强非线性多自由度动力系统主共振同伦分析法研究

应用同伦分析方法(HAM)解决强非线性多自由度系统在谐波激振力下的主共振问题．同伦分析方法的有效性独立于所考虑的方程中是否含有的小参数．同伦分析方法提供了一个简单的方法,通过一个辅助参数h-来调节和控制级数解的收敛区域．两个具体算例表明,同伦分析方法得出的结果与修正Linstedt-Poincaré法、增量谐波平衡法的解决方案得出的结果相吻合．     

二阶非线性微分方程边值问题的同伦分析解

研究了结合变量替换应用同伦分析方法,去求解二阶非线性微分方程的两点边值问题,并得到了逼近解析解的函数级数形式.给出了应用同伦分析方法求解二阶非线性问题的三个实例,显示了同伦分析方法可以比较有效地求解非线性问题.     

电池系统建模中Butler-Volmer方程的同伦分析求解

Butler-Volmer方程是电化学系统中描述电极动力学过程的本构方程,具有强非线性．为了对这一方程(耦合两个Ohm方程)进行解析求解,在同伦分析方法的框架下,发展了满足简单条件的广义非线性算子的算法,以取代原同伦分析中的非线性算子．该广义非线性算子的构造保证了高阶形变方程的线性特征．这一方法的有效性通过一些算例得到了验证．最后通过同伦分析方法对Butler-Volmer方程进行了求解,结果显示过电位和电流密度的级数解析解与数值解吻合很好,并有很好的收敛效率．     

广义泰勒定理:"同伦分析方法"之有效性的一个数理逻辑证明

推导了复变函数一个广义意义上的泰勒级数表达式,证明了有关的收敛性定理,大大增大摄动级数解的收敛区域。定理的证明亦为一种新的、求解非线性问题的解析方法（即“同伦分析方法”）的有效性奠定了一个坚实的数理逻辑基础。     

一类非线性双曲型发展方程的孤子解

研究了一类非线性发展方程.首先在无扰动情形下,利用待定函数和泛函同伦映射方法得到了非扰动发展方程的孤子精确解和扰动方程的任意次近似行波孤子解.接着引入一个同伦映射,并选取初始近似函数,再用同伦映射理论,依次求出非线性双曲型发展扰动方程孤子解的各次近似解析解.再利用摄动理论举例说明了用该方法得到的近似解析解的有效性和各次近似解的近似度.最后,简述了用同伦映射方法得到的近似解的意义,指出了用上述方法得到的各次近似解具有便于求解、精度高等优点.     

激光脉冲放大器增益通量耦合系统解

研究了一个激光脉冲放大器增益通量系统解的问题.首先讨论了较一般的系统, 然后引入一个同伦映射.再利用映射的性质, 引进一个人工参数, 将求解非线性问题转化为求解一系列线性问题.再逐次地求出对应的线性问题的解, 最后得到了原模型解的近似展开式.可以看出, 同伦映射方法是一个解析的方法.它是通过函数的解析运算并用初等函数来表达近似解,其不同于用离散数值运算的数值计算方法.因此通过同伦映射解, 还可以对它继续进行解析运算, 从而可以进行微分和积分等运算来得到与激光脉冲放大器增益通量相关的其他物理量的性态.     

一个广义非线性扰动Klein-Gordon方程孤子解

研究了一个广义非线性扰动Klein-Gordon方程.利用同伦映射方法,首先构造了相应的同伦映射;然后选取了适当的初始近似;并计算各阶相应孤子近似解.同时还考虑了一个微扰方程.     

单参数非线性问题中高阶奇异点的计算

本文考虑计算单参数非线性问题中高阶奇异点的数值方法，基于确定奇异点的一个普适的扩张系统，结合同伦参数的拟弧长延拓，给出了计算各类高阶奇异点的一个统一算法，数值例子表明了算法的有效性．     

Duffing简谐振子同伦分析法求解

利用同伦分析方法求解了Duffing简谐振子,数值确定了变形方程中的辅助参数,得到了一族响应和频率的近似周期解,该解与精确解符合很好,结果表明,同伦分析法在求解强非线性振子时,仍然是一种行之有效的方法.     

基于Bregman距离函数的可靠性分析

针对概率结构可靠性问题,引入Bregman距离函数,建立了基于同伦算法(HM)的可靠性分析模型.利用极限状态方程,将可靠性指标求解转化为一个非线性约束优化问题.结合同伦思想的基本理论和Bregman距离函数,构造同伦方程组,采用路径跟踪算法对该方程组进行求解.通过相应的数值算例探讨了不同函数形式以及不同程度非线性问题的可靠性计算,并与其他方法计算结果进行了对比,分析结果表明该模型能够有效求解概率结构可靠性问题.     

The homotopy analysis method for explicit analytical solutions of Jaulent–Miodek equations

In this work, the homotopy analysis method (HAM) is applied to obtain the explicit analytical solutions for system of the Jaulent–Miodek equations. The validity of the method is verified by comparing the approximation series solutions with the exact solutions. Unlike perturbation methods, the HAM does not depend on any small physical parameters at all. Thus, it is valid for both weakly and strongly nonlinear problems. Besides, different from all other analytic techniques, the HAM provides us a simple way to adjust and control the convergence region of the series solution by means of an auxiliary parameter ?. Briefly speaking, this work verifies the validity and the potential of the HAM for the study of nonlinear systems. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

The scaled boundary FEM for nonlinear problems

The traditional scaled boundary finite-element method (SBFEM) is a rather efficient semi-analytical technique widely applied in engineering, which is however valid mostly for linear differential equations. In this paper, the traditional SBFEM is combined with the homotopy analysis method (HAM), an analytic technique for strongly nonlinear problems: a nonlinear equation is first transformed into a series of linear equations by means of the HAM, and then solved by the traditional SBFEM. In this way, the traditional SBFEM is extended to nonlinear differential equations. A nonlinear heat transfer problem is used as an example to show the validity and computational efficiency of this new SBFEM.     

On the homotopy analysis method for backward/forward-backward stochastic differential equations

In this paper, an analytic approximation method for highly nonlinear equations, namely the homotopy analysis method (HAM), is employed to solve some backward stochastic differential equations (BSDEs) and forward-backward stochastic differential equations (FBSDEs), including one with high dimensionality (up to 12 dimensions). By means of the HAM, convergent series solutions can be quickly obtained with high accuracy for a FBSDE in a 6-dimensional case, within less than 1 % CPU time used by a currently reported numerical method for the same case [34]. Especially, as dimensionality enlarges, the increase of computational complexity for the HAM is not as dramatic as this numerical method. All of these demonstrate the validity and high efficiency of the HAM for the backward/forward-backward stochastic differential equations in science, engineering, and finance.     

Analytic approximate solutions for heat transfer of a micropolar fluid through a porous medium with radiation

This paper aims to present complete analytic solution to heat transfer of a micropolar fluid through a porous medium with radiation. Homotopy analysis method (HAM) has been used to get accurate and complete analytic solution. The analytic solutions of the system of nonlinear ordinary differential equations are constructed in the series form. The convergence of the obtained series solutions is carefully analyzed. The velocity and temperature profiles are shown and the influence of coupling constant, permeability parameter and the radiation parameter on the heat transfer is discussed in detail. The validity of our solutions is verified by the numerical results (fourth-order Runge–Kutta method and shooting method).     

Purely analytic approximate solutions for steady three-dimensional problem of condensation film on inclined rotating disk by homotopy analysis method

The similarity transform for the steady three-dimensional problem of a condensation film on an inclined rotating disk gives a system of nonlinear ordinary differential equations which are analytically solved by applying a newly developed method namely the homotopy analysis method (HAM). The analytic solutions of the system of nonlinear ordinary differential equations are constructed in the series form. The convergence of the obtained series solutions is carefully analyzed. The velocity and temperature profiles are shown and the influence of the Prandtl number on the heat transfer and the Nusselt number is discussed in detail. The validity of our results is verified by numerical results.     

Explicit series solutions of some linear and nonlinear Schrodinger equations via the homotopy analysis method

In this paper, by means of the homotopy analysis method (HAM), the solutions of some Schrodinger equations are exactly obtained in the form of convergent Taylor series. The HAM contains the auxiliary parameter ?, that provides a convenient way of controlling the convergent region of series solutions. This analytical method is employed to solve linear and nonlinear examples to obtain the exact solutions. HAM is a powerful and easy-to-use analytic tool for nonlinear problems.     

Analytic approximate solutions to Burgers,Fisher, Huxley equations and two combined forms of these equations

In this paper, we are giving analytic approximate solutions to a class of nonlinear PDEs using the homotopy analysis method (HAM). The Burgers, Fisher, Huxley, Burgers–Fisher and Burgers–Huxley equations are considered. We aim two goals: one is to highlight the efficiency of HAM in solving this class of PDEs and the other is that, although the considered equations have different combinations of nonlinear terms, when applying HAM, we use the same initial guess, the same auxiliary linear operator and the same auxiliary function for all of them.     

The explicit series solution of SIR and SIS epidemic models

In this paper the SIR and SIS epidemic models in biology are solved by means of an analytic technique for nonlinear problems, namely the homotopy analysis method (HAM). Both of the SIR and SIS models are described by coupled nonlinear differential equations. A one-parameter family of explicit series solutions are obtained for both models. This parameter has no physical meaning but provides us with a simple way to ensure convergent series solutions to the epidemic models. Our analytic results agree well with the numerical ones. This analytic approach is general and can be applied to get convergent series solutions of some other coupled nonlinear differential equations in biology.     

Analytical approximations for a population growth model with fractional order

In this paper, we apply the homotopy analysis method (HAM) to solve the fractional Volterra’s model for population growth of a species in a closed system. This technique is extended to give solutions for nonlinear fractional integro–differential equations. The whole HAM solution procedure for nonlinear fractional differential equations is established. Further, the accurate analytical approximations are obtained for the first time, which are valid and convergent for all time t. This indicates the validity and great potential of the homotopy analysis method for solving nonlinear fractional integro–differential equations.     

An approximation of the analytic solution of some nonlinear heat transfer in fin and 3D diffusion equations using HAM

In this article, the approximate solution of nonlinear heat diffusion and heat transfer equation are developed via homotopy analysis method (HAM). This method is a strong and easy‐to‐use analytic tool for investigating nonlinear problems, which does not need small parameters. HAM contains the auxiliary parameter ?, which provides us with a simple way to adjust and control the convergence region of solution series. By suitable choice of the auxiliary parameter ?, we can obtain reasonable solutions for large modulus. In this study, we compare HAM results, with those of homotopy perturbation method and the exact solutions. The first differential equation to be solved is a straight fin with a temperature‐dependent thermal conductivity and the second one is the two‐ and three‐dimensional unsteady diffusion problems. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010