Partial Differential Equations and Solitary Waves Theory is designed to serve as a text and a reference. The book is designed to be accessible to adwnnced undergraduate and beginning graduate students as well as research monograph to researchers in applied mathematics, science and engineering. This text is different from other texts in that it explains classical methods in a non abstract way and it introduces and explains how the newly developed methods provide more concise methods to provide efficient results.

Part Ⅰ Partial Differential Equations

1 Basic Concepts

 1.1 Introduction

 1.2 Definitions

1.2.1 Definition of a PDE

1.2.2 Order of a PDE

1.2.3 Linear and Nonlinear PDEs

1.2.4 Some Linear Partial Differential Equations

1.2.5 Some Nonlinear Partial Differential Equations

1.2.6 Homogeneous and Inhomogeneous PDEs

1.2.7 Solution of a PDE

1.2.8 Boundary Conditions

1.2.9 Initial Conditions

1.2.10 Well-posed PDEs

 1.3 Classifications of a Second-order PDE

 References

2 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

3  One Dimensional Heat Flow

 3.1 Introduction

 3.2 The Adomian Decomposition Method

3.2.1 Homogeneous Heat Equations

3.2.2 lnhomogeneous Heat Equations

 3.3 The Variational Iteration Method

3.3.1 Homogeneous Heat Equations

3.3.2 Inhomogeneous Heat Equations

 3.4 Method of Separation of Variables

3.4.1 Analysis of the Method

3.4.2 Inlaomogeneous Boundary Conditions

3.4.3 Equations with Lateral Heat Loss

 References

4 Higher Dimensional Heat Flow

 4.1 Introduction

 4.2 Adomian Decomposition Method

4.2.1 Two Dimensional Heat Flow

4.2.2 Three Dimensional Heat Flow

 4.3 Method of Separation of Variables

4.3.1 Two Dimensional Heat Flow

4.3.2 Three Dimensional Heat Flow

 References

5 One Dimensional Wave Equation

 5.1 Introduction

 5.2 Adomian Decomposition Method

5.2.1 Homogeneous Wave Equations

5.2.2 Inhomogeneous Wave Equations

5.2.3 Wave Equation in an Infinite Domain

 5.3 The Variational Iteration Method

5.3.1 Homogeneous Wave Equations

5.3.2 Inhomogeneous Wave Equations

5.3.3 Wave Equation in an Infinite Domain

 5.4 Method of Separation of Variables

5.4.1 Analysis of the Method

5.4.2 Inhomogeneous Boundary Conditions

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

 References

6 Higher Dimensional Wave Equation

 6.1 Introduction

 6.2 Adomian Decomposition Method

6.2.1 Two Dimensional Wave Equation

6.2.2 Three Dimensional Wave Equation

 ……

7 Laplace's Equation

8 Nonlinear Partial Differential Equations

9 Linear and Nonlinear Physical Models

10 Numerical Applications and Pade Approximants

11 Solitons and Compactons

Part Ⅱ Solitray Waves Theory

12 Solitary Waves Theory

13 The Family of the KdV Equations

14 KdV and mKdV Equations of Higher-orders

15 Family of KdV-type Equations

16 Boussinesq, Klein-Gordon and Liouville Equations

17 Burgers, Fisher and Related Equations

18 Families of Camassa-Holm and Schrodinger Equations

Appendix

A Indefinite Integrals

B Series

C Exact Solutions of Burgers' Equation

D Pade Approximants for Well-Known Functions

E The Error and Gamma Functions

F  Infinite Series

Answers

Index