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。
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书名 | 同伦分析方法与非线性微分方程(英文版)(精) |
分类 | 科学技术-自然科学-数学 |
作者 | 廖世俊 |
出版社 | 高等教育出版社 |
下载 | ![]() |
简介 | 编辑推荐 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。 内容推荐 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。 目录 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 |
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