Homotopy analysis method: Difference between revisions

Revision as of 02:16, 3 March 2012

Revision as of 13:44, 9 May 2012

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{{primarysourcesPrimary sources|date=June 2009}}

The '''homotopy analysis method (HAM)''' aims to solve [[nonlinear]] [[ordinary differential equation]]s and [[partial differential equation]]s analytically. The method distinguishes itself from other [[Mathematical analysis|analytical methods]] in the following four aspects. First, it is a [[series (mathematics)|series]] expansion method but it is independent of small physical parameters at all. Thus it is applicable for not only weakly but also strongly nonlinear problems. Secondly, the HAM is a unified method for the [[Aleksandr Lyapunov|Lyapunov]] [[artificial small parameter method]], the [[delta expansion method]] and the [[Adomian decomposition method]]. Thirdly, the HAM provides a simple way to ensure the [[limit of a sequence|convergence]] of the solution; also it provides freedom to choose the [[basis function|base function]] of the desired solution. Fourthly, the HAM can be combined with many other [[mathematics|mathematical]] methods—such as [[numerical method]]s, series expansion methods, [[integral transform]] methods and so forth.

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== References ==

*{{citation | last=Liao | first=S.J. | title=The proposed homotopy analysis technique for the solution of nonlinear problems | publisher=PhD thesis, Shanghai Jiao Tong University | year=1992 }}

*{{citation | last=Liao | first=S.J. | title=Beyond Perturbation: Introduction to the Homotopy Analysis Method | publisher=Chapman & Hall/ CRC Press | ___location=Boca Raton | year=2003 | isbn=158488407X1-58488-407-X }}

*{{citation | last=Liao | first=S.J. | title=An explicit, totally analytic approximation of Blasius’ viscous flow problems | journal=International Journal of Non-Linear Mechanics | volume=34 | issue=4 | pages=759–778 | year=1999 | doi=10.1016/S0020-7462(98)00056-0 |bibcode = 1999IJNLM..34..759L }}

*{{citation | last=Liao | first=S.J. | title=On the homotopy analysis method for nonlinear problems | url=http://numericaltank.sjtu.edu.cn/paper/AMC-HAM-2004.pdf | journal=Applied Mathematics and Computation | volume=147 | issue=2 | pages=499–513 | year=2004 | doi=10.1016/S0096-3003(02)00790-7 }}