Partial Differential Equations and Solitary Waves Theory

Preface

Contents

Part I Partial Differential Equations

Basic Concepts

1.1 Introduction

1.2 Definitions

1.3 Classifications of a Second-order PDE

References

First-order Partial Differential Equations

2.1 Introduction

2.2 Adomian Decomposition Method

2.3 The Noise Terms Phenomenon

2.4 The Modified Decomposition Method

2.5 The Variational Iteration Method

2.6 Method of Characteristics

2.7 Systems of Linear PDEs by Adomian Method

2.8 Systems of Linear PDEs by Variational Iteration Method

References

One Dimensional Heat Flow

3.1 Introduction

3.2 The Adomian Decomposition Method

3.3 The Variational Iteration Method

3.4 Method of Separation of Variables

References

Higher Dimensional Heat Flow

4.1 Introduction

4.2 Adomian Decomposition Method

4.3 Method of Separation of Variables

References

One DimensionalWave Equation

5.1 Introduction

5.2 Adomian Decomposition Method

5.3 The Variational Iteration Method

5.4 Method of Separation of Variables

5.5 Wave Equation in an Infinite Domain: D’Alembert Solution

References

Higher Dimensional Wave Equation

6.1 Introduction

6.2 Adomian Decomposition Method

6.3 Method of Separation of Variables

References

Laplace’s Equation

7.1 Introduction

7.2 Adomian Decomposition Method

7.3 The Variational Iteration Method

7.4 Method of Separation of Variables

7.5 Laplace’s Equation in Polar Coordinates

References

Nonlinear Partial Differential Equations

8.1 Introduction

8.2 Adomian Decomposition Method

8.3 Nonlinear ODEs by Adomian Method

8.4 Nonlinear ODEs by VIM

8.5 Nonlinear PDEs by Adomian Method

8.6 Nonlinear PDEs by VIM

8.7 Nonlinear PDEs Systems by Adomian Method

8.8 Systems of Nonlinear PDEs by VIM

References

Linear and Nonlinear Physical Models

9.1 Introduction

9.2 The Nonlinear Advection Problem

9.3 The Goursat Problem

9.4 The Klein-Gordon Equation

9.5 The Burgers Equation

9.6 The Telegraph Equation

9.7 Schrodinger Equation

9.8 Korteweg-deVries Equation

9.9 Fourth-order Parabolic Equation

References

Numerical Applications and Pad ´ e Approximants

10.1 Introduction

10.2 Ordinary Differential Equations

10.3 Partial Differential Equations

10.4 The Pad ´e Approximants

10.5 Pad ´e Approximants and Boundary Value Problems

References

Solitons and Compactons

11.1 Introduction

11.2 Solitons

11.3 Compactons

11.4 The Defocusing Branch of K(n,n)

References

Part II Solitray Waves Theory

Solitary Waves Theory

12.1 Introduction

12.2 Definitions

12.3 Analysis of the Methods

12.4 Conservation Laws

References

The Family of the KdV Equations

13.1 Introduction

13.2 The Family of the KdV Equations

13.3 The KdV Equation

13.4 The Modified KdV Equation

13.5 Singular Soliton Solutions

13.6 The Generalized KdV Equation

13.7 The Potential KdV Equation

13.8 The Gardner Equation

13.9 Generalized KdV Equation with Two Power Nonlinearities

13.10 Compactons: Solitons with Compact Support

13.11 Variants of the K(n,n) Equation

13.12 Compacton-like Solutions

References

KdV and mKdV Equations of Higher-orders

14.1 Introduction

14.2 Family of Higher-order KdV Equations

14.3 Fifth-order KdV Equations

14.4 Seventh-order KdV Equations

14.5 Ninth-order KdV Equations

14.6 Family of Higher-order mKdV Equations

14.7 Complex Solution for the Seventh-order mKdV Equations

14.8 The Hirota-Satsuma Equations

14.9 Generalized Short Wave Equation

References

Family of KdV-type Equations

15.1 Introduction

15.2 The Complex Modified KdV Equation

15.3 The Benjamin-Bona-Mahony Equation

15.4 The Medium Equal Width (MEW) Equation

15.5 The Kawahara and the Modified Kawahara Equations

15.6 The Kadomtsev-Petviashvili (KP) Equation

15.7 The Zakharov-Kuznetsov (ZK) Equation

15.8 The Benjamin-Ono Equation

15.9 The KdV-Burgers Equation

15.10 Seventh-order KdV Equation

15.11 Ninth-order KdV Equation

References

Boussinesq, Klein-Gordon and Liouville Equations

16.1 Introduction

16.2 The Boussinesq Equation

16.3 The Improved Boussinesq Equation

16.4 The Klein-Gordon Equation

16.5 The Liouville Equation

16.6 The Sine-Gordon Equation

16.7 The Sinh-Gordon Equation

16.8 The Dodd-Bullough-Mikhailov Equation

16.9 The Tzitzeica-Dodd-Bullough Equation

16.10 The Zhiber-Shabat Equation

References

Burgers, Fisher and Related Equations

17.1 Introduction

17.2 The Burgers Equation

17.3 The Fisher Equation

17.4 The Huxley Equation

17.5 The Burgers-Fisher Equation

17.6 The Burgers-Huxley Equation

17.7 The FitzHugh-Nagumo Equation

17.8 Parabolic Equation with Exponential Nonlinearity

17.9 The Coupled Burgers Equation

17.10 The Kuramoto-Sivashinsky (KS) Equation

References

Families of Camassa-Holm and Schrodinger Equations

18.1 Introduction

18.2 The Family of Camassa-Holm Equations

18.3 Schrodinger Equation of Cubic Nonlinearity

18.4 Schrodinger Equation with Power Law Nonlinearity

18.5 The Ginzburg-Landau Equation

References

Indefinite Integrals

A.1 Fundamental Forms

A.2 Trigonometric Forms

A.3 Inverse Trigonometric Forms

A.4 Exponential and Logarithmic Forms

A.5 Hyperbolic Forms

A.6 Other Forms

Series

B.1 Exponential Functions

B.2 Trigonometric Functions

B.3 Inverse Trigonometric Functions

B.4 Hyperbolic Functions

B.5 Inverse Hyperbolic Functions

Exact Solutions of Burgers’ Equation

Pade Approximants for Well-Known Functions

D.1 Exponential Functions

D.2 Trigonometric Functions

D.3 Hyperbolic Functions

D.4 Logarithmic Functions

The Error and Gamma Functions

E.1 The Error function

E.2 The Gamma function

Infinite Series

F.1 Numerical Series

F.2 Trigonometric Series

Answers

Index