同伦分析方法进展综述

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doi:10.6052/1000-0992-18-005

廖世俊^1,2,3,,,,
刘曾^2,5,6

¹海洋工程国家重点实验室, 上海 200240
²高新船舶与深海开发装备协同创新中心, 上海 200240
³上海交通大学船舶海洋与建筑工程学院, 上海 200240
⁴上海交通大学物理和天文学院, 上海 200240
⁵华中科技大学船舶与海洋工程学院, 武汉 430074
⁶船舶与海洋工程水动力湖北省重点实验室, 武汉 430074

基金项目:中国力学界许多学者对同伦分析方法的完善和应用做出了贡献,大大丰富了同伦分析方法的应用领域. 遗憾的是,本文由于篇幅限制不能一一介绍, 特在此表示衷心的感谢.该研究工作长期以来受到国家自然科学基金项目(50125923, 10572095,10872129, 11272209, 11432009, 51609090) 的资助

详细信息

作者简介:
通讯作者: † E-mail: sjliao@sjtu.edu.cn
作者简介: 廖世俊, 上海交通大学“春申”讲席教授, 博士生导师,现任职于上海交通大学船舶海洋与建筑工程学院,上海交通大学物理和天文学院,海洋工程国家重点实验室副主任(2001年—), 教育部长江奖励计划特聘教授(2001年), 国家杰出青年基金获得者(2001年).曾获“上海市第七届自然科学牡丹奖”(2009),“上海市自然科学一等奖”(2009 年, 唯一完成人),“国家自然科学二等奖”(2016年, 唯一完成人), “上海市科技精英”(2017年).廖世俊教授原创性地提出求解强非线性问题的“同伦分析方法” (Homotopyanalysis method, HAM), 撰写2本相关英文专著, 编辑一本英文专著,是“同伦分析方法”的创始人.廖世俊教授提出求解混沌动力系统的高精度数值方法(clean numericalsimulation, CNS),为非线性混沌动力系统提供了一个高精度的、全新的研究工具并与他人合作, 应用CNS和超级计算机,成功获得著名的三体问题两千多个全新的周期解. 迄今共发表150 余篇 SCI论文. 其博士论文、专著和杂志论文共被 SCI检索他引七千余次(H-index为49), 其中18 篇为ESI 高被引用论文,一篇论文入选“2009年中国百篇最具影响国际学术论文”,一篇论文入选“2010年中国百篇最具影响国际学术论文”.连续三年（2014—2016）入选全球高被引用科学家名单(highly-cited researchers).

刘曾, 华中科技大学船舶与海洋工程学院讲师, 硕士生导师.2008—2015年,上海交通大学船舶海洋与建筑工程学院攻读硕士和博士学位,师从廖世俊教授, 其间于2014年访学MIT海洋工程系一年.2015年获上海交通大学工学博士学位.研究方向为非线性海浪动力学、船舶与海洋工程水动力学、同伦分析方法及其在非线性微分方程中的应用.共发表10余篇SCI论文, 其中以第一作者在Journal of Fluid Mechanics发表3篇论文, 在Physics of Fluids上发表1篇论文

通讯作者:
廖世俊

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出版历程
- 收稿日期:2018-03-26
- 刊出日期:2019-02-08

A brief review of the homotopy analysis method

LIAO Shijun^1,2,3,,,,
LIU Zeng^2,5,6

¹State Key Laboratory of Ocean Engineering,Shanghai Jiao Tong University, Shanghai 200240, China
²Collaborative Innovation Center for Advanced Ship andDeep-Sea Exploration, Shanghai 200240, China
³School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
⁴School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
⁵School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
⁶Hubei Key Laboratory of Naval Architecture and Ocean Engineering Hydrodynamics, Huazhong University of Science andTechnology, Wuhan 430074, China

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Author Bio:
corresponding Author: † E-mail: sjliao@sjtu.edu.cn

Corresponding author:LIAO Shijun

摘要

摘要:本文简述同伦分析方法基本思想、最新理论进展及其在流体力学、固体力学、一般力学、量子力学、应用数学、金融等科学和工程领域的应用.同伦分析方法不依赖物理小参数, 适用范围更广,而且提供了一种简单的途径确保级数解收敛, 适用于强非线性问题.同伦分析方法已被成功应用于求解一些具有挑战性的力学问题,并获得一些全新的、从未见报道的解. 这些成功的应用,证明了同伦分析方法的普遍有效性和原创性.
- 同伦分析方法/
- 解析近似/
- 级数解/
- 非线性微分方程
Abstract:In this paper, a brief review of the current advances of the homotopy analysis method (HAM) in theory and applications is given. The HAM is an analytic approximation method for highly nonlinear problems. Traditionally, perturbation methods were widely used. However, perturbation methods are strongly dependent upon the existence of small physical parameters (called perturbation quantity), and besides perturbation approximations often become divergent as perturbation quantity enlarges. However, unlike perturbation methods, the HAM has nothing to do with the existence of small/large physical parameters, since it is based on the homotopy, a basic concept in topology. Especially, the HAM provides a convenient way to guarantee the convergence of solution series. In addition, the HAM provides great freedom to choose the base-functions and the equation-type of high-order equations so that good approximations can be obtained more efficiently. As illustrated in this paper, the HAM has been used to solve some challenging nonlinear problems in nonlinear mechanics, quantum mechanics, applied mathematics, finance and so on.
- homotopy analysis method/
- analytical approximation/
- serious solutions/
- non-perturbative approach

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