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>{{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, 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=710:379-418|year=2012|doi = 10.1017/jfm.2012.370}}</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}}</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| pages=109–113|year=2006 | doi=10.1016/j.physleta.2006.07.065}}</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 [[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 Fluid|volume=21|pages=106104|year=2009|doi=10.1063/1.3249752}}</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|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|year= 2011| pages=2730–2736|doi=10.1016/j.cnsns.2010.10.004}}</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 |year=2011|pages= 2501–2512|doi=10.1016/j.cnsns.2010.09.015}}</ref>, and others.
