Homotopy analysis method in nonlinear differential equations

00 Cover......Page 1
00 front......Page 2
Title Page\r......Page 3
Copyright Page\r......Page 4
Preface......Page 6
Table of Contents\r......Page 8
Acronyms......Page 14
01 Part 1......Page 15
1.1 Motivation and purpose......Page 16
1.2 Characteristic of homotopy analysis method......Page 18
1.3 Outline......Page 19
References......Page 21
2.1 Concept of homotopy......Page 28
2.2 Example 2.1: generalized Newtonian iteration formula......Page 32
2.3.1 Analysis of the solution characteristic......Page 39
2.3.2 Mathematical formulations......Page 44
2.3.3 Convergence of homotopy-series solution......Page 51
2.3.4 Essence of the convergence-control parameter c0......Page 61
2.3.5 Convergence acceleration by homotopy-Pade´ technique......Page 69
2.3.6 Convergence acceleration by optimal initial approximation......Page 72
2.3.7 Convergence acceleration by iteration......Page 76
2.3.8 Flexibility on the choice of auxiliary linear operator......Page 82
2.4 Concluding remarks and discussions......Page 88
Appendix 2.1 Derivation of δn in (2.57)......Page 92
Appendix 2.2 Derivation of (2.55) by the 2nd approach......Page 93
Appendix 2.3 Proof of Theorem 2.3......Page 95
Appendix 2.4 Mathematica code (without iteration) for Example 2.2......Page 96
Appendix 2.5 Mathematica code (with iteration) for Example 2.2......Page 100
References......Page 105
3.1 Introduction......Page 108
3.2.1 Basic ideas......Page 114
3.2.2 Different types of optimal methods......Page 117
3.2.2.1 Basic optimal HAM......Page 118
3.2.2.2 Three-parameter optimal HAM......Page 121
3.2.2.3 Infinite-parameter optimal HAM......Page 122
3.2.2.4 Finite-parameter optimal HAM......Page 126
3.3 Systematic description......Page 130
3.4 Concluding remarks and discussions......Page 134
Appendix 3.1 Mathematica code for Blasius flow......Page 135
3.2. Combination of the optimal HAM with iteration......Page 139
References......Page 140
4.1 Brief frame of the homotopy analysis method......Page 143
4.2 Properties of homotopy-derivative......Page 145
4.3.1 A brief history......Page 160
4.3.2 High-order deformation equations......Page 165
4.3.3 Examples......Page 177
4.4 Convergence theorems......Page 180
4.5 Solution expression......Page 185
4.5.1 Choice of initial approximation......Page 187
4.5.2 Choice of auxiliary linear operator......Page 188
4.6 Convergence control and acceleration......Page 191
4.6.1 Optimal convergence-control parameter......Page 192
4.6.3 Homotopy-iteration technique......Page 193
4.6.4 Homotopy-Pade´ technique......Page 194
4.7 Discussions and open questions......Page 195
References......Page 197
5.1 Introduction......Page 200
5.2 Generalized Taylor series......Page 201
5.3 Homotopy transform......Page 221
5.4 Relation between homotopy analysis method and Euler transform......Page 226
5.5 Concluding remarks......Page 230
References......Page 231
6.1 A brief history of the homotopy analysis method......Page 233
6.2 Homotopy perturbation method......Page 235
6.3 Optimal homotopy asymptotic method......Page 238
6.5 Generalized boundary element method......Page 240
6.6 Generalized scaled boundary finite element method......Page 241
6.7 Predictor homotopy analysis method......Page 242
References......Page 243
07 Part 2......Page 246
7.1 Introduction......Page 247
7.1.1 Scope......Page 250
7.1.2.1 Boundary-value problems in a finite interval......Page 251
7.1.2.3 Eigenvalue problems in a finite interval......Page 253
7.1.3 Choice of base function and initial guess......Page 255
7.1.4 Choice of the auxiliary linear operator......Page 258
7.1.5 Choice of the auxiliary function......Page 260
7.1.6 Choice of the convergence-control parameter c0......Page 261
7.2 Approximation and iteration of solutions......Page 262
7.2.1 Polynomials......Page 263
7.2.2 Trigonometric functions......Page 264
7.2.3 Hybrid-base functions......Page 265
7.3.1 Key modules......Page 268
7.3.2 Control parameters......Page 269
7.3.3 Input......Page 271
7.3.5 Global variables......Page 272
Appendix 7.1 Mathematica package BVPh (version 1.0)......Page 273
References......Page 286
8.1 Introduction......Page 293
8.2 Brief mathematical formulas......Page 294
8.3.1 Nonlinear diffusion-reaction model......Page 297
8.3.2 A three-point nonlinear boundary-value problem......Page 304
8.3.3 Channel flows with multiple solutions......Page 309
8.4 Concluding remarks......Page 314
Appendix 8.1 Input data of BVPh for Example 8.3.1......Page 315
Appendix 8.2 Input data of BVPh for Example 8.3.2......Page 317
Appendix 8.3 Input data of BVPh for Example 8.3.3......Page 318
8.4. Coupled nonlinear boundary-value problems in an infinite interval......Page 320
References......Page 321
9.1 Introduction......Page 323
9.2 Brief mathematical formulas......Page 325
9.3.1 Non-uniform beam acted by axial load......Page 330
9.3.1.1 Uniform beam......Page 332
9.3.1.2 Non-uniform beam......Page 340
9.3.2 Gelfand equation......Page 341
9.3.3 Equation with singularity and varying coefficient......Page 345
9.3.4 Multipoint boundary-value problem with multiple solutions......Page 350
9.3.5 Orr-Sommerfeld stability equation with complex coefficient......Page 354
9.4 Concluding remarks......Page 358
Appendix 9.1 Input data of BVPh for Example 9.3.1......Page 359
Appendix 9.2 Input data of BVPh for Example 9.3.2......Page 361
Appendix 9.3 Input data of BVPh for Example 9.3.3......Page 362
Appendix 9.4 Input data of BVPh for Example 9.3.4......Page 363
Appendix 9.5 Input data of BVPh for Example 9.3.5......Page 365
9.2. Coupled nonlinear eigenvalue problems......Page 366
References......Page 367
10.1 Introduction......Page 370
10.2 Exponentially decaying solutions......Page 372
10.3 Algebraically decaying solutions......Page 376
10.4 Concluding remarks......Page 383
Appendix 10.1 Input data of BVPh for exponentially decaying solution......Page 384
Appendix 10.2 Input data of BVPh for algebraically decaying solution......Page 385
References......Page 387
11.1 Introduction......Page 389
11.2 Brief mathematical formulas......Page 393
11.3 Homotopy-series solution......Page 398
11.4 Concluding remarks......Page 402
Appendix 11.1 Input data of BVPh......Page 403
References......Page 405
12.1 Introduction......Page 408
12.2 Perturbation approximation......Page 411
12.3.1 Brief mathematical formulas......Page 413
12.3.2 Homotopy-approximation......Page 417
12.4 Concluding remarks......Page 422
Appendix 12.1 Input data of BVPh......Page 423
References......Page 425
13 Part 3......Page 427
13.1 Mathematical modeling......Page 428
13.2 Brief mathematical formulas......Page 431
13.3 Validity of the explicit homotopy-approximations......Page 439
Example 13.1......Page 440
Example 13.2......Page 444
13.4 A practical code for businessmen......Page 446
13.5 Concluding remarks......Page 447
Appendix 13.1 Detailed derivation of fn(τ ) and gn(τ )......Page 449
Appendix 13.2 Mathematica code for American put option......Page 451
Appendix 13.3 Mathematica code APOh for businessmen......Page 457
References......Page 460
14.1 Introduction......Page 463
14.2.1 Brief mathematical formulas......Page 464
14.2.2 Homotopy-approximations......Page 470
14.3 Homotopy-approximations of 3D Gelfand equation......Page 476
14.4 Concluding remarks......Page 482
Appendix 14.1 Mathematica code of 2D Gelfand equation......Page 483
Appendix 14.2 Mathematica code of 3D Gelfand equation......Page 487
References......Page 491
15.1 Introduction......Page 494
15.2.1 Original boundary-value equation......Page 495
15.2.2 Dubreil-Jacotin transformation......Page 497
15.3.1 Solution expression......Page 498
15.3.2 Zeroth-order deformation equation......Page 499
15.3.3 High-order deformation equation......Page 501
15.3.4 Successive solution procedure......Page 503
15.4 Homotopy approximations......Page 505
15.5 Concluding remarks......Page 516
Appendix 15.1 Mathematica code of wave-current interaction......Page 517
References......Page 522
16.1 Introduction......Page 524
16.2.1 Brief Mathematical formulas......Page 526
16.2.2 Non-resonant waves......Page 534
16.2.3 Resonant waves......Page 539
16.3.1 Resonance criterion of small-amplitude waves......Page 548
16.3.2 Resonance criterion of large-amplitude waves......Page 551
16.4 Concluding remark and discussions......Page 554
Appendix 16.1 Detailed derivation of high-order equation......Page 556
Index......Page 564