Fractional variational iteration method and its application

Fractional differential equations have been investigated by variational iteration method. However, the previous works avoid the term of fractional derivative and handle them as a restricted variation. We propose herein a fractional variational iteration method with modified Riemann Liouville derivative which is more efficient to solve the fractional differential equations.     

Variational iteration method for solving time-fractional diffusion equations in porous the medium

<正>The variational iteration method is successfully extended to the case of solving fractional differential equations, and the Lagrange multiplier of the method is identified in a more accurate way.Some diffusion models with fractional derivatives are investigated analytically,and the results show the efficiency of the new Lagrange multiplier for fractional differential equations of arbitrary order.     

A Tutorial Review on Fractal Spacetime and Fractional Calculus

This tutorial review of fractal-Cantorian spacetime and fractional calculus begins with Leibniz’s notation for derivative without limits which can be generalized to discontinuous media like fractal derivative and q-derivative of quantum calculus. Fractal spacetime is used to elucidate some basic properties of fractal which is the foundation of fractional calculus, and El Naschie’s mass-energy equation for the dark energy. The variational iteration method is used to introduce the definition of fractional derivatives. Fractal derivative is explained geometrically and q-derivative is motivated by quantum mechanics. Some effective analytical approaches to fractional differential equations, e.g., the variational iteration method, the homotopy perturbation method, the exp-function method, the fractional complex transform, and Yang-Laplace transform, are outlined and the main solution processes are given.     

The fractional variational iteration method using He's polynomials

In this Letter, by introducing He's polynomials in the correct functional, we propose a new fractional variational iteration method to solve nonlinear time-fractional partial differential equations involving Jumarie's modified Riemann-Liouville derivative. Several examples have been solved to illustrate the proposed method is quite effective and convenient for solving kinds of nonlinear fractional order problems.     

Variational iteration method — a promising technique for constructing equivalent integral equations of fractional order

The variational iteration method is newly used to construct various integral equations of fractional order. Some iterative schemes are proposed which fully use the method and the predictor-corrector approach. The fractional Bagley-Torvik equation is then illustrated as an example of multi-order and the results show the efficiency of the variational iteration method’s new role.     

Conformable variational iteration method,conformable fractional reduced differential transform method and conformable homotopy analysis method for non-linear fractional partial differential equations

In this paper, we introduce conformable variational iteration method (C-VIM), conformable fractional reduced differential transform method (CFRDTM) and conformable homotopy analysis method (C-HAM). Between these methods, the C-VIM is introduced for the first time for fractional partial differential equations (FPDEs). These methods are new versions of well-known VIM, RDTM and HAM. In addition, above-mentioned techniques are based on new defined conformable fractional derivative to solve linear and non-linear conformable FPDEs. Firstly, we present some basic definitions and general algorithm for proposal methods to solve linear and non-linear FPDEs. Secondly, to understand better, the presented new methods are supported by some examples. Finally, the obtained results are illustrated by the aid of graphics and the tables. The applications show that these new techniques C-VIM, CFRDTM and C-HAM are extremely reliable and highly accurate and it provides a significant improvement in solving linear and non-linear FPDEs.     

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.     

He's variational iteration method applied to the solution of the prey and predator problem with variable coefficients

In this Letter, a mathematical model of the problem of prey and predator is presented and He's variational iteration method is employed to compute an approximation to the solution of the system of nonlinear differential equations governing the problem. The results are compared with the results obtained by Adomian decomposition method and homotopy perturbation method. Comparison of the methods show that He's variational iteration method is a powerful method for obtaining approximate solutions to nonlinear equations and their systems.     

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).     

Compacton and solitary pattern solutions for nonlinear dispersive KdV-type equations involving Jumarie?s fractional derivative

In this Letter, the fractional variational iteration method using He?s polynomials is implemented to construct compacton solutions and solitary pattern solutions of nonlinear time-fractional dispersive KdV-type equations involving Jumarie?s modified Riemann-Liouville derivative. The method yields solutions in the forms of convergent series with easily calculable terms. The obtained results show that the considered method is quite effective, promising and convenient for solving fractional nonlinear dispersive equations. It is found that the time-fractional parameter significantly changes the soliton amplitude of the solitary waves.     

On the new exact traveling wave solutions of the time-space fractional strain wave equation in microstructured solids via the variational method

In this paper, we mainly study the time-space fractional strain wave equation in microstructured solids. He’s variational method, combined with the two-scale transform are implemented to seek the solitary and periodic wave solutions of the time-space strain wave equation. The main advantage of the variational method is that it can reduce the order of the differential equation, thus simplifying the equation, making the solving process more intuitive and avoiding the tedious solving process.Finally, the numerical results are shown in the form of 3D and 2D graphs to prove the applicability and effectiveness of the method. The obtained results in this work are expected to shed a bright light on the study of fractional nonlinear partial differential equations in physics.     

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     

强阻尼广义sine-Gordon方程特征问题的变分迭代法

研究了在数学、力学中广泛出现的一类非线性强阻尼广义sine-Gordon扰动微分方程问题. 首先, 引入行波变换, 求出退化方程的精确解. 再构造一个泛函, 创建了一个变分迭代算法, 最后, 求出原非线性强阻尼广义sine-Gordon扰动微分方程问题的近似行波解析解. 用变分迭代法可得到的各次近似解, 具有便于求解、精度高等特点. 求得的近似解析解弥补了单纯用数值方法的模拟解的不足.     

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.     

厄尔尼诺大气物理机理的变分迭代解法

研究了一个厄尔尼诺大气物理机理振荡.利用变分迭代理论，简捷地得到了该模型解的近似展开式.通过与特殊情形下模型的精确解的比较，得到的一次近似解具有完全符合的精确度. 关键词： 非线性 变分迭代 厄尔尼诺现象     

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     

Numerical simulations of systems of PDEs by variational iteration method

In this Letter, a general framework of the variational iteration method (VIM) is presented for solving systems of linear and nonlinear partial differential equations (PDEs). In VIM, a correction functional is constructed by a general Lagrange's multiplier which can be identified via a variational theory. VIM yields an approximate solution in the form of a rapid convergent series. Comparison with the exact solutions shows that VIM is a powerful method for the solution of linear and nonlinear systems of PDEs.     

The residue harmonic balance for fractional order van der Pol like oscillators

When seeking a solution in series form, the number of terms needed to satisfy some preset requirements is unknown in the beginning. An iterative formulation is proposed so that when an approximation is available, the number of effective terms can be doubled in one iteration by solving a set of linear equations. This is a new extension of the Newton iteration in solving nonlinear algebraic equations to solving nonlinear differential equations by series. When Fourier series is employed, the method is called the residue harmonic balance. In this paper, the fractional order van der Pol oscillator with fractional restoring and damping forces is considered. The residue harmonic balance method is used for generating the higher-order approximations to the angular frequency and the period solutions of above mentioned fractional oscillator. The highly accurate solutions to angular frequency and limit cycle of the fractional order van der Pol equations are obtained analytically. The results that are obtained reveal that the proposed method is very effective for obtaining asymptotic solutions of autonomous nonlinear oscillation systems containing fractional derivatives. The influence of the fractional order on the geometry of the limit cycle is investigated for the first time.     

A discrete time method to the first variation of fractional order variational functionals

The fact that the first variation of a variational functional must vanish along an extremizer is the base of most effective solution schemes to solve problems of the calculus of variations. We generalize the method to variational problems involving fractional order derivatives. First order splines are used as variations, for which fractional derivatives are known. The Grünwald-Letnikov definition of fractional derivative is used, because of its intrinsic discrete nature that leads to straightforward approximations.     

Modified variational iteration method for an El Ni no Southern Oscillation delayed oscillator

This paper studies a delayed air-sea coupled oscillator describing the physical mechanism of El Niño Southern Oscillation. The approximate expansions of the delayed differential equation's solution are obtained successfully by the modified variational iteration method. The numerical results illustrate the effectiveness and correctness of the method by comparing with the exact solution of the reduced model.