偏微分方程与孤波理论

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
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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