Application of He's variational iteration method to nonlinear heat transfer equations

Instead of finding a small parameter for solving nonlinear problems through perturbation method, a new analytical method called He's variational iteration method (VIM) is introduced to be applied to solve nonlinear heat transfer equations in this Letter. In this research, variational iteration method is used to solve an unsteady nonlinear convective-radiative equation and a nonlinear convective-radiative-conduction equation containing two small parameters of ε₁ and ε₂ and evaluate the efficiency of straight fins. VIM can apply to the nonlinear equations with boundary or initial conditions defined in different points just with developing the correction functional using the extra parameters such as Cn, as used in this Letter.     

Assessment of two analytical approaches in some nonlinear problems arising in engineering sciences

In this research, two powerful analytical methods are introduced to handle nonlinear good Boussinesq, heat transfer and coupled Burgers' equations. One is the homotopy-perturbation method (HPM) and the other is the variational iteration method (VIM). VIM is used to construct correction functionals using general Lagrange multipliers identified optimally via the variational theory. HPM converts a difficult problem into a simple one, which can be easily handled. The results attained in this paper confirm the idea that HPM and VIM are powerful mathematical tools and that they can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering.     

A Variational Iteration Solving Method for a Class of Generalized Boussinesq Equations

We study a generalized nonlinear Boussinesq equation by introducing a proper functional and constructing the variational iteration sequence with suitable initial approximation. The approximate solution is obtained for the solitary wave of the Boussinesq equation with the variational iteration method.     

一类非线性扰动发展方程的广义迭代解

利用广义变分迭代方法研究了一类非线性发展扰动方程.首先引入一个泛函.然后求其变分,最后构造方程解的迭代关系式.得到了问题的近似解和精确解析解. 关键词： 发展方程 扰动 变分迭代     

Higher accuracy analytical approximations to a nonlinear oscillator with discontinuity by He's homotopy perturbation method

He's homotopy perturbation method is used to calculate higher-order approximate periodic solutions of a nonlinear oscillator with discontinuity for which the elastic force term is proportional to sgn(x). We find He's homotopy perturbation method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. Only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate period of less than 1.56% for all values of oscillation amplitude, while this relative error is 0.30% for the second iteration and as low as 0.057% when the third-order approximation is considered. Comparison of the result obtained using this method with those obtained by different harmonic balance methods reveals that He's homotopy perturbation method is very effective and convenient.     

一类非线性扰动Burgers方程的孤子变分迭代解法

研究了一类非线性扰动Burgers方程的求解问题. 利用变分迭代方法, 首先引入一个泛函, 然后计算它的变分, 最后构造方程的迭代关系式, 得到了相应方程的孤子解的近似展开式.     

Application of He's parameter-expansion method for oscillators with smooth odd nonlinearities

This Letter applies He's parameter-expansion method to oscillators with smooth nonlinearities. The method does not depend upon small parameter assumption, hence it is very better than the perturbation method. In parameter-expansion method the solution and unknown frequency of oscillation are expanded in a series by a bookkeeping parameter. By imposing the non-secularity condition at each order in the expansion the method provides different approximations to both the solution and the frequency of oscillation. One iteration step provides an approximate solution which is valid for the whole solution domain. The method can be easily extended to other nonlinear oscillations.     

Exact Solutions of the Harry-Dym Equation

The aim of this paper is to generate exact travelling wave solutions of the Harry-Dym equation through the methods of Adomian decomposition, He's variational iteration, direct integration, and power series. We show that the two later methods are more successful than the two former to obtain more solutions of the equation.     

Prescribing a multistage analytical method to a prey-predator dynamical system

This article discusses the effectiveness of a fresh analytical method in solving a prey-predator problem, which is described as a system of two nonlinear ordinary differential equations. The method of interest is the multistage variational iteration method (MVIM), which provides a slight modification of the classical variational iteration method (VIM). We shall compare solutions of the classical VIM along with MVIM and match them against the conventional numerical method, Runge-Kutta (RK4) (fourth-order).     

Modified variational iteration method

In this Letter, we introduce a modified variational iteration method by inserting some unknown parameters into the correctional functional. The main advantage of this method is that one can avoid the uncontrollability problems, of the nonzero endpoint conditions, encountered in the traditional variational iteration method. Moreover, the method is applied to some nonlinear equations and the numerical solutions reveal that the modified method is accurate and efficient to solve a large class of nonlinear differential equations. Furthermore, the method does not share the drawbacks of the conventional variational iteration method, namely the restriction of the order of the nonlinearity term or even the form of the boundary conditions.     

一类非线性尘埃等离子体孤波解

研究了一类非线性尘埃等离子体孤波解. 首先构造一个变分迭代, 其次决定系统的初始近似, 最后通过变分迭代方法得到了对应模型的各次近似解. 关键词： 等离子体 变分迭代 近似解     

The variational iteration method for solving <Emphasis Type="Italic">n</Emphasis>-th order fuzzy differential equations

The variational iteration method (VIM) proposed by Ji-Huan He is a new analytical method for solving linear and nonlinear equations. In this paper, the variational iteration method has been applied in solving nth-order fuzzy linear differential equations with fuzzy initial conditions. This method is illustrated by solving several examples.     

三类大气基本波动的广义变分原理

半反推法是He为了寻求数学物理问题的变分原理而提出的, 可避免由拉氏乘子法引起的临界变分现象. 应用半反推法分别获得了动力气象中Rossby波、大气声波和重力外波等三类基本大气波动的广义变分原理, 并验证了它们的正确性.     

赤道东太平洋SST的海-气振子模型

研究了一个海-气振子的非线性耦合系统的模型.利用变分迭代原理，首先构造了相应的一组泛函.其次选取其Lagrange乘子.再用广义变分迭代方法得到了海-气振子模型一组解的近似序列. 关键词： 海-气振子 非线性 耦合系统 厄尔尼诺模型     

Approximate analytical solutions to a conservative oscillator using global residue harmonic balance method

In the current research paper, a conservative system comprising of a mass grounded by linear and nonlinear springs in series connection is studied. The equation of motion for the aforementioned system has been derived as a nonlinear ordinary differential equation with inertia and static–type cubic nonlinearities. The global residue harmonic balance method is applied to obtain an approximate analytical frequency and periodic solution of the problem. Using the obtained analytical expressions, the influences of the hardening and softening nonlinear spring on the non–dimensional frequency are investigated. The results show that developing the system nonlinearity leads the displacement of the mass and the deflection of linear spring to approach each other. Moreover, comparison of the results obtained using the proposed procedure with those achieved by other methods such as numerical method, variational iteration method and harmonic balance approach demonstrates the accuracy and advantages of the current approach.     

用广义变分迭代理论求一类相对转动动力学方程的解

研究了一类具有非线性阻尼力和强迫周期力项的相对转动扰动动力学方程. 首先利用变分原理构造了广义变分迭代, 其次决定方程的初始近似, 最后通过迭代表示式得到了对应方程的任意次近似解. 关键词： 相对转动 动力系统 变分迭代     

Traveling Wave Solutions for Generalized Bretherton Equation

This paper studies the Generalized Bretherton equation using trigonometric function method including the sech-function method, the sine-cosine function method, and the tanh-function method, and He's semi-inverse method (He's variational method). Various traveling wave solutions are obtained, revealing anintrinsic relationship among the amplitude, frequency, and wave speed.     

(3+1)维Burgers扰动系统孤波的解法

研究了在物理模型中的一类扰动高维非线性Burgers系统. 利用经过改进的广义变分迭代方法, 构造了相应迭代关系式. 得到了扰动系统的孤波近似解.     

一类大气浅水波系统的广义变分迭代行波近似解

本文研究了一类非线性浅水波系统,首先构造了相应的泛函. 其次选取Lagrange乘子,再用改进的广义变分迭代方法,得到了相应模型的行波近似解析解. 关键词： 浅水波 非线性 变分迭代 近似解     

Generalized Variational Iteration Solution of Soliton for Disturbed KdV Equation

The corresponding solution for a class of disturbed KdV equation is considered using the analytic method. From the generalized variational iteration theory, the problem of solving soliton for the corresponding equation translates into the problem of variational iteration. And then the approximate solution of the soliton for the equation is obtained.