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→Frequency Response Analysis for Nonlinear Oscillators: fixing these incorrect capitals as required by WP:MOS |
Someone wrote "Navier–Stokes" with a hyphen instead of an en-dash, and elsewhere a hyphen was used instead of a minus sign and without proper spacing. |
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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=710:379–418|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 }}</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| 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|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 [[
== Brief mathematical description ==
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:<math>
u(x) = u_0(x) + \sum_{m=1}^
</math>
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</math>
called the ''m''<sup>th</sup>-order deformation equation, where <math>\chi_1 = 0</math> and <math>\chi_k = 1</math> for ''k'' > 1, and the right-hand side ''R''<sub>''m''</sub> is dependent only upon the known results ''u''<sub>0</sub>, ''u''<sub>1</sub>, ..., ''u''<sub>''m''
Since the HAM is based on a homotopy, one has great freedom to choose the initial guess ''u''<sub>0</sub>(''x''), the auxiliary linear operator <math>\mathcal{L}</math>, and the convergence-control parameter ''c''<sub>0</sub> in the zeroth-order deformation equation. Thus, the HAM provides the mathematician 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''<sub>0</sub> 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''<sub>0</sub> 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.
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