The homotopy analysis method (HAM) is a semi-analytical technique to solve nonlinear ordinary/partial differential equations. The homotopy analysis method employs the concept of the homotopy from topology to generate a convergent series solution for nonlinear systems. This is enabled by utilizing a homotopy-Maclaurin series to deal with the nonlinearities in the system. The HAM was first applied to calculate the ballistic transport of hot electrons in inhomogeneous sub-micron structures by solving a system of nonlinear differential equations in 1988 by Claus Hillebrand, RWTH Aachen. This was published in his thesis in 1989, see reference [4]. The HAM was first devised in 1992 by Liao Shijun of Shanghai Jiaotong University in his PhD dissertation and further modified 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. 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.
Characteristics The HAM distinguishes itself from various other analytical methods in four important aspects. First, it is a 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 methods. Second, the HAM is a unified method for the Lyapunov artificial small parameter method, the delta expansion method, the Adomian decomposition method, and the homotopy perturbation method. 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 basis functions of the desired solution and the corresponding auxiliary linear operator of the homotopy. Finally, unlike the other analytic approximation techniques, the HAM provides a simple way to ensure the convergence of the solution series. The homotopy analysis method is also able to combine with other techniques employed in nonlinear differential equations such as spectral methods and Padé approximants. 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 homotopy continuation, 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 infinite set 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.
Applications In the last twenty years, the HAM has been applied to solve a growing number of nonlinear ordinary/partial differential equations in science, finance, and engineering. For example, multiple steady-state resonant waves in deep and finite water depth 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, 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, the limit cycle of nonlinear dynamic systems, the American put option, the exact Navier–Stokes equation, the option pricing under stochastic volatility, the electrohydrodynamic flows, the Poisson–Boltzmann equation for semiconductor devices, and others.
Brief mathematical description
Consider a general nonlinear differential equation
N [ u ( x ) ] = 0 {\displaystyle {\mathcal {N}}[u(x)]=0} , where N {\displaystyle {\mathcal {N}}} is a nonlinear operator. Let L {\displaystyle {\mathcal {L}}} 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,
( 1 − q ) L [ U ( x ; q ) − u 0 ( x ) ] = c 0 q N [ U ( x ; q ) ] , {\displaystyle (1-q){\mathcal {L}}[U(x;q)-u_{0}(x)]=c_{0}\,q\,{\mathcal {N}}[U(x;q)],}
called the zeroth-order deformation equation, whose solution varies continuously with respect to the embedding parameter q ∈ [0,1]. This is the linear equation
L [ U ( x ; q ) − u 0 ( x ) ] = 0 , {\displaystyle {\mathcal {L}}[U(x;q)-u_{0}(x)]=0,}
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