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 Inhomogeneot, s 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 Inhomogeneous 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 Inhomogeneous 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: DAlembert 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
6.3 Method of Separation of Variables
6.3.1 Two Dimensional Wave Equation
6.3.2 Three Dimensional Wave Equation
References
7 Laplaces Equation
7.1 Introduction
7.2 Adomian Decomposition Method
7.2.1 Two Dimensional Laplaces Equation ...
7.3 The Variational Iteration Method
7.4 Method of Separation of Variables
7.4.1 Laplaces Equation in Two Dimensions..
7.4.2 Laplaces Equation in Three Dimensions
7.5 Laplaces Equation in Polar Coordinates
7.5.1 Laplaces Equation for a Disc
7.5.2 Laplaces Equation for an Annulus
References
8 Nonlinear Partial Differential Equations
8.1 Introduction
8.2 Adomian Decomposition Method
8.2.1 Calculation of Adomian Polynomials ...
8.2.2 Alternative Algorithm for Calculating Adomian Polynomials
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
9 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
9.4.1 Linear Klein-Gordon Equation
9.4.2 Nonlinear Klein-Gordon Equation
9.4.3 The Sine-Gordon Equation
9.5 The Burgers Equation
9.6 The Telegraph Equation
9.7 Schrodinger Equation
9.7.1 The Linear Schrodinger Equation
9.7.2 The Nonlinear Schrodinger Equation
9.8 Korteweg-deVries Equation
9.9 Fourth-order Parabolic Equation
9.9.1 Equations with Constant Coefficients
9.9.2 Equations with Variable Coefficients
References
10 Numerical Applications and Pade Approximants
10.1 Introduction
10.2 Ordinary Differential Equations
10.2.1 Perturbation Problems
10.2.2 Nonperturbed Problems
10.3 Partial Differential Equations
10.4 The Pade Approximants
10.5 Pad6 Approximants and Boundary Value Problems
References
11 Solitons and Compaetons
11.1 Introduction
11.2 Solitons
11.2.1 The KdV Equation
11.2.2 The Modified KdV Equation
11.2.3 The Generalized KdV Equation
11.2.4 The Sine-Gordon Equation
11.2.5 The Boussinesq Equation
11.2.6 The Kadomtsev-Petviashvili Equation
11.3 Compactons
11.4 The Defocusing Branch of K(n,n)
References
Part HSolitray Waves Theory
12 Solitary Waves Theory
12.1 Introduction
12.2 Definitions
12.2.1 Dispersion and Dissipation
12.2.2 Types of Travelling Wave Solutions
12.2.3 Nonanalytic Solitary Wave Solutions
12.3 Analysis of the Methods
12.3.1 The Tanh-coth Method
12.3.2 The Sine-cosine Method
12.3.3 Hirotas Bilinear Method
12.4 Conservation Laws
References
Partial Differential Equations and Solitary Waves Theory is a self-containedbook divided into two parts： Part I is a coherent survey bringing together newlydeveloped methods for solving PDEs. While some traditional techniques are pre-sented, this part does not require thorough understanding of abstract theories orcompact concepts. Well-selected worked examples and exercises shall guide thereader through the text. Part II provides an extensive exposition of the solitarywaves theory. This part handles nonlinear evolution equations by methods suchas Hirotas bilinear method or the tanh-coth method. A self-contained treatmentis presented to discuss complete integrability of a wide class of nonlinear equa-tions. This part presents in an accessible manner a systematic presentation ofsolitons, multi-soliton solutions, kinks, peakons, cuspons, and compactons.
While the whole book can be used as a text for advanced undergraduate andgraduate students in applied mathematics, physics and engineering, Part II will be most useful for graduate students and researchers in mathematics, engineer-ing, and other related fields.
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