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<title>Homotopy analysis method</title>

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Homotopy analysis method

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The homotopy analysis method (HAM) is a semi-analytical technique to solve <a href="Nonlinear" class="mw-redirect" title="Nonlinear">nonlinear</a> <a href="Ordinary_differential_equations" class="mw-redirect" title="Ordinary differential equations">ordinary</a>/<a href="Partial_differential_equations" class="mw-redirect" title="Partial differential equations">partial</a> <a href="Differential_equations" class="mw-redirect" title="Differential equations">differential equations</a>. The homotopy analysis method employs the concept of the <a href="Homotopy" title="Homotopy">homotopy</a> from <a href="Topology" title="Topology">topology</a> to generate a convergent series solution for nonlinear systems. This is enabled by utilizing a homotopy-<a href="Taylor_series" title="Taylor series">Maclaurin series</a> to deal with the nonlinearities in the system.

The HAM was first devised in 1992 by <a href="Liao_Shijun" title="Liao Shijun">Liao Shijun</a> of <a href="Shanghai_Jiaotong_University" class="mw-redirect" title="Shanghai Jiaotong University">Shanghai Jiaotong University</a> in his PhD dissertation<a href="#cite_note-1">[1]</a> and further modified<a href="#cite_note-2">[2]</a> in 1997 to introduce a non-zero auxiliary parameter, referred to as the convergence-control parameter, c0, to construct a homotopy on a differential system in general form.<a href="#cite_note-3">[3]</a> 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.

<div class="mw-heading mw-heading2"><h2 id="Characteristics">Characteristics</h2></div>

The HAM distinguishes itself from various other <a href="Mathematical_analysis" title="Mathematical analysis">analytical methods</a> in four important aspects. First, it is a <a href="Series_(mathematics)" title="Series (mathematics)">series</a> 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 <a href="Perturbation_theory" title="Perturbation theory">perturbation methods</a>. Second, the HAM is a unified method for the <a href="Aleksandr_Lyapunov" title="Aleksandr Lyapunov">Lyapunov</a> artificial small parameter method, the delta expansion method, the <a href="Adomian_decomposition_method" title="Adomian decomposition method">Adomian decomposition method</a>,<a href="#cite_note-Adomian94-4">[4]</a> and the <a href="Homotopy_perturbation_method" class="mw-redirect" title="Homotopy perturbation method">homotopy perturbation method</a>.<a href="#cite_note-5">[5]</a><a href="#cite_note-6">[6]</a> 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 <a href="Basis_functions" class="mw-redirect" title="Basis functions">basis functions</a> of the desired solution and the corresponding auxiliary <a href="Linear_operator" class="mw-redirect" title="Linear operator">linear operator</a> of the homotopy. Finally, unlike the other analytic approximation techniques, the HAM provides a simple way to ensure the <a href="Limit_of_a_sequence" title="Limit of a sequence">convergence</a> of the solution series.

The homotopy analysis method is also able to combine with other techniques employed in nonlinear differential equations such as <a href="Spectral_methods" class="mw-redirect" title="Spectral methods">spectral methods</a><a href="#cite_note-7">[7]</a> and <a href="Pad%C3%A9_approximant" title="Padé approximant">Padé approximants</a>. It may further be combined with computational methods, such as the <a href="Boundary_element_method" title="Boundary element method">boundary element method</a> to allow the linear method to solve nonlinear systems. Different from the numerical technique of <a href="Numerical_continuation" title="Numerical continuation">homotopy continuation</a>, 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 <a href="Infinite_set" title="Infinite set">infinite set</a> 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.

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

In the last twenty years, the HAM has been applied to solve a growing number of nonlinear <a href="Ordinary_differential_equations" class="mw-redirect" title="Ordinary differential equations">ordinary</a>/<a href="Partial_differential_equation" title="Partial differential equation">partial differential equations</a> in science, finance, and engineering.<a href="#cite_note-HAM_in_NDEs-8">[8]</a><a href="#cite_note-9">[9]</a>

For example, multiple steady-state resonant waves in deep and finite water depth<a href="#cite_note-10">[10]</a> were found with the <a href="Wave_resonance" class="mw-redirect" title="Wave resonance">wave resonance</a> criterion of arbitrary number of traveling <a href="Gravity_waves" class="mw-redirect" title="Gravity waves">gravity waves</a>; this agreed with Phillips' criterion for four waves with small amplitude. Further, a unified wave model applied with the HAM,<a href="#cite_note-11">[11]</a> 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 <a href="Heat_transfer" title="Heat transfer">heat transfer</a>,<a href="#cite_note-12">[12]</a> the <a href="Limit_cycle" title="Limit cycle">limit cycle</a> of nonlinear dynamic systems,<a href="#cite_note-13">[13]</a> the American <a href="Put_option" title="Put option">put option</a>,<a href="#cite_note-14">[14]</a> the exact <a href="Navier%E2%80%93Stokes_equation" class="mw-redirect" title="Navier–Stokes equation">Navier–Stokes equation</a>,<a href="#cite_note-15">[15]</a> the option pricing under <a href="Stochastic_volatility" title="Stochastic volatility">stochastic volatility</a>,<a href="#cite_note-16">[16]</a> the <a href="Electrohydrodynamic" class="mw-redirect" title="Electrohydrodynamic">electrohydrodynamic</a> flows,<a href="#cite_note-17">[17]</a> the <a href="Poisson%E2%80%93Boltzmann_equation" title="Poisson–Boltzmann equation">Poisson–Boltzmann equation</a> for semiconductor devices,<a href="#cite_note-18">[18]</a> and others.

<div class="mw-heading mw-heading2"><h2 id="Brief_mathematical_description">Brief mathematical description</h2></div>

Consider a general nonlinear differential equation

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}[u(x)]=0}</annotation>

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</math><img src="./84af72c9cf8bab045e170e63a49a9024e8c1d97a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:12.36ex; height:3.009ex;" alt="{\displaystyle {\mathcal {N}}[u(x)]=0}" loading="lazy">,</dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}}</annotation>

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</math><img src="./b7551c7bed2cd2ee83e10536d157c94a5f8f72fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}" loading="lazy"> is a nonlinear operator. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation>

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</math><img src="./9027196ecb178d598958555ea01c43157d83597c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}" loading="lazy"> denote an auxiliary linear operator, u0(x) an initial guess of u(x), and c0 a constant (called the convergence-control parameter), respectively. Using the embedding parameter q ∈ [0,1] from homotopy theory, one may construct a family of equations,

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<annotation encoding="application/x-tex">{\displaystyle (1-q){\mathcal {L}}[U(x;q)-u_{0}(x)]=c_{0}\,q\,{\mathcal {N}}[U(x;q)],}</annotation>

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</math><img src="./5767ae1eeb4b006ce7d522d3627d75a5a7d2ed3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.411ex; height:3.009ex;" alt="{\displaystyle (1-q){\mathcal {L}}[U(x;q)-u_{0}(x)]=c_{0}\,q\,{\mathcal {N}}[U(x;q)],}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}[U(x;q)-u_{0}(x)]=0,}</annotation>

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</math><img src="./c7c0236d8004644ac2a3de478b64026f19645c08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.194ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}[U(x;q)-u_{0}(x)]=0,}" loading="lazy"></dd></dl>

with known initial guess U(x; 0) = u0(x) when q = 0, but is equivalent to the original nonlinear equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}[u(x)]=0}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}[u(x)]=0}</annotation>

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</math><img src="./84af72c9cf8bab045e170e63a49a9024e8c1d97a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:12.36ex; height:3.009ex;" alt="{\displaystyle {\mathcal {N}}[u(x)]=0}" loading="lazy">, 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 zeroth-order deformation equation varies (or deforms) from the chosen initial guess u0(x) to the solution u(x) of the considered nonlinear equation.

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

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<annotation encoding="application/x-tex">{\displaystyle U(x;q)=u_{0}(x)+\sum _{m=1}^{\infty }u_{m}(x)\,q^{m}.}</annotation>

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</math><img src="./fe89906744aeea50c51fefd338caf9c45b4b5aa0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.349ex; height:6.843ex;" alt="{\displaystyle U(x;q)=u_{0}(x)+\sum _{m=1}^{\infty }u_{m}(x)\,q^{m}.}" loading="lazy"></dd></dl>

Assuming that the so-called convergence-control parameter c0 of the zeroth-order deformation equation is properly chosen that the above series is convergent at q = 1, we have the homotopy-series solution

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<annotation encoding="application/x-tex">{\displaystyle u(x)=u_{0}(x)+\sum _{m=1}^{\infty }u_{m}(x).}</annotation>

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</math><img src="./f3691b9cd036502ab034c42c2a13df8366a7faae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.651ex; height:6.843ex;" alt="{\displaystyle u(x)=u_{0}(x)+\sum _{m=1}^{\infty }u_{m}(x).}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}[u_{m}(x)-\chi _{m}u_{m-1}(x)]=c_{0}\,R_{m}[u_{0},u_{1},\ldots ,u_{m-1}],}</annotation>

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</math><img src="./62c359c3f9055352a8c42dd1ced445a18052d4cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.267ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}[u_{m}(x)-\chi _{m}u_{m-1}(x)]=c_{0}\,R_{m}[u_{0},u_{1},\ldots ,u_{m-1}],}" loading="lazy"></dd></dl>

called the mth-order deformation equation, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{1}=0}">

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<annotation encoding="application/x-tex">{\displaystyle \chi _{1}=0}</annotation>

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</math><img src="./5bc722bd3d1fb6745251da48a086b3dc91c8e1ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.77ex; height:2.509ex;" alt="{\displaystyle \chi _{1}=0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{k}=1}">

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<annotation encoding="application/x-tex">{\displaystyle \chi _{k}=1}</annotation>

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</math><img src="./c8464575c70588984a479ca2c47da428faacd98f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.805ex; height:2.509ex;" alt="{\displaystyle \chi _{k}=1}" loading="lazy"> for k > 1, and the right-hand side Rm is dependent only upon the known results u0, u1, ..., um − 1 and can be obtained easily using <a href="Computer_algebra" title="Computer algebra">computer algebra</a> 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.

Since the HAM is based on a homotopy, one has great freedom to choose the initial guess u0(x), the auxiliary linear operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation>

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</math><img src="./9027196ecb178d598958555ea01c43157d83597c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}" loading="lazy">, and the convergence-control parameter c0 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 c0 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 c0 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.

<div class="mw-heading mw-heading2"><h2 id="The_HAM_and_computer_algebra">The HAM and computer algebra</h2></div>

The HAM is an analytic approximation method designed for the computer era with the goal of "computing with functions instead of numbers." In conjunction with a <a href="Computer_algebra_system" title="Computer algebra system">computer algebra system</a> such as <a href="Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a> or <a href="Maple_(software)" title="Maple (software)">Maple</a>, one 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, has been made available online for solving nonlinear boundary-value problems <a rel="nofollow" class="external autonumber" href="http://numericaltank.sjtu.edu.cn/BVPh.htm">[4]</a>. 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 includes support for certain types of nonlinear PDEs.<a href="#cite_note-HAM_in_NDEs-8">[8]</a> Another HAM-based Mathematica code, APOh, has been produced to solve for an explicit analytic approximation of the optimal exercise boundary of American put option, which is also available online <a rel="nofollow" class="external autonumber" href="http://numericaltank.sjtu.edu.cn/APO.htm">[5]</a>.

<div class="mw-heading mw-heading2"><h2 id="Frequency_response_analysis_for_nonlinear_oscillators">Frequency response analysis for nonlinear oscillators</h2></div>

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.<a href="#cite_note-19">[19]</a><a href="#cite_note-20">[20]</a> These analytical equations are also useful in prediction of chaos in nonlinear systems.<a href="#cite_note-21">[21]</a>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFLiao1992" class="citation cs2">Liao, S.J. (1992), The proposed homotopy analysis technique for the solution of nonlinear problems, PhD thesis, Shanghai Jiao Tong University</cite>

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