Homotopy analysis method: Difference between revisions

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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 to introduced a non-zero auxiliary parameter, referred to as the '''convergence-control parameter''', '''''c'''''_'''0''', 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.

== 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 ana 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=4057–4064|year=2009 | doi=10.1016/j.cnsns.2009.02.016|bibcode = 2009CNSNS..14.4057L }}</ref><ref>{{citation | last1=Sajid | first1=M. |last2=Hayat |first2=T. | title= Comparison of HAM and HPM methods in nonlinear heat conduction and convection equations | journal=Nonlinear Analysis: Real World Applications| volume=9| pages=2296–2301|year=2008 | doi=10.1016/j.nonrwa.2007.08.007}}</ref> The greater generality of the method often allows for strong convergence of the solution over larger spatial and parameter domains. Third, the HAM gives excellent flexibility in the expression of the solution and how the solution is explicitly obtained. It provides great freedom to choose the [[basis functions]] of the desired solution and the corresponding auxiliary [[linear operator]] of the homotopy. Finally, unlike the other analytic approximation techniques, the HAM provides a simple way to ensure the [[limit of a sequence|convergence]] of the solution series.

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.

== Applications ==