廖世俊所著的《同伦分析方法与非线性微分方程(英文版)(精)》不依赖任何物理小参数，适用范围广；提供简单、有效的途径确保解析级数解之收敛；可自由选择相关线性子问题的方程类型和解的基函数。本书适合于应用数学、物理、非线性力学、金融和工程等领域对强非线性问题解析近似解感兴趣的科研人员和研究生。关键词：非线性，微分方程，解析近似，美式期权，波浪共振。

　　 廖世俊所著的《同伦分析方法与非线性微分方程(英文版)(精)》介绍同伦分析方法的基本思想、理论上的发展与完善以及新的应用。全书分三个部分。第一部分描述同伦分析方法的基本思想和相关理论。第二部分给出基于同伦分析方法和计算机数学软件Mathematica开发的软件包BVPh 1.0及其应用举例。该软件包可以求解具有多解、奇性、多点边界条件的多种类型的非线性边值问题。第三部分给出同伦分析方法求解非线性偏微分方程的一些典型例子，如美式期权问题、任意多个波浪的共振条件等。本书提供可免费下载的Mathematica程序，以方便读者更好地理解和应用该方法。　　《同伦分析方法与非线性微分方程(英文版)(精)》适合于应用数学、物理、非线性力学、金融和工程等领域对强非线性问题解析近似解感兴趣的科研人员和研究生。关键词：非线性，微分方程，解析近似，美式期权，波浪共振。

Part Ⅰ Basic Ideas and Theorems

 1 Introduction

1.1 Motivation and purpose

1.2 Characteristic of homotopy analysis method

1.3 Outline References

 2 Basic Ideas of the Homotopy Analysis Method

2.1 Concept ofhomotopy

2.2 Example 2.1 : general_ized Newtonian iteration formula

2.3 Example 2.2 : nonlinear oscillation

 2.3.1 Analysis of the solution characteristic

 2.3.2 Mathematical formulations

 2.3.3 Convergence of homotopy-series solution

 2.3.4 Essence of the convergence-controlparameter co

 2.3.5 Convergence acceleration by homotopy-Pade technique

 2.3.6 Convergence acceleration by optimalinitial approximation

 2.3.7 Convergence acceleration by iteration

 2.3.8 Flexibility on the choice of auxiliary linear operator

2.4 Concluding remarks and discussions

Appendix 2.1 Derivation of & in （2.5 7）

Appendix 2.2 Derivation of （2.5 5） by the 2nd approach

Appendix 2.3 Proof of Theorem 2.3

Appendix 2.4 Mathematica code （withoutiteration） for Example 2.2

Appendix 2.5 Mathematica code （with iteration） for Example 2.2

Problems

References

 3 Optimal Homotopy Analysis Method

3.1 Introduction

3.2 An illustrative description

 3.2.1 Basic ideas

 3.2.2 Different types of optimal methods

3.3 Systematic description

3.4 Concluding remarks and discussions

Appendix 3.1 Mathematica code for Blasius flow

Problems N N

References

 4 Systematic Descriptions and Related Theorems

4.1 Brief frame of the homotopy analysis method

4.2 Properties of homotopy-derivative

4.3 Deformation equations

 4.3.1 A briefhistory

 4.3.2 High-order deformation equations

 4.3.3 Examples

4.4 Convergence theorems

4.5 Solution expression

 4.5.1 Choice of initial approximation

 4.5.2 Choice of auxiliary linear operator

4.6 Convergence control and acceleration

 4.6.1 Optimal convergence-controlparameter

 4.6.2 Optimal initial approximation

 4.6.3 Homotopy-iteration technique

 4.6.4 Homotopy-Pade technique

4.7 Discussions and open questions

References

 5 Relationship to Euler Transform

5.1 Introduction

5.2 Generalized Taylor series

5.3 Homotopytransform

5.4 Relation between homotopy analysis method and Eulertransform

5.5 Concluding remarks References

 6 Some Methods Based on the HAM

6.1 A brief history of the homotopy analysis method

6.2 Homotopy perturbation method

6.3 Optimal homotopy asymptotic method

6.4 Spectral homotopy analysis method

6.5 Generalized boundary element method

6.6 Generalized scaled boundary finite element method

6.7 Predictor homotopy analysis method

References

Part II Mathematica Package BVPh and Its Applications

 Mathematica Package BVPh

 7.1 Introduction

7.1.1 Scope

7.1.2 Brief mathematical formulas

7.1.3 Choice of base function and initial guess

7.1.4 Choice of the auxiliary linear operator

7.1.5 Choice of the auxiliary function

7.1.6 Choice of the convergence-control parameter co

 7.2 Approximation and iteration of solutions

7.2.1 Polynomials

7.2.2 Trigonometric functions

7.2.3 Hybrid-base functions

 7.3 A simple users guide of the BVPh 1.0

7.3.1 Key modules

7.3.2 Control parameters

7.3.3 Input

7.3.4 Output

7.3.5 Global variables

 Appendix 7.1 Mathematica package BVPh (version 1.0)

 References

 Nonlinear Boundary-value Problems with Multiple Solutions

 8.1 Introduction

 8.2 Brief mathematical formulas

 8.3 Examples

8.3.1 Nonlinear diffusion-reaction model

8.3.2 A three-point nonlinear boundary-value problem

8.3.3 Channel flows with multiple solutions

 8.4 Concluding remarks

 Appendix 8.1 Input data of BVPh for Example 8.3.1

 Appendix 8.2 Input data of BVPh for Example 8.3.2

 Appendix 8.3 Input data of BVPh for Example 8.3.3

 Problems

 References

 Nonlinear Eigenvalue Equations with Varying Coefficients

 9.1 Introduction

 9.2 Brief mathematical formulas

 9.3 Examples

9.3.1 Non-uniform beam acted by axial load

9.3.2 Gelfand equation

9.3.3 Equation with singularity and varying coefficient

9.3.4 Multipoint boundary-value problem with multiple

solutions

9.3.5 Orr-Sommerfeld stability equation with complex

coefficient

 9.4 Concluding remarks

 Appendix 9.1 Input data of BVPh for Example 9.3.1

 Appendix 9.2 Input data of BVPh for Example 9.3.2

 Appendix 9.3 Input data of BVPh for Example 9.3.3

 Appendix 9.4 Input data of BVPh for Example 9.3.4

 Appendix 9.5 Input data of BVPh for Example 9.3.5

 Problems

 References

 10 A Boundary-layer Flow with an Infinite Number of Solutions

10.1 Introduction

10.2 Exponentially decaying solutions

10.3 Algebraically decaying solutions

10.4 Concluding remarks

Appendix 10.1 Input data of BVPh for exponentially decaying solution

Appendix 10.2 Input data of BVPh for algebraically decaying solution

References

 11 Non-similarity Boundary-layer Flows

11.1 Introduction

11.2 Brief mathematical formulas

11.3 Homotopy-series solution

11.4 Concluding remarks

Appendix 11.1 Input data of BVPh

References

 12 Unsteady Boundary-layer Flows

12.1 Introduction

12.2 Perturbation approximation

12.3 Homotopy-series solution

 12.3.1 Brief mathematical formulas

 12.3.2 Homotopy-approximation

12.4 Concluding remarks

Appendix 12.1 Input data of BVPh

References

Part III Applications in Nonlinear Partial Differential Equations

 13 Applications in Finance: American Put Options

13.1 Mathematical modeling

13.2 Brief mathematical formulas

13.3 Validity of the explicit homotopy-approximations

13.4 A practical code for businessmen

13.5 Concluding remarks

Appendix 13.1 Detailed derivation of fn(t) and gn(t)

Appendix 13.2 Mathematica code for American put option

Appendix 13.3 Mathematica code APOh for businessmen

References

 14 Two and Three Dimensional Gelfand Equation

14.1 Introduction

14.2 Homotopy-approximations of 2D Gelfand equation

 14.2.1 Brief mathematical formulas

 14.2.2 Homotopy-approximations

14.3 Homotopy-approximations of 3D Gelfand equation

14.4 Concluding remarks

Appendix 14.1 Mathematica code of 2D Gelfand equation

Appendix 14.2 Mathematica code of 3D Gelfand equation

References

 15 Interaction of Nonlinear Water Wave and Nonuniform Currents

15.1 Introduction

15.2 Mathematical modeling

 15.2.1 Original boundary-value equation

 15.2.2 Dubreil-Jacotin transformation

15.3 Brief mathematical formulas

 15.3.1 Solution expression

 15.3.2 Zeroth-order deformation equation

 15.3.3 High-order deformation equation

 15.3.4 Successive solution procedure

15.4 Homotopy approximations

15.5 Concluding remarks

Appendix 15.1 Mathematica code of wave-current interaction

References

 16 Resonance of Arbitrary Number of Periodic Traveling Water

Waves

16.1 Introduction

16.2 Resonance criterion of two small-amplitude primary waves

 16.2.1 Brief Mathematical formulas

 16.2.2 Non-resonant waves

 16.2.3 Resonant waves

16.3 Resonance criterion of arbitrary number of primary waves

 16.3.1 Resonance criterion of small-amplitude waves

 16.3.2 Resonance criterion of large-amplitude waves

16.4 Concluding remark and discussions

Appendix 16.1 Detailed derivation of high-order equation

References

Index