Content deleted Content added
u_0 should not depend on q. |
Erel Segal (talk | contribs) No edit summary |
||
| (27 intermediate revisions by 19 users not shown) | |||
Line 1:
{{Distinguish|text=[[Homotopy method]], a method for computing fixed points of functions, devised in 1972 by B. Eaves Curtis}}{{Like resume|date=May 2020}}
[[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
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
== 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
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
== 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 |
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=
== Brief mathematical description ==
Line 49 ⟶ 51:
:<math>
u(x) = u_0(x) + \sum_{m=1}^
</math>
Line 55 ⟶ 57:
:<math>
\mathcal{L}[u_m(x) - \chi_m u_{m-1}(x) ] = c_0 \, R_m[u_0, u_1, \
</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.
Line 64 ⟶ 66:
== The HAM and computer algebra ==
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 [[computer algebra system]] such as [[Mathematica]] or [[Maple (software)|Maple]], 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 [http://numericaltank.sjtu.edu.cn/BVPh.htm]. 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.<ref name="HAM in NDEs"/> 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 [http://numericaltank.sjtu.edu.cn/APO.htm].
== Frequency
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
== References ==
| |||