Partial Differential Equations and Solitary Waves Theory (eBook)

Preface 7
Contents 10
Part I Partial Differential Equations 19
Basic Concepts 20
1.1 Introduction 20
1.2 Definitions 21
1.3 Classifications of a Second-order PDE 31
References 34
First-order Partial Differential Equations 35
2.1 Introduction 35
2.2 Adomian Decomposition Method 35
2.3 The Noise Terms Phenomenon 52
2.4 The Modified Decomposition Method 57
2.5 The Variational Iteration Method 63
2.6 Method of Characteristics 70
2.7 Systems of Linear PDEs by Adomian Method 75
2.8 Systems of Linear PDEs by Variational Iteration Method 79
References 84
One Dimensional Heat Flow 85
3.1 Introduction 85
3.2 The Adomian Decomposition Method 86
3.3 The Variational Iteration Method 99
3.4 Method of Separation of Variables 105
References 122
Higher Dimensional Heat Flow 123
4.1 Introduction 123
4.2 Adomian Decomposition Method 124
4.3 Method of Separation of Variables 140
References 156
One DimensionalWave Equation 158
5.1 Introduction 158
5.2 Adomian Decomposition Method 159
5.3 The Variational Iteration Method 177
5.4 Method of Separation of Variables 189
5.5 Wave Equation in an Infinite Domain: D’Alembert Solution 205
References 209
Higher Dimensional Wave Equation 210
6.1 Introduction 210
6.2 Adomian Decomposition Method 210
6.3 Method of Separation of Variables 235
References 251
Laplace’s Equation 252
7.1 Introduction 252
7.2 Adomian Decomposition Method 253
7.3 The Variational Iteration Method 262
7.4 Method of Separation of Variables 266
7.5 Laplace’s Equation in Polar Coordinates 282
References 298
Nonlinear Partial Differential Equations 300
8.1 Introduction 300
8.2 Adomian Decomposition Method 302
8.3 Nonlinear ODEs by Adomian Method 316
8.4 Nonlinear ODEs by VIM 327
8.5 Nonlinear PDEs by Adomian Method 334
8.6 Nonlinear PDEs by VIM 349
8.7 Nonlinear PDEs Systems by Adomian Method 356
8.8 Systems of Nonlinear PDEs by VIM 362
References 366
Linear and Nonlinear Physical Models 367
9.1 Introduction 367
9.2 The Nonlinear Advection Problem 368
9.3 The Goursat Problem 374
9.4 The Klein-Gordon Equation 384
9.5 The Burgers Equation 395
9.6 The Telegraph Equation 402
9.7 Schrodinger Equation 408
9.8 Korteweg-deVries Equation 415
9.9 Fourth-order Parabolic Equation 419
References 427
Numerical Applications and Pad ´ e Approximants 428
10.1 Introduction 428
10.2 Ordinary Differential Equations 429
10.3 Partial Differential Equations 440
10.4 The Pad ´e Approximants 443
10.5 Pad ´e Approximants and Boundary Value Problems 452
References 468
Solitons and Compactons 469
11.1 Introduction 469
11.2 Solitons 471
11.3 Compactons 481
11.4 The Defocusing Branch of K(n,n) 486
References 487
Part II Solitray Waves Theory 488
Solitary Waves Theory 489
12.1 Introduction 489
12.2 Definitions 490
12.3 Analysis of the Methods 501
12.4 Conservation Laws 506
References 512
The Family of the KdV Equations 513
13.1 Introduction 513
13.2 The Family of the KdV Equations 515
13.3 The KdV Equation 517
13.4 The Modified KdV Equation 528
13.5 Singular Soliton Solutions 533
13.6 The Generalized KdV Equation 536
13.7 The Potential KdV Equation 538
13.8 The Gardner Equation 543
13.9 Generalized KdV Equation with Two Power Nonlinearities 552
13.10 Compactons: Solitons with Compact Support 554
13.11 Variants of the K(n,n) Equation 557
13.12 Compacton-like Solutions 563
References 565
KdV and mKdV Equations of Higher-orders 567
14.1 Introduction 567
14.2 Family of Higher-order KdV Equations 568
14.3 Fifth-order KdV Equations 572
14.4 Seventh-order KdV Equations 586
14.5 Ninth-order KdV Equations 592
14.6 Family of Higher-order mKdV Equations 595
14.7 Complex Solution for the Seventh-order mKdV Equations 601
14.8 The Hirota-Satsuma Equations 602
14.9 Generalized Short Wave Equation 607
References 612
Family of KdV-type Equations 614
15.1 Introduction 614
15.2 The Complex Modified KdV Equation 615
15.3 The Benjamin-Bona-Mahony Equation 621
15.4 The Medium Equal Width (MEW) Equation 624
15.5 The Kawahara and the Modified Kawahara Equations 626
15.6 The Kadomtsev-Petviashvili (KP) Equation 629
15.7 The Zakharov-Kuznetsov (ZK) Equation 635
15.8 The Benjamin-Ono Equation 638
15.9 The KdV-Burgers Equation 639
15.10 Seventh-order KdV Equation 641
15.11 Ninth-order KdV Equation 643
References 646
Boussinesq, Klein-Gordon and Liouville Equations 647
16.1 Introduction 647
16.2 The Boussinesq Equation 649
16.3 The Improved Boussinesq Equation 654
16.4 The Klein-Gordon Equation 656
16.5 The Liouville Equation 657
16.6 The Sine-Gordon Equation 659
16.7 The Sinh-Gordon Equation 665
16.8 The Dodd-Bullough-Mikhailov Equation 666
16.9 The Tzitzeica-Dodd-Bullough Equation 667
16.10 The Zhiber-Shabat Equation 669
References 670
Burgers, Fisher and Related Equations 672
17.1 Introduction 672
17.2 The Burgers Equation 673
17.3 The Fisher Equation 677
17.4 The Huxley Equation 678
17.5 The Burgers-Fisher Equation 680
17.6 The Burgers-Huxley Equation 680
17.7 The FitzHugh-Nagumo Equation 682
17.8 Parabolic Equation with Exponential Nonlinearity 683
17.9 The Coupled Burgers Equation 685
17.10 The Kuramoto-Sivashinsky (KS) Equation 687
References 688
Families of Camassa-Holm and Schrodinger Equations 689
18.1 Introduction 689
18.2 The Family of Camassa-Holm Equations 692
18.3 Schrodinger Equation of Cubic Nonlinearity 695
18.4 Schrodinger Equation with Power Law Nonlinearity 696
18.5 The Ginzburg-Landau Equation 698
References 702
Indefinite Integrals 704
A.1 Fundamental Forms 704
A.2 Trigonometric Forms 705
A.3 Inverse Trigonometric Forms 705
A.4 Exponential and Logarithmic Forms 706
A.5 Hyperbolic Forms 706
A.6 Other Forms 707
Series 708
B.1 Exponential Functions 708
B.2 Trigonometric Functions 708
B.3 Inverse Trigonometric Functions 709
B.4 Hyperbolic Functions 709
B.5 Inverse Hyperbolic Functions 709
Exact Solutions of Burgers’ Equation 710
Pade Approximants for Well-Known Functions 712
D.1 Exponential Functions 712
D.2 Trigonometric Functions 712
D.3 Hyperbolic Functions 714
D.4 Logarithmic Functions 714
The Error and Gamma Functions 716
E.1 The Error function 716
E.2 The Gamma function 716
Infinite Series 717
F.1 Numerical Series 717
F.2 Trigonometric Series 718
Answers 720
Index 743

"Part I Partial Differential Equations (p. 1-3)

Chapter 1 Basic Concepts

1.1 Introduction

It is well known that most of the phenomena that arise in mathematical physics and engineering fields can be described by partial differential equations (PDEs). In physics for example, the heat flow and the wave propagation phenomena are well described by partial differential equations [1–4]. In ecology, most population models are governed by partial differential equations [5–6].

The dispersion of a chemically reactive material is characterized by partial differential equations. In addition, most physical phenomena of fluid dynamics, quantum mechanics, electricity, plasma physics, propagation of shallow water waves, and many other models are controlled within its domain of validity by partial differential equations. Partial differential equations have become a useful tool for describing these natural phenomena of science and engineering models.

Therefore, it becomes increasingly important to be familiar with all traditional and recently developed methods for solving partial differential equations, and the implementation of these methods. However, in this text, we will restrict our analysis to solve partial differential equations along with the given conditions that characterize the initial conditions and the boundary conditions of the dependent variable [7].We fill focus our concern on deriving solutions to PDEs and not on the derivation of these equations.

In this text, our presentation will be based on applying the recent developments in this field and on applying some of the traditional methods as well. The formulation of partial differential equations and the scientific interpretation of the models will not be discussed. It is to be noted that several methods are usually used in solving PDEs.

The newly developed Adomian decomposition method and the related improvements of the modified technique and the noise terms phenomena will be effectively used. Moreover, the variational iteration method that was recently developed will be used as well. The recently developed techniques have been proved to be reliable, accurate and effective in both the analytic and the numerical purposes.

The Adomian decomposition method and the variational iteration method were formally proved to provide the solution in terms of a rapid convergent infinite series that may yield the exact solution in many cases. As will be seen in part I of this text, both methods require the use of conditions such as initial conditions. The other related modifications were shown to be powerful in that it accelerate the rapid convergence of the solution. However, some of the traditional methods, such as the separation of variables method and the method of characteristics will be applied as well.

Moreover, the other techniques, such as integral transforms, perturbation methods, numerical methods and other traditional methods, that are usually used in other texts, will not be used in this text. In Part II of this text, we will focus our work on nonlinear evolution equations that describe a variety of physical phenomena. The Hirota’s bilinear formalism and the tanh-coth method will be employed in the second part. These methods will be used to determine soliton solutions andmultiple-soliton solutions, for completely integrable equations, as well. Several well-known nonlinear evolution equations such as the KdV equation, Burgers equation, Boussinesq equation, Camassa-Holm equation, sine-Gordon equation, and many others will be investigated."