Homotopy analysis method: Difference between revisions

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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’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 introduce 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=978-1-58488-407-X1 }}[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==

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 a 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| issue=12 | 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| issue=5 | 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| issue=7 | 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 opposed to a 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 ==

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}} [https://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}} [https://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=379–418|year=2012|doi = 10.1017/jfm.2012.370|bibcode = 2012JFM...710..379X }}</ref> were found with the [[wave resonance]] criterion of arbitrary number of traveling [[gravity waves]]; this agreed with Phillips' criterion for four waves with small amplitude. Further, a unified wave model applied with the HAM,<ref>{{citation | last=Liao | first=S.J. | title= Do peaked solitary water waves indeed exist? | journal=Communications in Nonlinear Science and Numerical Simulation|year=2013 | doi=10.1016/j.cnsns.2013.09.042|arxiv = 1204.3354 |bibcode = 2014CNSNS..19.1792L | volume=19 | issue=6 | pages=1792–1821}}</ref> admits not only the traditional smooth progressive periodic/solitary waves, but also the progressive solitary waves with peaked crest in finite water depth. This model shows peaked solitary waves are consistent solutions along with the known smooth ones. Additionally, the HAM has been applied to many other nonlinear problems such as nonlinear [[heat transfer]],<ref>{{citation | last1=Abbasbandy | first1=S. | title= The application of homotopy analysis method to nonlinear equations arising in heat transfer | journal=Physics Letters A| volume=360| issue=1 | pages=109–113|year=2006 | doi=10.1016/j.physleta.2006.07.065|bibcode = 2006PhLA..360..109A }}</ref> the [[limit cycle]] of nonlinear dynamic systems,<ref>{{citation|last1= Chen|first1=Y.M.|first2=J.K. |last2=Liu|title=Uniformly valid solution of limit cycle of the Duffing–van der Pol equation|journal = Mechanics Research Communications|volume= 36|issue=7|year= 2009|pages= 845–850|doi=10.1016/j.mechrescom.2009.06.001}}</ref> the American [[put option]],<ref>{{citation | last1=Zhu | first1=S.P. | title= An exact and explicit solution for the valuation of American put options | journal=Quantitative Finance| volume=6| pages=229–242|year=2006 | issue=3 | doi=10.1080/14697680600699811}}</ref> the exact [[Navier–Stokes equation]],<ref>{{citation|last=Turkyilmazoglu|first=M.|title=Purely analytic solutions of the compressible boundary layer flow due to a porous rotating disk with heat transfer|journal=Physics of Fluids|volume=21|issue=10|pages=106104106104–106104–12|year=2009|doi=10.1063/1.3249752|bibcode = 2009PhFl...21j6104T }}</ref> the option pricing under [[stochastic volatility]],<ref>{{citation|last1=Park|first1=Sang-Hyeon|last2=Kim|first2=Jeong-Hoon|title=Homotopy analysis method for option pricing under stochastic volatility|journal=Applied Mathematics Letters|volume= 24|issue=10|year= 2011|pages= 1740–1744|doi=10.1016/j.aml.2011.04.034}}</ref> the [[electrohydrodynamic]] flows,<ref>{{citation|last=Mastroberardino|first=A.|title=Homotopy analysis method applied to electrohydrodynamic flow|journal = Commun. Nonlinear. Sci. Numer. Simulat.| volume=16|issue=7|year= 2011| pages=2730–2736|doi=10.1016/j.cnsns.2010.10.004|bibcode = 2011CNSNS..16.2730M }}</ref> the [[Poisson–Boltzmann equation]] for semiconductor devices,<ref>{{citation|last1=Nassar|first1= Christopher J.| first2= Joseph F. |last2=Revelli|first3=Robert J. |last3=Bowman|title=Application of the homotopy analysis method to the Poisson–Boltzmann equation for semiconductor devices |journal = Commun Nonlinear Sci Numer Simulat |volume=16 |issue= 6|year=2011|pages= 2501–2512|doi=10.1016/j.cnsns.2010.09.015|bibcode = 2011CNSNS..16.2501N }}</ref> and others.

== Brief mathematical description ==

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== Frequency response analysis for nonlinear oscillators ==

The HAM has recently been reported to be useful for obtaining analytical solutions for nonlinear frequency response equations. Such solutions are able to capture various nonlinear behaviors such as hardening-type, softening-type or mixed behaviors of the oscillator,.<ref>{{cite journal|last1=Tajaddodianfar|first1=Farid|title=Nonlinear dynamics of MEMS/NEMS resonators: analytical solution by the homotopy analysis method|journal=Microsystem Technologies|volume=23|issue=6|pages=1913–1926|doi=10.1007/s00542-016-2947-7|urlyear=https://link.springer.com/article/10.1007/s00542-016-2947-72017}}</ref><ref>{{cite journal|last1=Tajaddodianfar|first1=Farid|title=On the dynamics of bistable micro/nano resonators: Analytical solution and nonlinear behavior|journal=Communications in Nonlinear Science and Numerical Simulation|volume=20|issue=3|doi=10.1016/j.cnsns.2014.06.048|url=http://www.sciencedirect.com/science/article/pii/S1007570414003116|pages=1078–1089|bibcode=2015CNSNS..20.1078T|date=March 2015}}</ref> These analytical equations are also useful in prediction of chaos in nonlinear systems.<ref>{{cite journal|last1=Tajaddodianfar|first1=Farid|title=Prediction of chaos in electrostatically actuated arch micro-nano resonators: Analytical approach|journal=Communications in Nonlinear Science and Numerical Simulation|volume=30|issue=1–3|doi=10.1016/j.cnsns.2015.06.013|url=http://www.sciencedirect.com/science/article/pii/S1007570415002142|pages=182–195|date=January 2016}}</ref>

== References ==