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Homotopy analysis method
Homotopy analysis method
Homotopy analysis method
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The two dashed paths shown above are homotopic relative to their endpoints. The animation represents one possible homotopy.

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 devised in 1992 by Liao Shijun of Shanghai Jiaotong University in his PhD dissertation[1] and further modified[2] 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.[3] 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

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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,[4] and the homotopy perturbation method.[5][6] 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[7] 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

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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.[8][9] For example, multiple steady-state resonant waves in deep and finite water depth[10] 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,[11] 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,[12] the limit cycle of nonlinear dynamic systems,[13] the American put option,[14] the exact Navier–Stokes equation,[15] the option pricing under stochastic volatility,[16] the electrohydrodynamic flows,[17] the Poisson–Boltzmann equation for semiconductor devices,[18] and others.

Brief mathematical description

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An isotopy of a coffee cup into a doughnut (torus).

Consider a general nonlinear differential equation

,

where is a nonlinear operator. Let 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,

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

with known initial guess U(x; 0) = u0(x) when q = 0, but is equivalent to the original nonlinear equation , 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

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

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

called the mth-order deformation equation, where and 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 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.

Since the HAM is based on a homotopy, one has great freedom to choose the initial guess u0(x), the auxiliary linear operator , 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.

The HAM and computer algebra

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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, 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 [4]. 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.[8] 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 [5].

Frequency response analysis for nonlinear oscillators

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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.[19][20] These analytical equations are also useful in prediction of chaos in nonlinear systems.[21]

References

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Homotopy analysis method

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The homotopy analysis method (HAM) is an analytical approximation technique for solving nonlinear problems in and , developed by Shijun Liao in as a general way to obtain solutions independent of small physical parameters. Unlike traditional perturbation methods, which require problems to be weakly nonlinear or depend on small parameters for expansion, HAM leverages the concept of from to continuously deform an initial guess solution into the exact solution of a nonlinear equation through a family of intermediate equations. This method introduces flexibility in choosing the form of the solution and auxiliary parameters to ensure convergence, making it particularly suitable for strongly nonlinear problems where other analytic approaches fail. HAM's core formulation involves the zero-order deformation equation, which embeds the original nonlinear problem into a homotopy parameter qq (ranging from 0 to 1), transitioning from an auxiliary linear problem at q=0q = 0 to the full nonlinear problem at q=1q = 1. The equation is generally expressed as (1−q)L[ϕ(r,t;q)−u0(r,t)]=c0qH(r,t){N[ϕ(r,t;q)]−f(r,t)}(1 - q) \mathcal{L}[\phi(\mathbf{r}, t; q) - u_0(\mathbf{r}, t)] = c_0 q \mathcal{H}(\mathbf{r}, t) \left\{ \mathcal{N}[\phi(\mathbf{r}, t; q)] - f(\mathbf{r}, t) \right\}, where L\mathcal{L} is an auxiliary linear operator, u0u_0 is the initial guess, ϕ\phi is the homotopy solution, N\mathcal{N} represents the nonlinear operator, ff is a known analytic function (often 0 for homogeneous problems), H\mathcal{H} is an auxiliary function (often 1), and c0c_0 is a convergence-control parameter (often denoted as ℏ\hbar) that adjusts the convergence radius by optimizing the solution series. Higher-order approximations are derived by expanding ϕ\phi in a Taylor series with respect to qq and differentiating successively, providing a systematic way to refine solutions order by order. One of HAM's key advantages is its built-in convergence control, achieved by plotting the ℏ\hbar-curve (a graph of squared residuals versus ℏ\hbar) to select optimal values within a valid region, ensuring the series converges to the exact solution even for large nonlinearity. This contrasts with methods like the Adomian decomposition or Lyapunov's small parameter expansion, which lack such guarantees and often diverge for strong nonlinearities. Since its inception, HAM has been extensively applied in fluid mechanics, including viscous flows over spheres and cylinders (valid up to Reynolds numbers around 30), nonlinear water waves revealing multiple equilibria, and plasma physics problems like shock waves. Its versatility extends to other fields, such as fractional differential equations, quantum mechanics, and finance, with thousands of publications demonstrating its impact as of 2025.

Introduction

History and Development

The homotopy analysis method (HAM) was first introduced by Shijun Liao in his PhD thesis titled "The Homotopy Analysis Method and Its Applications in Mechanics," completed in 1992 at . This work laid the foundational framework for an analytic technique that builds on the concept of from to address nonlinear problems without relying on small parameter assumptions typical of traditional perturbation methods. In the , Liao applied HAM to solve nonlinear boundary-value problems in , marking its initial practical demonstrations. These early efforts highlighted the method's potential for generating series solutions to strongly nonlinear equations. The publication of Liao's book Beyond Perturbation: Introduction to the Homotopy Analysis Method in 2003 provided a comprehensive exposition, solidifying HAM as a distinct approach. By 2004, Liao published a paper that further improved and systematically described the homotopy analysis method for nonlinear problems, enhancing its theoretical foundation. Liao's key contributions include the development of the auxiliary parameter ħ, which enables explicit control over the convergence of solution series. He also established the Numerical project at , a initiative that leverages computation tools to apply HAM extensively in simulations. By 2010, HAM had gained widespread adoption in , as evidenced by its use in solving complex viscous flow problems and reviewed in contemporary literature.

Core Concepts and Overview

The (HAM) is an analytic approximation technique developed for solving nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs), enabling the construction of continuous deformations from a known initial guess to approximate the exact solution. Introduced by Shijun Liao in his 1992 PhD dissertation, HAM provides a framework for generating reliable series solutions to highly nonlinear problems without assuming the existence of small or large physical parameters. At its core, relies on the construction of a —a continuous mapping that deforms a simple initial linear problem into the target nonlinear equation—ensuring that the solution evolves smoothly as an embedding parameter qq varies from 0 (corresponding to the initial guess) to 1 (corresponding to the original nonlinear problem). This deformation process incorporates auxiliary choices, such as linear operators and convergence-control parameters, to guarantee the convergence of the resulting . Unlike traditional perturbation methods, which depend on the presence of small or large parameters in the physical problem, HAM introduces an artificial embedding parameter q∈[0,1]q \in [0, 1] to build the homotopy independently of any inherent problem scales, thereby offering greater flexibility for a wide range of nonlinear systems. The method yields an infinite series solution of the form u(τ)=∑m=0∞um(τ)u(\tau) = \sum_{m=0}^{\infty} u_m(\tau), where the terms um(τ)u_m(\tau) are obtained iteratively through successive deformation steps, allowing for approximations of arbitrary order.

Mathematical Foundations

Homotopy Construction

The homotopy analysis method addresses a general nonlinear problem given by N[u(τ)]=0N[u(\tau)] = 0, where NN denotes a nonlinear operator acting on the unknown function u(τ)u(\tau) depending on the independent variable τ\tau. This setup requires an initial approximation u0(τ)u_0(\tau), selected to satisfy a simple linear problem L[u0(τ)−g(τ)]=0L[u_0(\tau) - g(\tau)] = 0, with LL as an auxiliary linear operator chosen based on the problem's linear part and g(τ)g(\tau) as a known function typically incorporating boundary or initial conditions. The choice of u0(τ)u_0(\tau) and LL provides flexibility, ensuring the initial guess aligns with the physical context without needing small parameters. The homotopy construction embeds the initial linear problem and the target nonlinear problem into a continuous deformation via an q∈[0,1]q \in [0,1], where q=0q = 0 yields the initial approximation and q=1q = 1 recovers the original nonlinear equation N[u(τ)]=0N[u(\tau)] = 0. This deformation blends the linear and nonlinear components progressively, enabling a smooth transition from the solvable initial guess to the full solution through a family of intermediate problems. To enhance convergence, particularly in problems with localized features, an optional auxiliary function H(τ)H(\tau) (non-zero almost everywhere) weights the deformation, influencing the emphasis on specific regions of the domain. For instance, in boundary layer flows, H(τ)=1−exp⁡(−τ2)H(\tau) = 1 - \exp(-\tau^2) concentrates the homotopy near the boundary surface τ=0\tau = 0, improving accuracy in thin-layer regions. This framework culminates in the zeroth-order deformation , which governs the embedding process.

Zeroth-Order Deformation Equation

The zeroth-order deformation forms the core of the , providing a continuous mapping from an initial to the exact solution of a nonlinear problem. This is given by (1−q)L[ϕ(τ;q)−u0(τ)]=qℏH(τ)N[ϕ(τ;q)],(1 - q) L \left[ \phi(\tau; q) - u_0(\tau) \right] = q \hbar H(\tau) N \left[ \phi(\tau; q) \right], where q∈[0,1]q \in [0, 1] is the homotopy , ϕ(τ;q)\phi(\tau; q) represents the solution function that varies continuously with qq, LL is an auxiliary linear operator, u0(τ)u_0(\tau) is the initial guess satisfying the original boundary or initial conditions, NN denotes the nonlinear operator of the original problem, H(τ)H(\tau) is an auxiliary function (often taken as 1 for simplicity), and ℏ\hbar is the convergence-control . The boundary or initial conditions associated with this equation ensure that ϕ(τ;0)=u0(τ)\phi(\tau; 0) = u_0(\tau), recovering the initial linear problem, and ϕ(τ;1)=u(τ)\phi(\tau; 1) = u(\tau), yielding the exact solution to the nonlinear equation when q=1q = 1. This setup, derived from the construction, allows for a deformation process that bridges the gap between a solvable linear case and the target nonlinear problem without relying on small physical parameters. Physically, the equation at q=0q = 0 enforces a linear subproblem governed solely by the auxiliary operator LL, providing a starting point for the approximation series, while at q=1q = 1, it fully incorporates the nonlinear effects through NN, resulting in the desired exact solution. The auxiliary parameter ℏ\hbar, introduced by Liao in 1997, is typically a negative chosen to guarantee the convergence of the resulting series solution; its optimal value is determined by analyzing ℏ\hbar-curves, which plot the residual error against ℏ\hbar to identify the region where the converges most effectively. This flexibility in selecting ℏ\hbar distinguishes HAM from traditional perturbation methods and enhances its applicability to strongly nonlinear problems.

Solution Procedure and Convergence

High-Order Deformation Equations

To derive the high-order deformation equations in the (HAM), the zeroth-order deformation equation is differentiated successively with respect to the embedding parameter qq and then evaluated at q=0q = 0. This process yields a sequence of linear equations that provide higher-order approximations to the solution of the original nonlinear problem. The solution ϕ(τ;q)\phi(\tau; q) to the zeroth-order deformation equation is assumed to admit a expansion in powers of the embedding parameter q∈[0,1]q \in [0, 1]: ϕ(τ;q)=u0(τ)+∑m=1∞um(τ)qm,\phi(\tau; q) = u_0(\tau) + \sum_{m=1}^\infty u_m(\tau) q^m, where u0(τ)u_0(\tau) is the initial guess, and the higher-order terms are defined by the mm-th order deformation derivatives: um(τ)=1m!∂mϕ(τ;q)∂qm∣q=0,m≥1.u_m(\tau) = \frac{1}{m!} \left. \frac{\partial^m \phi(\tau; q)}{\partial q^m} \right|_{q=0}, \quad m \geq 1.
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