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{{POV|date=September 2013}}
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[[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-[[Mclaurin series]] to deal with the nonlinearities in the system.
The HAM was first devised in 1992 by Dr. Shijun Liao 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}}</ref> in 1997 to introduced 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=1-58488-407-X }}[http://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.
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== Characteristics of the HAM ==
The HAM distinguishes itself from various other [[Mathematical analysis|analytical methods]] in four important aspects. First, it is a [[series (mathematics)|series]] expansion method that is not directly dependent on small or large physical parameters. Thus, it is applicable for not only weakly but also strongly nonlinear problems, going beyond some of the inherent limitations of the standard [[Perturbation theory|perturbation methods]]. Second, the HAM is an unified method for the [[Aleksandr Lyapunov|Lyapunov]] artificial small parameter method, the delta expansion method, the [[Adomian decomposition method]],<ref name="Adomian94">{{cite book |title=Solving Frontier problems of Physics: The decomposition method|first=G.|last=Adomian|publisher=Kluwer Academic Publishers|year=1994|isbn=|page=}}</ref> and the [[homotopy perturbation method]].<ref>{{citation | last1=Liang | first1=Songxin |last2=Jeffrey |first2=David J. | title= Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation | journal=Communications in Nonlinear Science and Numerical Simulation| volume=14| pages=
The homotopy analysis method is also able to combine with other techniques employed in nonlinear differential equations such as [[spectral methods]]<ref>{{citation | last1=Motsa | first1=S.S. | last2=Sibanda|first2=P.| last3=Awad| first3=F.G.| last4 = Shateyi| first4 = S.| title= A new spectral-homotopy analysis method for the MHD Jeffery–Hamel problem | journal=Computers & Fluids| volume=39| pages=1219–1225|year=2010 | doi=10.1016/j.compfluid.2010.03.004}}</ref> and [[Padé approximant]]s. It may further be combined with computational methods, such as the [[boundary element method]] to allow the linear method to solve nonlinear systems. Different from the numerical technique of [[Numerical continuation|homotopy continuation]], the homotopy analysis method is an analytic approximation method as apposed to discrete computational method. Further, the HAM uses the homotopy parameter only on a theoretical level to demonstrate that a nonlinear system may be split into an infinite set of linear systems which are solved analytically, while the continuation methods require solving a discrete linear system as the homotopy parameter is varied to solve the nonlinear system.
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In the last twenty years, the HAM has been applied to solve a growing number of nonlinear [[ordinary differential equations|ordinary]]/[[partial differential equation]]s in science, finance, and engineering.<ref name="HAM in NDEs">{{citation | last=Liao | first=S.J. | title=Homotopy Analysis Method in Nonlinear Differential Equations| publisher=Springer & Higher Education Press| ___location=Berlin & Beijing | year=2012 | isbn=978-7-04-032298-9}} [http://www.amazon.com/Homotopy-Analysis-Nonlinear-Differential-Equations/dp/3642251315]</ref><ref>{{citation | last1=Vajravelu | first1=K. | last2= Van Gorder| title= Nonlinear Flow Phenomena and Homotopy Analysis| publisher=Springer & Higher Education Press| ___location=Berlin & Beijing | year=2013 | isbn=978-3-642-32102-3}} [http://www.amazon.com/Nonlinear-Flow-Phenomena-Homotopy-Analysis/dp/3642321011/ref=sr_1_1?s=books&ie=UTF8&qid=1384402655&sr=1-1]</ref>
For example, multiple steady-state resonant waves in deep and finite water depth<ref>{{citation|last1=Xu|first1=D.L.|last2=Lin|first2=Z.L.|last3=Liao|first3=S.J.|last4=Stiassnie|first4=M.|title=On the steady-state fully resonant progressive waves in water of finite depth|journal =Journal of Fluid Mechanics|volume = 710|pages=710:
== Brief mathematical description ==
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[[File:Mug and Torus morph.gif|thumb|top|200px|An isotopy of a coffee cup into a doughnut ([[torus]]).]]
Consider a general nonlinear differential equation
:<math>
\mathcal{N}[u(x)] = 0
</math>,
where <math>\mathcal{N}</math> is a nonlinear operator. Let <math>\mathcal{L}</math> denote an auxiliary linear operator, ''u''<sub>0</sub>(''x'') an initial guess of ''u''(''x''), and ''c''<sub>0</sub> a constant (called the convergence-control parameter), respectively. Using the embedding parameter ''q'' ∈ [0,1] from homotopy theory, one may construct a family of equations,
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</math>
with known initial guess ''U''(''x''; 0) = ''u''<sub>0</sub>(''x'') when ''q'' = 0, but is equivalent to the original nonlinear equation <math>\mathcal{N}[u(x)] = 0</math>, when ''q'' = 1, i.e. ''U''(''x''; 1) = ''u''(''x'')). Therefore, as ''q'' increases from 0 to 1, the solution ''U''(''x''; ''q'') of the zeroth-order deformation equation varies (or deforms) from the chosen initial guess ''u''<sub>0</sub>(''x'') to the solution ''u''(''x'') of the considered nonlinear equation.
Expanding ''U''(''x''; ''q'') in a Taylor series about ''q'' = 0, we have the homotopy-Maclaurin series
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</math>
From the zeroth-order deformation equation, one can directly derive the governing equation of ''u''<sub>m</sub>(''x'')
:<math>
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== The HAM and computer algebra ==
The HAM is an analytic approximation method designed for the computer era with the goal of "computing with functions instead of numbers." In conjunction with a computer algebra system such as [[Mathematica]] or [[Maple]], one can gain analytic approximations of a highly nonlinear problem to arbitrarily high order by means of the HAM in only a few seconds. Inspired by the recent successful applications of the HAM in different fields, a Mathematica package based on the HAM, called BVPh, has been made available online for solving nonlinear boundary-value problems.[http://numericaltank.sjtu.edu.cn/BVPh.htm] BVPh is a solver package for highly nonlinear ODEs with singularities, multiple solutions, and multipoint boundary conditions in either a finite or an infinite interval, and includes support for certain types of nonlinear PDEs.<ref name="HAM in NDEs"/> Another HAM-based Mathematica code, APOh, has been produced to solve for an explicit analytic approximation of the optimal exercise boundary of American put option, which is also available online.[http://numericaltank.sjtu.edu.cn/APO.htm]
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== External links ==
* http://numericaltank.sjtu.edu.cn/BVPh.htm
* http://numericaltank.sjtu.edu.cn/APO.htm
[[Category:Asymptotic analysis]]
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