Content deleted Content added
m Open access bot: doi added to citation with #oabot. |
No edit summary |
||
Line 1:
{{Like resume|date=May 2020}}
[[File:HomotopySmall.gif|thumb|top|200px|The two dashed paths shown above are homotopic relative to their endpoints. The animation represents one possible homotopy.]]
The '''homotopy analysis method''' ('''HAM''') is a semi-analytical technique to solve [[nonlinear]] [[ordinary differential equations|ordinary]]/[[partial differential equations|partial]] [[differential equations]]. The homotopy analysis method employs the concept of the [[homotopy]] from [[topology]] to generate a convergent series solution for nonlinear systems. This is enabled by utilizing a homotopy-[[Taylor series|Maclaurin series]] to deal with the nonlinearities in the system.
The HAM was first devised in 1992 by [[Liao Shijun]] of [[Shanghai Jiaotong University]] in his PhD dissertation<ref>{{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 }}</ref> and further modified<ref>{{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 }}</ref> in 1997 {{Advert inline}} to introduce a non-zero auxiliary parameter, referred to as the '''convergence-control parameter''', '''''c'''''<sub>'''0'''</sub>, to construct a homotopy on a differential system in general form.<ref>{{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=978-1-58488-407-1 }}[https://www.amazon.com/Beyond-Perturbation-Introduction-Mechanics-Mathematics/dp/158488407X]</ref> The convergence-control parameter is a non-physical variable that provides a simple way to verify and enforce convergence of a solution series. The capability of the HAM to naturally show convergence of the series solution is unusual in analytical and semi-analytic approaches to nonlinear partial differential equations.
== Characteristics==
| |||