Sinh-Gordon方程的同伦近似解

本文利用同伦分析方法得到了Sinh-Gordon方程的近似解.在所得到的解中包含一个辅助参数,可以有效地控制级数解的收敛范围和收敛速度. 关键词： 同伦分析方法 Sinh-Gordon方程 近似解     

微重力下圆管毛细流动解析近似解研究

应用同伦分析法研究微重力环境下圆管毛细流动解析近似解问题, 给出了级数解的表达公式. 不同于其他解析近似方法, 该方法从根本上克服了摄动理论对小参数的过分依赖, 其有效性与所研究的非线性问题是否含有小参数无关, 适用范围广. 同伦分析法提供了选取基函数的自由, 可以选取较好的基函数, 更有效地逼近问题的解, 通过引入辅助参数和辅助函数来调节和控制级数解的收敛区域和收敛速度, 同伦分析法为圆管毛细流动问题的解析近似求解开辟了一个全新的途径. 通过具体算例, 将同伦分析法与四阶龙格库塔方法数值解做了比较, 结果表明, 该方法具有很高的计算精度. 关键词： 圆管 微重力 毛细流动 同伦分析法     

一类Lorenz系统的同伦映射解法

研究了一个Lorenz方程的求解问题. 首先构造一组同伦映射,其次决定系统的初始近似,最后通过同伦映射得到了对应模型的各次近似解. 同伦映射方法是一个解析方法,得到的解还能够继续进行解析运算. 关键词： 洛伦兹方程 同伦映射 近似解 厄尔尼诺和拉尼娜现象     

扰动KdV方程孤子的同伦映射解

利用同伦映射方法研究了一类非线性KdV(Korteweg de Vries)方程. 首先引入一个同伦变换,使相应的方程求孤子解问题转化为映射变换问题.然后利用映射特性得到了原方程孤子的近似解. 关键词： 孤子 扰动 同伦映射     

微重力环境下无限长柱体内角毛细流动解析近似解研究

应用同伦分析法研究无限长柱体内角毛细流动解析近似解问题,给出了级数解的递推公式.不同于其他解析近似方法,该方法从根本上克服了摄动理论对小参数的过分依赖,其有效性与所研究的非线性问题是否含有小参数无关,适用范围广.同伦分析法提供了选取基函数的自由,可以选取较好的基函数,更有效地逼近问题的解,通过引入辅助参数和辅助函数来调节和控制级数解的收敛区域和收敛速度,同伦分析法为内角毛细流动问题的解析近似求解开辟了一个全新的途径.通过具体算例,将同伦分析法与四阶龙格库塔方法数值解做了比较,结果表明,该方法具有很高的计算精度.     

广义Landau-Ginzburg-Higgs方程孤子解的扰动理论

利用扰动方法研究了一类广义Landau-Ginzburg-Higgs方程.引入一个同伦映射，将原方程的解表示为渐近展开式，然后用相应的线性方程的解来近似地表示.讨论了方程的解与得到的近似解的关系. 关键词： 孤子 扰动 同伦映射     

一类扰动非线性发展方程的类孤子同伦近似解析解

采用一个简单而有效的技巧,研究了一类扰动非线性发展方程.首先,引入一个相应典型方程的孤子近似解.然后,利用同伦映射方法得到了原扰动非线性发展方程的近似解. 关键词： 孤子 扰动发展方程 同伦映射     

同伦分析法在求解非线性演化方程中的应用

利用同伦分析法求解了(2+1)维改进的 Zakharov-Kuznetsov方程, 得到了它的近似周期解,该解与精确解符合很好. 结果表明，同伦分析法在求解高维非线性演化方程时, 仍然是一种行之有效的方法. 同时,还对该方法进行了一定的扩展， 经过扩展后的方法能够更方便地求解更多非线性演化方程的高精度近似解析解. 关键词： 同伦分析法 改进的 Zakharov-Kuznetsov方程 周期解     

海-气振子ENSO模型的同伦解法

研究了一个厄尔尼诺-南方海涛(ENSO)的时滞振子的模型.利用同伦映射方法求出了ENSO模型的近似解 关键词： 非线性 同伦 ENSO模型     

一类燃烧模型的同伦分析解法

研究了一类非线性燃烧模型.利用同伦分析方法,得到了该模型的近似解. 关键词： 非线性方程 燃烧模型 同伦分析法 近似解     

Approximate homotopy similarity reduction for the generalized Kawahara equation via Lie symmetry method and direct method

<正>This paper studies the generalized Kawahara equation in terms of the approximate homotopy symmetry method and the approximate homotopy direct method.Using both methods it obtains the similarity reduction solutions and similarity reduction equations of different orders,showing that the approximate homotopy direct method yields more general approximate similarity reductions than the approximate homotopy symmetry method.The homotopy series solutions to the generalized Kawahara equation are consequently derived.     

Approximate homotopy symmetry method: Homotopy series solutions to the sixth-order Boussinesq equation

An approximate homotopy symmetry method for nonlinear problems is proposed and applied to the sixth-order Boussinesq equation,which arises from fluid dynamics.We summarize the general formulas for similarity reduction solutions and similarity reduction equations of different orders,educing the related homotopy series solutions.Zero-order similarity reduction equations are equivalent to the Painlevé IV type equation or Weierstrass elliptic equation.Higher order similarity solutions can be obtained by solving...     

Approximate Homotopy Symmetry Reduction Method: Infinite Series Reductions to Kawahara Equation

The Kawahara equation is studied through the approximate homotopy symmetry method. Under this method we get the similarity reduction solutions of the Kawahara equation, leading to the corresponding homotopy series solutions. Furthermore, the similarity solutions of the corresponding reduced linear ordinary differential equations are also considered.     

Approximate Homotopy Direct Reduction Method: Infinite Series Reductions to Perturbed mKdV Equations

An approximate homotopy direct reduction method is proposed and applied to two perturbed modified Korteweg- de Vries （mKdV） equations with fourth-order dispersion and second-order dissipation. The similarity reduction equations are derived to arbitrary orders. The method is valid not only for single soliton solutions but also for the Painlevd Ⅱ waves and periodic waves expressed by Jacobi elliptic functions for both fourth-order dispersion and second-order dissipation. The method is also valid for strong perturbations.     

Approximate direct reduction method: infinite series reductions to the perturbed mKdV equation

The approximate direct reduction method is applied to the perturbed mKdV equation with weak fourth order dispersion and weak dissipation. The similarity reduction solutions of different orders conform to formal coherence, accounting for infinite series reduction solutions to the original equation and general formulas of similarity reduction equations. Painlevé II type equations, hyperbolic secant and Jacobi elliptic function solutions are obtained for zero-order similarity reduction equations. Higher order similarity reduction equations are linear variable coefficient ordinary differential equations.     

Application of Homotopy Analysis Method for Solving Systems of Volterra Integral Equations

In this paper, we prove the convergence of homotopy analysis method (HAM). We also apply the homotopy analysis method to obtain approximate analytical solutions of systems of the second kind Volterra integral equations. The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of series solutions. It is shown that the solutions obtained by the homotopy-perturbation method (HPM) are only special cases of the HAM solutions. Several examples are given to illustrate the efficiency and implementation of the method.     

Approximate Similarity Reduction for Perturbed Kaup-Kupershmidt Equation via Lie Symmetry Method and Direct Method

The perturbed Kaup-Kupershmidt equation is investigated in terms of the approximate symmetry perturbation method and the approximate direct method. The similarity reduction solutions of different orders are obtained for both methods, series reduction solutions are consequently derived. Higher order similarity reduction equations are linear variable coefficients ordinary differential equations. By comparison, it is find that the results generated from the approximate direct method are more general than the results generated from the approximate symmetry perturbation method.     

六组点堆中子动力学方程组的同伦分析解

鉴于目前六组点堆中子动力学方程仍然无法获得解析解,本文尝试将同伦分析方法应用于六组缓发中子动力方程组的求解,获得了它的级数解析解,并对级数解析解算法的有效性进行了检验.结果表明,该级数解析解算法从计算时间和精度上都能达到工程应用的要求,可适宜于反应堆中子动力学控制的设计分析和仿真计算.     

Series solutions of stagnation slip flow and heat transfer by the homotopy analysis method

An analytical approximation for the similarity solutions of the two- and three-dimensional stagnation slip flow and heat transfer is obtained by using the homotopy analysis method. This method is a series expansion method, but it is different from the perturbation technique, because it is independent of small physical parameters at all. Instead, it is based on a continuous mapping in topology so that it is applicable for not only weakly but also strongly nonlinear flow phenomena. Convergent [m,m] homotopy Padé approximants are obtained and compared with the numerical results and the asymptotic approximations. It is found that the homotopy Padé approximants agree well with the numerical results. The effects of the slip length ℓ and the thermal slip constant β on the heat transfer characteristics are investigated and discussed. Supported by the National Natural Science Foundation of China (Grant No. 10872129)