A brief review of the homotopy analysis method

LIAO Shijun; LIU Zeng

doi:10.6052/1000-0992-18-005

Volume 49 Issue 1

Feb. 2019

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LIAO Shijun, LIU Zeng. A brief review of the homotopy analysis method[J]. Advances in Mechanics, 2019, 49(1): 201902. doi: 10.6052/1000-0992-18-005

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LIAO Shijun, LIU Zeng. A brief review of the homotopy analysis method[J]. Advances in Mechanics, 2019, 49(1): 201902. doi: 10.6052/1000-0992-18-005

LIAO Shijun, LIU Zeng. A brief review of the homotopy analysis method[J]. Advances in Mechanics, 2019, 49(1): 201902. doi: 10.6052/1000-0992-18-005

Citation:

LIAO Shijun, LIU Zeng. A brief review of the homotopy analysis method[J]. Advances in Mechanics, 2019, 49(1): 201902. doi: 10.6052/1000-0992-18-005

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A brief review of the homotopy analysis method

doi: 10.6052/1000-0992-18-005

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

¹ State Key Laboratory of Ocean Engineering,Shanghai Jiao Tong University, Shanghai 200240, China
² Collaborative Innovation Center for Advanced Ship andDeep-Sea Exploration, Shanghai 200240, China
³ School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
⁴ School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
⁵ School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
⁶ Hubei 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
Received Date: 2018-03-26
Publish Date: 2019-02-08

Abstract

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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References(109)

References

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