Homotopy analysis method: Difference between revisions

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{{Distinguish|text=[[Homotopy method]], a method for computing fixed points of functions, devised in 1972 by B. Eaves Curtis}}{{Like resume|date=May 2020}}

{{POV|date=September 2013}}

{{Primary sources|date=June 2009}}

[[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-[[MclaurinTaylor series|Maclaurin series]] to deal with the nonlinearities in the system.

The HAM was first devised in 1992 by Dr.[[Liao 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’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 introducedintroduce 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 |year=2003 |___location=Boca Raton |publisher=Chapman & Hall/ CRC Press | ___location=Boca Raton | year=2003 | isbn=978-1-58488-407-X 1}}[httphttps://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 but itthat is entirelynot independentdirectly ofdependent 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 wellof knownthe instandard [[Perturbation theory|perturbation methods]]. SecondlySecond, 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| issue=12 | pages=4057-40644057–4064|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| pagesissue=22965 –| 2301pages=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 spacialspatial and parameter domains. ThirdlyThird, the HAM gives excellent flexibility in the expression of the solution and how the solution is explicitly obtained. It also provides great freedom to choose the [[basis functions]] of the desired solution and the corresponding auxiliary [[linear operator]] of the homotopy. EspeciallyFinally, unlike allthe other analytic approximation techniques, the HAM provides a simple way to ensure the [[limit of a sequence|convergence]] of the solution series. This differs the HAM from all other analytic approximation methods.

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 essentially an analytic approximation method, sinceas itopposed givesto analytica approximationsdiscrete throughoutcomputational the ___domainmethod. TheFurther, 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 numerousa 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 | publisheryear=Springer2012 & Higher Education Press| ___location=Berlin & Beijing | yearpublisher=2012Springer |& Higher Education Press |isbn=978-7-04-032298-9}} [httphttps://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 | publisheryear=Springer2013 & Higher Education Press| ___location=Berlin & Beijing | yearpublisher=2013Springer |& Higher Education Press |isbn=978-3-642-32102-3 |last2=Van Gorder}} [httphttps://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, by means of the HAM, the 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=710:379-418379–418|year=2012|doi = 10.1017/jfm.2012.370|bibcode = 2012JFM...710..379X |s2cid=122094345 }}</ref> were found, andwith the [[wave resonance]] criterion of arbitrary number of traveling [[gravity waves]]; wasthis obtained,agreed which logically contains the famouswith Phillips' criterion for four waves with small amplitude. Using the HAMFurther, ana unified wave model (UWM)applied waswith proposedthe by LiaoHAM,<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| s2cid=119203215 }}</ref> which admits not only the traditional smooth progressive periodic/solitary waves, but also the progressive solitary waves with peaked crest in finite water depth. AccordingThis tomodel the UWM, theshows peaked solitary waves are consistent withsolutions thealong smooth ones. Thus,with the peaked solitary waves are as acceptable as the traditionalknown smooth ones. In additionAdditionally, 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| s2cid=121851109 }}</ref> the exact [[Navier-StokesNavier–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 FluidFluids|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|doi-access=free}}</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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[[File:Mug and Torus morph.gif|thumb|top|200px|An isotopy of a coffee cup into a doughnut ([[torus]]).]]

ForConsider example,a let us consider ageneral 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''₀(''x'') an initial guess of ''u''(''x''), and ''c''₀ a constant (called the convergence-control parameter), respectively. Using the so-called embedding parameter ''q'' ∈ [0,1] infrom homotopy theory, one constructsmay suchconstruct a family of equations,

:<math>

(1 - q) \mathcal{L}[U(x; q) - u_0(x; q)] = c_0 \, q \, \mathcal{N}[U(x;q)],

</math>

called the zeroth-order deformation equation, whose solution varies continuously with respect to the embedding parameter ''q'' ∈ [0,1]:. it This is the linear equation

:<math>

\mathcal{L}[U(x; q) - u_0(x; q)] = 0,

</math>

with known initial guess ''U''(''x''; 0) = ''u''₀(''x'') when ''q'' = 0, but is equivalent to the consideredoriginal 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 so-called zeroth-order deformation equation varies (or deforms) from the chosen initial guess ''u''₀(''x'') to the solution ''u''(''x'') of the considered nonlinear equation.

Expanding ''U''(''x''; ''q'') in a Taylor series atabout ''q'' = 0, we have the so-called homotopy-Maclaurin series

:<math>

Line 50 ⟶ 48:

</math>

Assuming that the so-called convergence-control parameter ''c''₀ of the zeroth-order deformation equation is so properly chosen that the above series is convergent at ''q'' = 1, we have the homotopy-series solution

:<math>

u(x) = u_0(x) + \sum_{m=1}^{\infty} u_m(x).

</math>

From the zeroth-order deformation equation, one can directly derive the governing equation of ''u''_m(''x'')

:<math>

\mathcal{L}[u_m(x) - \chi_m u_{m-1}(x) ] = c_0 \, R_m[u_0, u_1, \cdotsldots, u_{m-1}],

</math>

called the ''m''^th-order deformation equation, where <math>\chi_1 = 0</math> and <math>\chi_k = 1</math> for ''k'' > 1, and the right-hand side ''R''_''m'' is dependent only upon the known results ''u''₀, ''u''₁, ..., ''u''_{''m''-&nbsp;−&nbsp;1} and can be obtained easily by means ofusing [[computer algebra]] software. In this way, the original nonlinear equation is transferred into an infinite number of linear ones, but without the assumption of any small/large physical parameters.

It should be emphasized that, in the frame ofSince the HAM that is based on a homotopy in topology, one has great freedom to choose the initial guess ''u''₀(''x''), the auxiliary linear operator <math>\mathcal{L}</math>, and the so-called convergence-control parameter ''c''₀ in the zeroth-order deformation equation. Thus, the HAM provides usthe greatmathematician freedom to choose the equation-type of the high-order deformation equation and the base functions of its solution. The optimal value of the convergence-control parameter ''c''₀ is determined by the minimum of the squared residual error of governing equations and/or boundary conditions after the general form has been solved for the chosen initial guess and linear operator. Thus, the convergence-control parameter ''c''₀ is a simple way to guarantee the convergence of the homotopy series solution and differentiates the HAM from other analytic approximation methods. The method overall gives a useful generalization of the concept of homotopy.

== The HAM and computer algebra ==

It should be emphasized that the convergence-control parameter ''c''₀ greatly generalizes the traditional concept of the homotopy. In fact, the convergence-control parameter ''c''₀ provides us a simple way to guarantee the convergence of the homotopy series solution. Essentially the convergence-control parameter ''c''₀ is what significantly differentiates the HAM from other analytic approximation methods.

InspiritedThe byHAM sois manyan successfulanalytic applicationsapproximation ofmethod thedesigned HAMfor inthe differentcomputer fieldsera and also bywith the abilitygoal of "computing with functions instead of numbers." of In conjunction with a [[computer algebra systemssystem]] such as [[Mathematica]], or [[MatlabMaple (software)|Maple]], andone [[Maple]]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, washas issuedbeen made available online for solving nonlinear boundary-value problems. It is available online.[http://numericaltank.sjtu.edu.cn/BVPh.htm]. The HAM-based Mathematica package 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 evenincludes support for somecertain types of nonlinear PDEs.<ref name="HAM in NDEs"/> Another HAM-based Mathematica code, APOh, has been produced to solve for an explicit analytic approximationsapproximation of the optimal exercise boundary of American put option, which is also available online. [http://numericaltank.sjtu.edu.cn/APO.htm].

== The HAM and computer algebra ==

In conjunction with computer algebra system such as Mathematica, Maple and so on, one can gain analytic approximations of a highly nonlinear problem at a high enough order by means of the HAM. Replacing pencil, paper and traditional calculator by keyboard, hard disk and CPUs of a laptop computer, one can easily read, calculate and save analytic results obtained by the HAM in only a few seconds. Thus, the HAM is in essence an analytic approximation method for the computer era.

== Frequency response analysis for nonlinear oscillators ==

Inspirited by so many successful applications of the HAM in different fields and also by the ability of "computing with functions instead of numbers" of computer algebra systems such as [[Mathematica]], [[Matlab]], and [[Maple]], a Mathematica package based on the HAM, called BVPh, was issued for solving nonlinear boundary-value problems. It is available online.[http://numericaltank.sjtu.edu.cn/BVPh.htm] The HAM-based Mathematica package 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 even for some types of nonlinear PDEs. Another HAM-based Mathematica code APOh has been produced to solve for an explicit analytic approximations of the optimal exercise boundary of American put option, which is also available online.[http://numericaltank.sjtu.edu.cn/APO.htm]

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|year=2017|bibcode=2017MiTec..23.1913T |s2cid=113216381}}</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|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|pages=182–195|date=January 2016|doi-access=free}}</ref>

== References ==

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== External links ==

* http://numericaltank.sjtu.edu.cn/BVPh.htm

* http://numericaltank.sjtu.edu.cn/APO.htm

[[Category:Asymptotic analysis]]