Part Ⅰ Partial Differential Equations 1
1 Basic Concepts 3
1.1 Introduction 3
1.2 Definitions 4
1.2.1 Definition of a PDE 4
1.2.2 Order of a PDE 5
1.2.3 Linear and Nonlinear PDEs 6
1.2.4 Some Linear Partial Differential Equations 7
1.2.5 Some Nonlinear Partial Differential Equations 7
1.2.6 Homogeneous and Inhomogeneous PDEs 9
1.2.7 Solution of a PDE 9
1.2.8 Boundary Conditions 11
1.2.9 Initial Conditions 12
1.2.10 Well-posed PDEs 12
1.3 Classifications of a Second-order PDE 14
References 17
2 First-order Partial Differential Equations 19
2.1 Introduction 19
2.2 Adomian Decomposition Method 19
2.3 The Noise Terms Phenomenon 36
2.4 The Modified Decomposition Method 41
2.5 The Variational Iteration Method 47
2.6 Method of Characteristics 54
2.7 Systems of Linear PDEs by Adomian Method 59
2.8 Systems of Linear PDEs by Variational Iteration Method 63
References 68
3 One Dimensional Heat Flow 69
3.1 Introduction 69
3.2 The Adomian Decomposition Method 70
3.2.1 Homogeneous Heat Equations 73
3.2.2 Inhomogeneous Heat Equations 80
3.3 The Variational Iteration Method 83
3.3.1 Homogeneous Heat Equations 84
3.3.2 Inhomogeneous Heat Equations 87
3.4 Method of Separation of Variables 89
3.4.1 Analysis of the Method 89
3.4.2 Inhomogeneous Boundary Conditions 99
3.4.3 Equations with Lateral Heat Loss 102
References 106
4 Higher Dimensional Heat Flow 107
4.1 Introduction 107
4.2 Adomian Decomposition Method 108
4.2.1 Two Dimensional Heat Flow 108
4.2.2 Three Dimensional Heat Flow 116
4.3 Method of Separation of Variables 124
4.3.1 Two Dimensional Heat Flow 124
4.3.2 Three Dimensional Heat Flow 134
References 140
5 One Dimensional Wave Equation 143
5.1 Introduction 143
5.2 Adornian Decomposition Method 144
5.2.1 Homogeneous Wave Equations 146
5.2.2 Inhomogeneous Wave Equations 152
5.2.3 Wave Equation in an Infinite Domain 157
5.3 The Variational Iteration Method 162
5.3.1 Homogeneous Wave Equations 162
5.3.2 Inhomogeneous Wave Equations 168
5.3.3 Wave Equation in an Infinite Domain 170
5.4 Method of Separation of Variables 174
5.4.1 Analysis of the Method 174
5.4.2 Inhomogeneous Boundary Conditions 184
5.5 Wave Equation in an Infinite Domain:D'Alembert Solution 190
References 194
6 Higher Dimensional Wave Equation 195
6.1 Introduction 195
6.2 Adomian Decomposition Method 195
6.2.1 Two Dimensional Wave Equation 196
6.2.2 Three Dimensional Wave Equation 210
6.3 Method of Separation of Variables 220
6.3.1 Two Dimensional Wave Equation 221
6.3.2 Three Dimensional Wave Equation 231
References 236
7 Laplace's Equation 237
7.1 Introduction 237
7.2 Adomian Decomposition Method 238
7.2.1 Two Dimensional Laplace's Equation 238
7.3 The Variational Iteration Method 247
7.4 Method of Separation of Variables 251
7.4.1 Laplace's Equation in Two Dimensions 251
7.4.2 Laplace's Equation in Three Dimensions 259
7.5 Laplace's Equation in Polar Coordinates 267
7.5.1 Laplace's Equation for a Disc 268
7.5.2 Laplace's Equation for an Annulus 275
References 283
8 Nonlinear Partial Differential Equations 285
8.1 Introduction 285
8.2 Adomian Decomposition Method 287
8.2.1 Calculation of Adomian Polynomials 288
8.2.2 Alternative Algorithm for Calculating Adomian Polynomials 292
8.3 Nonlinear ODEs by Adomian Method 301
8.4 Nonlinear ODEs by VIM 312
8.5 Nonlinear PDEs by Adomian Method 319
8.6 Nonlinear PDEs by VIM 334
8.7 Nonlinear PDEs Systems by Adomian Method 341
8.8 Systems of Nonlinear PDEs by VIM 347
References 351
9 Linear and Nonlinear Physical Models 353
9.1 Introduction 353
9.2 The Nonlinear Advection Problem 354
9.3 The Goursat Problem 360
9.4 The Klein-Gordon Equation 370
9.4.1 Linear Klein-Gordon Equation 371
9.4.2 Nonlinear Klein-Gordon Equation 375
9.4.3 The Sine-Gordon Equation 378
9.5 The Burgers Equation 381
9.6 The Telegraph Equation 388
9.7 Schrodinger Equation 394
9.7.1 The Linear Schrodinger Equation 394
9.7.2 The Nonlinear Schrodinger Equation 397
9.8 Korteweg-de Vries Equation 401
9.9 Fourth-order Parabolic Equation 405
9.9.1 Equations with Constant Coefficients 405
9.9.2 Equations with Variable Coefficients 408
References 413
10 Numerical Applications and Padé Approximants 415
10.1 Introduction 415
10.2 Ordinary Differential Equations 416
10.2.1 Perturbation Problems 416
10.2.2 Nonperturbed Problems 421
10.3 Partial Differential Equations 427
10.4 The Padé Approximants 430
10.5 Padé Approximants and Boundary Value Problems 439
References 455
11 Solitons and Compactons 457
11.1 Introduction 457
11.2 Solitons 459
11.2.1 The KdV Equation 460
11.2.2 The Modified KdV Equation 462
11.2.3 The Generalized KdV Equation 464
11.2.4 The Sine-Gordon Equation 464
11.2.5 The Boussinesq Equation 465
11.2.6 The Kadomtsev-Petviashvili Equation 467
11.3 Compactons 469
11.4 The Defocusing Branch of K(n,n) 474
References 475
Part Ⅱ Solitray Waves Theory 479
12 Solitary Waves Theory 479
12.1 Introduction 479
12.2 Definitions 480
12.2.1 Dispersion and Dissipation 482
12.2.2 Types of Travelling Wave Solutions 484
12.2.3 Nonanalytic Solitary Wave Solutions 490
12.3 Analysis of the Methods 491
12.3.1 The Tanh-coth Method 491
12.3.2 The Sine-cosine Method 493
12.3.3 Hirota's Bilinear Method 494
12.4 Conservation Laws 496
References 502
13 The Family of the KdV Equations 503
13.1 Introduction 503
13.2 The Family of the KdV Equations 505
13.2.1 Third-order KdV Equations 505
13.2.2 The K(n,n) Equation 507
13.3 The KdV Equation 507
13.3.1 Using the Tanh-coth Method 508
13.3.2 Using the Sine-cosine Method 510
13.3.3 Multiple-soliton Solutions of the KdV Equation 510
13.4 The Modified KdV Equation 518
13.4.1 Using the Tanh-coth Method 519
13.4.2 Using the Sine-cosine Method 520
13.4.3 Multiple-soliton Solutions of the mKdV Equation 521
13.5 Singular Soliton Solutions 523
13.6 The Generalized KdV Equation 526
13.6.1 Using the Tanh-coth Method 526
13.6.2 Using the Sine-cosine Method 528
13.7 The Potential KdV Equation 528
13.7.1 Using the Tanh-coth Method 529
13.7.2 Multiple-soliton Solutions of the Potential KdV Equation……531 13.8 The Gardner Equation 533
13.8.1 The Kink Solution 533
13.8.2 The Soliton Solution 534
13.8.3 N-soliton Solutions of the Positive Gardner Equation 535
13.8.4 Singular Soliton Solutions 537
13.9 Generalized KdV Equation with Two Power Nonlinearities 542
13.9.1 Using the Tanh Method 543
13.9.2 Using the Sine-cosine Method 544
13.10 Compactons:Solitons with Compact Support 544
13.10.1 The K(n,n)Equation 546
13.11 Variants of the K(n,n) Equation 547
13.11.1 First Variant 548
13.11.2 Second Variant 549
13.11.3 Third Variant 551
13.12 Compacton-like Solutions 553
13.12.1 The Modified KdV Equation 553
13.12.2 The Gardner Equation 554
13.12.3 The Modified Equal Width Equation 554
References 555
14 KdV and mKdV Equations of Higher-orders 557
14.1 Introduction 557
14.2 Family of Higher-order KdV Equations 558
14.2.1 Fifth-order KdV Equations 558
14.2.2 Seventh-order KdV Equations 561
14.2.3 Ninth-order KdV Equations 562
14.3 Fifth-order KdV Equations 562
14.3.1 Using the Tanh-coth Method 563
14.3.2 The First Condition 564
14.3.3 The Second Condition 566
14.3.4 N-soliton Solutions of the Fifth-order KdV Equations 567
14.4 Seventh-order KdV Equations 576
14.4.1 Using the Tanh-coth Method 576
14.4.2 N-soliton Solutions of the Seventh-order KdV Equations 578
14.5 Ninth-order KdV Equations 582
14.5.1 Using the Tanh-coth Method 583
14.5.2 The Soliton Solutions 584
14.6 Family of Higher-order mKdV Equations 585
14.6.1 N-soliton Solutions for Fifth-order mKdV Equation 586
14.6.2 Singular Soliton Solutions for Fifth-order mKdV Equation 587
14.6.3 N-soliton Solutions for the Seventh-order mKdV Equation 589
14.7 Complex Solution for the Seventh-order mKdV Equations 591
14.8 The Hirota-Satsuma Equations 592
14.8.1 Using the Tanh-coth Method 593
14.8.2 N-soliton Solutions of the Hirota-Satsuma System 594
14.8.3 N-soliton Solutions by an Alternative Method 596
14.9 Generalized Short Wave Equation 597
References 602
15 Family of KdV-type Equations 605
15.1 Introduction 605
15.2 The Complex Modified KdV Equation 606
15.2.1 Using the Sine-cosine Method 607
15.2.2 Using the Tanh-coth Method 608
15.3 The Benjamin-Bona-Mahony Equation 612
15.3.1 Using the Sine-cosine Method 612
15.3.2 Using the Tanh-coth Method 613
15.4 The Medium Equal Width(MEW)Equation 615
15.4.1 Using the Sine-cosine Method 615
15.4.2 Using the Tanh-coth Method 616
15.5 The Kawahara and the Modified Kawahara Equations 617
15.5.1 The Kawahara Equation 618
15.5.2 The Modified Kawahara Equation 619
15.6 The Kadomtsev-Petviashvili(KP)Equation 620
15.6.1 Using the Tanh-coth Method 621
15.6.2 Multiple-soliton Solutions of the KP Equation 622
15.7 The Zakharov-Kuznetsov(ZK)Equation 626
15.8 The Benjamin-Ono Equation 629
15.9 The KdV-Burgers Equation 630
15.10 Seventh-order KdV Equation 632
15.10.1 The Sech Method 632
15.11 Ninth-order KdV Equation 634
15.11.1 The Sech Method 634
References 637
16 Boussinesq,Klein-Gordon and Liouville Equations 639
16.1 Introduction 639
16.2 The Boussinesq Equation 641
16.2.1 Using the Tanh-coth Method 641
16.2.2 Multiple-soliton Solutions of the Boussinesq Equation 643
16.3 The Improved Boussinesq Equation 646
16.4 The Klein-Gordon Equation 648
16.5 The Liouville Equation 649
16.6 The Sine-Gordon Equation 651
16.6.1 Using the Tanh-coth Method 651
16.6.2 Using the B？cklund Transformation 654
16.6.3 Multiple-soliton Solutions for Sine-Gordon Equation 655
16.7 The Sinh-Gordon Equation 657
16.8 The Dodd-Bullough-Mikhailov Equation 658
16.9 The Tzitzeica-Dodd-Bullough Equation 659
16.10 The Zhiber-Shabat Equation 661
References 662
17 Burgers,Fisher and Related Equations 665
17.1 Introduction 665
17.2 The Burgers Equation 666
17.2.1 Using the Tanh-coth Method 667
17.2.2 Using the Cole-Hopf Transformation 668
17.3 The Fisher Equation 670
17.4 The Huxley Equation 671
17.5 The Burgers-Fisher Equation 673
17.6 The Burgers-Huxley Equation 673
17.7 The FitzHugh-Nagumo Equation 675
17.8 Parabolic Equation with Exponential Nonlinearity 676
17.9 The Coupled Burgers Equation 678
17.10 The Kuramoto-Sivashinsky(KS)Equation 680
References 681
18 Families of Camassa-Holm and Schrodinger Equations 683
18.1 Introduction 683
18.2 The Family of Camassa-Holm Equations 686
18.2.1 Using the Tanh-coth Method 686
18.2.2 Using an Exponential Algorithm 688
18.3 Schrodinger Equation of Cubic Nonlinearity 689
18.4 Schrodinger Equation with Power Law Nonlinearity 690
18.5 The Ginzburg-Landau Equation 692
18.5.1 The Cubic Ginzburg-Landau Equation 693
18.5.2 The Generalized Cubic Ginzburg-Landau Equation 694
18.5.3 The Generalized Quintic Ginzburg-Landau Equation 695
References 696
Appendix 699
A Indefinite Integrals 699
A.1 Fundamental Forms 699
A.2 Trigonometric Forms 700
A.3 Inverse Trigonometric Forms 700
A.4 Exponential and Logarithmic Forms 701
A.5 Hyperbolic Forms 701
A.6 Other Forms 702
B Series 703
B.1 Exponential Functions 703
B.2 Trigonometric Functions 703
B.3 Inverse Trigonometric Functions 704
B.4 Hyperbolic Functions 704
B.5 Inverse Hyperbolic Functions 704
C Exact Solutions of Burgers' Equation 705
D Padé Approximants for Well-Known Functions 707
D.1 Exponential Functions 707
D.2 Trigonometric Functions 707
D.3 Hyperbolic Functions 709
D.4 Logarithmic Functions 709
E The Error and Gamma Functions 711
E.1 The Error function 711
E.2 The Gamma function Г（x） 711
F Infinite Series 712
F.1 Numerical Series 712
F.2 Trigonometric Series 713
Answers 715
Index 739