. Equations containing the first derivative with respect to t…….Page 581
9.2.5-2. Equations containing the second derivative with respect to t…….Page 582
9.3.1-2. The function and its first derivatives are prescribed at the sides of a rectangle:……Page 583
9.3.1-5. The second and third derivatives are prescribed at the sides of a rectangle:……Page 584
9.3.2-2. Domain: Cauchy problem…….Page 585
9.3.2-6. Domain: Mixed boundary conditions are set at the sides:……Page 586
9.3.3-1. Three-dimensional case. Cauchy problem…….Page 587
9.3.3-4. n-dimensional case. Boundary value problem…….Page 588
9.3.4-3. The first and third derivatives are prescribed at the sides of a rectangle:……Page 589
9.3.5-2. The function and its first derivatives are prescribed at the sides of a rectangle:……Page 590
9.4.1-1. Two-dimensional equation. Particular solutions…….Page 591
9.4.1-3. Two-dimensional boundary value problems for the upper half-plane…….Page 592
9.4.1-5. Three-dimensional equation…….Page 593
9.4.1-6. n-dimensional equation…….Page 594
9.4.2-2. Domain: Boundary value problem…….Page 595
9.4.3-1. Homogeneous equation…….Page 596
9.4.3-4. Domain: Eigenvalue problem with…….Page 597
9.4.4-1. Homogeneous equation…….Page 598
9.4.5-1. Particular solutions of the homogeneous equation:……Page 599
9.4.6-1. Stokes equation for the stream function in the spherical coordinate system…….Page 600
9.4.6-2. Stokes equation in the bipolar coordinate system…….Page 602
9.5.1-1. Domain:……Page 603
9.5.1-3. Solution of the Cauchy problem for general initial conditions…….Page 604
9.5.2-2. Elliptic differential operator of general form…….Page 605
9.5.2-4. Fundamental solution of a general elliptic equation…….Page 606
9.5.4-1. Equations with two independent variables…….Page 607
9.5.4-2. Equations with many independent variables…….Page 609
9.5.5. Some Special-Type Equations……Page 610
9.6.1-2. The case of general homogeneous boundary conditions. The Green’s function…….Page 614
9.6.1-3. The case of nonhomogeneous boundary conditions. Preliminary transformations…….Page 615
9.6.1-4. The case of special nonhomogeneous boundary conditions…….Page 616
9.6.1-5. The case of general nonhomogeneous boundary conditions…….Page 617
9.6.2-2. The case of nonhomogeneous initial and boundary conditions…….Page 618
9.6.3-1. Equations with the first-order partial derivative with respect to t…….Page 619
9.6.3-2. Equations with the second-order partial derivative with respect to t…….Page 620
9.6.4. Some Special-Type Equations……Page 621
A.1.2. Binomial Coefficients……Page 624
A.2.2. Exponential Integral……Page 625
A.3.1. Sine Integral……Page 626
A.3.3. Fresnel Integrals……Page 627
A.4.1-2. Some formulas…….Page 628
A.5.1. Incomplete Gamma Function……Page 629
A.6.1-1. The Bessel functions of the first and the second kinds…….Page 630
A.6.1-4. The Bessel functions for v = n, where n = 0, 1, 2, ………Page 631
A.6.2-2. Integrals with Bessel functions:……Page 632
A.6.3-2. Orthogonality properties of Bessel functions…….Page 633
A.6.4. Hankel Functions (Bessel Functions of the Third Kind)……Page 634
A.7.1-4. Modified Bessel functions for v = n, where n = 0, 1, 2, ………Page 635
A.7.2-3. Asymptotic expansions as :……Page 636
A.8.2-2. Asymptotic expansions as…….Page 637
A.9.1-1. The degenerate hypergeometric functions and…….Page 638
A.9.1-4. Degenerate hypergeometric functions for n = 0, 1, 2, …:……Page 639
A.9.2-3. Asymptotic expansion as |x | :……Page 640
A.11. Whittaker Functions……Page 641
A.12.3. Associated Legendre Functions……Page 643
A.14.1-1. Mathieu equation and Mathieu functions…….Page 644
A.14.1-2. Properties of the Mathieu functions…….Page 645
A.16. Orthogonal Polynomials……Page 646
A.16.1-2. Generalized Laguerre polynomials…….Page 647
A.16.3. Hermite Polynomial……Page 648
A.16.4. Jacobi Polynomials……Page 649
B.1.1. Preliminary Remarks……Page 650
B.1.2. Simple Cases of Variable Separation in Nonlinear Equations……Page 651
B.1.3. Examples of Nontrivial Variable Separation in Nonlinear Equations……Page 652
B.2.1-2. General form of functional differential equations…….Page 654
B.2.2-2. Examples of constructing exact generalized separable solutions…….Page 655
B.2.3-1. Preliminary remarks. Description of the method…….Page 657
B.2.3-2. Solutions of simple functional equations and their application…….Page 658
B.2.4-1. Description of the simplified scheme…….Page 660
B.2.4-2. Examples of constructing exact solutions of higher-order equations…….Page 661
B.3.2. Special Functional Separable Solutions……Page 662
B.3.2-1. Solutions of the form (1) with z linear in one of the independent variables…….Page 663
B.3.2-2. Solution by reduction to equations with quadratic nonlinearities…….Page 664
B.3.3-2. Examples of constructing functional separable solutions…….Page 665
B.3.4-1. Splitting method. Reduction to a standard functional equation…….Page 669
B.3.5-2. The functional equation where…….Page 670
B.3.5-3. The functional equation where…….Page 673
B.3.5-4. Equation…….Page 674
B.4.2. Individual Equations……Page 675
B.5.1-1. Equations of the form……Page 678
B.5.1-2. Equations of the form……Page 679
B.5.1-3. Equations of the form……Page 683
B.5.1-4. Equations of the form……Page 685
B.5.1-5. Equations of the form……Page 687
B.5.1-7. Equations with three independent variables…….Page 689
B.5.2-1. Equations of the form……Page 690
B.5.2-2. Equations of the form……Page 694
B.5.3-1. Equations of the form……Page 696
B.5.3-2. Equations of the form……Page 698
B.5.3-3. Other equations with two independent variables…….Page 699
B.5.3-4. Equations with three independent variables…….Page 701
B.5.4-1. Monge–Ampère equations…….Page 702
B.5.4-2. Other equations with quadratic nonlinearities…….Page 704
B.5.5-1. Equations of the form……Page 705
B.5.5-2. Equations of the form……Page 707
B.6.1. Stationary Hydrodynamic Boundary Layer Equations……Page 708
B.6.2. Nonstationary Hydrodynamic Boundary Layer Equations……Page 710
B.7.1. Stationary Hydrodynamic Equations (Navier–Stokes Equations)……Page 718
B.7.2. Nonstationary Hydrodynamic Equations……Page 721
B.8.1. Equations of the Form……Page 726
B.8.2. Equations of the Form……Page 730
B.8.3. Other Equations……Page 734
REFERENCES……Page 737