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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
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where <math>\mathcal{N}</math> is a nonlinear operator. Let <math>\mathcal{L}</math> denote an auxiliary linear operator, ''u''<sub>0</sub>(''x'') an initial guess of ''u''(''x''), and ''c''<sub>0</sub> 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,
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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
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</math>
with known initial guess ''U''(''x''; 0) = ''u''<sub>0</sub>(''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''<sub>0</sub>(''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
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</math>
Assuming that the so-called convergence-control parameter ''c''<sub>0</sub> 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
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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''-1</sub> 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''<sub>0</sub>(''x''), the auxiliary linear operator <math>\mathcal{L}</math>, and the so-called convergence-control parameter ''c''<sub>0</sub> 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''<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.
It should be emphasized that the convergence-control parameter ''c''<sub>0</sub> greatly generalizes the traditional concept of the homotopy. In fact, the convergence-control parameter ''c''<sub>0</sub> provides us a simple way to guarantee the convergence of the homotopy series solution. Essentially the convergence-control parameter ''c''<sub>0</sub> is what significantly differentiates the HAM from other analytic approximation methods.
== The HAM and computer algebra ==
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