Wazwaz Partial Differential Equations and Solitary Waves Theory

1. Auflage 2010
ISBN: 978-3-642-00251-9
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 700 Seiten

Reihe: Nonlinear Physical Science

'Partial Differential Equations and Solitary Waves Theory' is a self-contained book divided into two parts: Part I is a coherent survey bringing together newly developed methods for solving PDEs. While some traditional techniques are presented, this part does not require thorough understanding of abstract theories or compact concepts. Well-selected worked examples and exercises shall guide the reader through the text. Part II provides an extensive exposition of the solitary waves theory. This part handles nonlinear evolution equations by methods such as Hirota's bilinear method or the tanh-coth method. A self-contained treatment is presented to discuss complete integrability of a wide class of nonlinear equations. This part presents in an accessible manner a systematic presentation of solitons, multi-soliton solutions, kinks, peakons, cuspons, and compactons. While the whole book can be used as a text for advanced undergraduate and graduate students in applied mathematics, physics and engineering, Part II will be most useful for graduate students and researchers in mathematics, engineering, and other related fields. Dr. Abdul-Majid Wazwaz is a Professor of Mathematics at Saint Xavier University, Chicago, Illinois, USA.

Autoren/Hrsg.

Wazwaz, Abdul-Majid

Inhaltsverzeichnis

1;Preface;7
2;Contents;10
3;Part I Partial Differential Equations;19
3.1;Basic Concepts;20
3.1.1;1.1 Introduction;20
3.1.2;1.2 Definitions ;21
3.1.3;1.3 Classifications of a Second-order PDE;31
3.1.4;References;34
3.2;First-order Partial Differential Equations;35
3.2.1;2.1 Introduction;35
3.2.2;2.2 Adomian Decomposition Method;35
3.2.3;2.3 The Noise Terms Phenomenon;52
3.2.4;2.4 The Modified Decomposition Method;57
3.2.5;2.5 The Variational Iteration Method;63
3.2.6;2.6 Method of Characteristics;70
3.2.7;2.7 Systems of Linear PDEs by Adomian Method;75
3.2.8;2.8 Systems of Linear PDEs by Variational Iteration Method;79
3.2.9;References;84
3.3;One Dimensional Heat Flow;85
3.3.1;3.1 Introduction;85
3.3.2;3.2 The Adomian Decomposition Method;86
3.3.3;3.3 The Variational Iteration Method;99
3.3.4;3.4 Method of Separation of Variables;105
3.3.5;References;122
3.4;Higher Dimensional Heat Flow;123
3.4.1;4.1 Introduction;123
3.4.2;4.2 Adomian Decomposition Method;124
3.4.3;4.3 Method of Separation of Variables;140
3.4.4;References;156
3.5;One DimensionalWave Equation;158
3.5.1;5.1 Introduction;158
3.5.2;5.2 Adomian Decomposition Method;159
3.5.3;5.3 The Variational Iteration Method;177
3.5.4;5.4 Method of Separation of Variables;189
3.5.5;5.5 Wave Equation in an Infinite Domain: D’Alembert Solution;205
3.5.6;References;209
3.6;Higher Dimensional Wave Equation;210
3.6.1;6.1 Introduction;210
3.6.2;6.2 Adomian Decomposition Method;210
3.6.3;6.3 Method of Separation of Variables;235
3.6.4;References;251
3.7;Laplace’s Equation;252
3.7.1;7.1 Introduction;252
3.7.2;7.2 Adomian Decomposition Method;253
3.7.3;7.3 The Variational Iteration Method;262
3.7.4;7.4 Method of Separation of Variables;266
3.7.5;7.5 Laplace’s Equation in Polar Coordinates;282
3.7.6;References;298
3.8;Nonlinear Partial Differential Equations;300
3.8.1;8.1 Introduction;300
3.8.2;8.2 Adomian Decomposition Method;302
3.8.3;8.3 Nonlinear ODEs by Adomian Method;316
3.8.4;8.4 Nonlinear ODEs by VIM;327
3.8.5;8.5 Nonlinear PDEs by Adomian Method;334
3.8.6;8.6 Nonlinear PDEs by VIM;349
3.8.7;8.7 Nonlinear PDEs Systems by Adomian Method;356
3.8.8;8.8 Systems of Nonlinear PDEs by VIM;362
3.8.9;References;366
3.9;Linear and Nonlinear Physical Models;367
3.9.1;9.1 Introduction;367
3.9.2;9.2 The Nonlinear Advection Problem;368
3.9.3;9.3 The Goursat Problem;374
3.9.4;9.4 The Klein-Gordon Equation;384
3.9.5;9.5 The Burgers Equation;395
3.9.6;9.6 The Telegraph Equation;402
3.9.7;9.7 Schrodinger Equation;408
3.9.8;9.8 Korteweg-deVries Equation;415
3.9.9;9.9 Fourth-order Parabolic Equation;419
3.9.10;References;427
3.10;Numerical Applications and Pad ´ e Approximants;428
3.10.1;10.1 Introduction;428
3.10.2;10.2 Ordinary Differential Equations ;429
3.10.3;10.3 Partial Differential Equations;440
3.10.4;10.4 The Pad ´e Approximants;443
3.10.5;10.5 Pad ´e Approximants and Boundary Value Problems;452
3.10.6;References;468
3.11;Solitons and Compactons;469
3.11.1;11.1 Introduction;469
3.11.2;11.2 Solitons;471
3.11.3;11.3 Compactons;481
3.11.4;11.4 The Defocusing Branch of K(n,n);486
3.11.5;References;487
4;Part II Solitray Waves Theory;488
4.1;Solitary Waves Theory;489
4.1.1;12.1 Introduction;489
4.1.2;12.2 Definitions;490
4.1.3;12.3 Analysis of the Methods;501
4.1.4;12.4 Conservation Laws;506
4.1.5;References;512
4.2;The Family of the KdV Equations;513
4.2.1;13.1 Introduction;513
4.2.2;13.2 The Family of the KdV Equations;515
4.2.3;13.3 The KdV Equation;517
4.2.4;13.4 The Modified KdV Equation;528
4.2.5;13.5 Singular Soliton Solutions;533
4.2.6;13.6 The Generalized KdV Equation;536
4.2.7;13.7 The Potential KdV Equation;538
4.2.8;13.8 The Gardner Equation;543
4.2.9;13.9 Generalized KdV Equation with Two Power Nonlinearities;552
4.2.10;13.10 Compactons: Solitons with Compact Support;554
4.2.11;13.11 Variants of the K(n,n) Equation;557
4.2.12;13.12 Compacton-like Solutions;563
4.2.13;References;565
4.3;KdV and mKdV Equations of Higher-orders;567
4.3.1;14.1 Introduction;567
4.3.2;14.2 Family of Higher-order KdV Equations;568
4.3.3;14.3 Fifth-order KdV Equations;572
4.3.4;14.4 Seventh-order KdV Equations;586
4.3.5;14.5 Ninth-order KdV Equations;592
4.3.6;14.6 Family of Higher-order mKdV Equations;595
4.3.7;14.7 Complex Solution for the Seventh-order mKdV Equations;601
4.3.8;14.8 The Hirota-Satsuma Equations;602
4.3.9;14.9 Generalized Short Wave Equation;607
4.3.10;References;612
4.4;Family of KdV-type Equations;614
4.4.1;15.1 Introduction;614
4.4.2;15.2 The Complex Modified KdV Equation;615
4.4.3;15.3 The Benjamin-Bona-Mahony Equation;621
4.4.4;15.4 The Medium Equal Width (MEW) Equation;624
4.4.5;15.5 The Kawahara and the Modified Kawahara Equations;626
4.4.6;15.6 The Kadomtsev-Petviashvili (KP) Equation;629
4.4.7;15.7 The Zakharov-Kuznetsov (ZK) Equation;635
4.4.8;15.8 The Benjamin-Ono Equation;638
4.4.9;15.9 The KdV-Burgers Equation;639
4.4.10;15.10 Seventh-order KdV Equation;641
4.4.11;15.11 Ninth-order KdV Equation;643
4.4.12;References;646
4.5;Boussinesq, Klein-Gordon and Liouville Equations;647
4.5.1;16.1 Introduction;647
4.5.2;16.2 The Boussinesq Equation;649
4.5.3;16.3 The Improved Boussinesq Equation;654
4.5.4;16.4 The Klein-Gordon Equation;656
4.5.5;16.5 The Liouville Equation;657
4.5.6;16.6 The Sine-Gordon Equation;659
4.5.7;16.7 The Sinh-Gordon Equation;665
4.5.8;16.8 The Dodd-Bullough-Mikhailov Equation;666
4.5.9;16.9 The Tzitzeica-Dodd-Bullough Equation;667
4.5.10;16.10 The Zhiber-Shabat Equation;669
4.5.11;References;670
4.6;Burgers, Fisher and Related Equations;672
4.6.1;17.1 Introduction;672
4.6.2;17.2 The Burgers Equation;673
4.6.3;17.3 The Fisher Equation;677
4.6.4;17.4 The Huxley Equation;678
4.6.5;17.5 The Burgers-Fisher Equation;680
4.6.6;17.6 The Burgers-Huxley Equation;680
4.6.7;17.7 The FitzHugh-Nagumo Equation;682
4.6.8;17.8 Parabolic Equation with Exponential Nonlinearity;683
4.6.9;17.9 The Coupled Burgers Equation;685
4.6.10;17.10 The Kuramoto-Sivashinsky (KS) Equation;687
4.6.11;References;688
4.7;Families of Camassa-Holm and Schrodinger Equations;689
4.7.1;18.1 Introduction;689
4.7.2;18.2 The Family of Camassa-Holm Equations;692
4.7.3;18.3 Schrodinger Equation of Cubic Nonlinearity;695
4.7.4;18.4 Schrodinger Equation with Power Law Nonlinearity;696
4.7.5;18.5 The Ginzburg-Landau Equation;698
4.7.6;References;702
5;Indefinite Integrals;704
5.1;A.1 Fundamental Forms;704
5.2;A.2 Trigonometric Forms;705
5.3;A.3 Inverse Trigonometric Forms;705
5.4;A.4 Exponential and Logarithmic Forms;706
5.5;A.5 Hyperbolic Forms;706
5.6;A.6 Other Forms;707
6;Series;708
6.1;B.1 Exponential Functions;708
6.2;B.2 Trigonometric Functions;708
6.3;B.3 Inverse Trigonometric Functions;709
6.4;B.4 Hyperbolic Functions;709
6.5;B.5 Inverse Hyperbolic Functions;709
7;Exact Solutions of Burgers’ Equation;710
8;Pade Approximants for Well-Known Functions;712
8.1;D.1 Exponential Functions;712
8.2;D.2 Trigonometric Functions;712
8.3;D.3 Hyperbolic Functions;714
8.4;D.4 Logarithmic Functions;714
9;The Error and Gamma Functions;716
9.1;E.1 The Error function;716
9.2;E.2 The Gamma function;716
10;Infinite Series;717
10.1;F.1 Numerical Series;717
10.2;F.2 Trigonometric Series;718
11;Answers;720
12;Index;743

Leseproben

"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."