The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method

In this paper, a scheme is developed to study numerical solution of the space- and time-fractional Burgers equations with initial conditions by the variational iteration method (VIM). The exact and numerical solutions obtained by the variational iteration method are compared with that obtained by Adomian decomposition method (ADM). The results show that the variational iteration method is much easier, more convenient, and more stable and efficient than Adomian decomposition method. Numerical solutions are calculated for the fractional Burgers equation to show the nature of solution as the fractional derivative parameter is changed.     

Schrodinger方程的变分迭代解法

该文以Ｓｃｈｒｏｄｉｎｇｅｒ方程为例,分析变分迭代法的一些基本特点．在该方法中引进了一广义拉氏乘子构造了一迭代格式,拉氏乘子可由变分理论最佳识别．由于在识别拉氏乘子是应用了限制变分的概念,所以只能通过迭代才能得到收敛解．为了加快收敛速度,可以在初始近似引入待定常数,而待定常数又可用各种方式最佳识别．文中初步分析了该方法的收敛性,对于Ｓｃｈｒｏｄｉｎｇｅｒ方程,其一阶近似即可得到Ｊｏｓｔ解．     

Solution of twelfth-order boundary value problems by variational iteration technique

In this paper, we implement a relatively new analytical technique which is called the variational iteration method for solving the twelfth-order boundary value problems. The analytical results of the problems have been obtained in terms of convergent series with easily computable components. Comparisons are made to verify the reliability and accuracy of the proposed algorithm. Several examples are given to check the efficiency of the suggested technique. The fact that variational iteration method solves nonlinear problems without using the Adomian’s polynomials is a clear advantage of this technique over the decomposition method.     

Applications of He’s principles to partial differential equations

This paper applies the variational iteration method (VIM) and semi-inverse variational principle to obtain solutions of linear and nonlinear partial differential equations. The nonlinear model is considered from gas dynamics, fluid dynamics and Burgers equation. The linear model is the heat transfer (diffusion) equation. Results show that variational iteration method is a powerful mathematical tool for solving linear and nonlinear partial differential equations, and therefore, can be widely applied to engineering problems.     

Solving Schrödinger equation by using modified variational iteration and homotopy analysis methods

In this paper, a nonlinear Schr ö dinger equation is solved by using the variational iteration method (VIM), modified variational iteration method (MVIM) and homotopy analysis method (HAM) numerically. For each method, the approximate solution of this equation is calculated based on a recursive relation which its components are computed easily. The existence and uniqueness of the solution and the convergence of the proposed methods are proved. A numerical example is studied to demonstrate the accuracy of the given algorithms     

Stabilizing the pull-in instability of an electro-statically actuated micro-beam using piezoelectric actuation

In the present article an investigation is presented into the stability of an electro-statically deflected clamped–clamped micro-beam sandwiched by two piezoelectric layers undergoing a parametric excitation applying an AC voltage to these layers. Applying an electrostatic actuation not only deflects the micro-beam but also decreases the bending stiffness of the structure, which can lead the structure to an unstable position by undergoing a saddle node bifurcation. Utilizing an appropriate AC actuation voltage to the piezoelectric layers produces a time varying axial force, which can play the role of a stabilizer exciting the system parameter. The governing equation of the motion is a nonlinear electro-mechanically coupled type PDE, which is derived using variational principle and discretized, applying Eigen-function expansion method. The resultant is a Mathieu type equation in its damped form. Using Floquet theory for single degree of freedom system the stable and unstable regions of the problem are investigated. The effects of viscous damping and electrostatic actuation on the stable regions of the problem are also studied.     

A characteristic description of orthonormal wavelet on subspace of

In this paper, by means of variational iteration method numerical and explicit solutions are computed for some fifth-order Korteweg-de Vries equations, without any linearization or weak nonlinearity assumptions. These equations are the Kawahara equation, Lax’s fifth-order KdV equation and Sawada–Kotera equation. Comparison with Adomian decomposition method reveals that the variational iteration method is easier to be implemented. We conclude that the method is a promising method to various kinds of fifth-order Korteweg-de Vries equations.     

分数阶变分迭代法处理分数阶类Boussinesq方程

分数阶变分迭代法（FVIM）是一种处理分数阶微分方程的有效工具.用分数阶变分迭代法求解了时间分数阶类Boussinesq方程,并且作为一种特殊情况,得到了类Boussinesq方程B（2.2）的单孤子解.     

A variational iteration method for solving Troesch’s problem

Troesch’s problem is an inherently unstable two-point boundary value problem. A new and efficient algorithm based on the variational iteration method and variable transformation is proposed to solve Troesch’s problem. The underlying idea of the method is to convert the hyperbolic-type nonlinearity in the problem into polynomial-type nonlinearities by variable transformation, and the variational iteration method is then directly used to solve this transformed problem. Only the second-order iterative solution is required to provide a highly accurate analytical solution as compared with those obtained by other analytical and numerical methods.     

Approximate solutions of fractional Zakharov–Kuznetsov equations by VIM

This paper presents the approximate analytical solution of a fractional Zakharov–Kuznetsov equation with the help of the powerful variational iteration method. The fractional derivatives are described in the Caputo sense. Several examples are given and the results are compared to exact solutions. The results show that the variational iteration method is very effective, convenient and simple to use.     

Dynamics of the Hantavirus infection through variational iteration method

This paper studies the dynamics of the Hantavirus infection model, which was originally developed by Abramson and Kenkre [G. Abramson, V.M. Kenkre, Spatiotemporal patterns in the hantavirus infection, Phys. Rev. E 66 (2002) 011912], by using a simple analytical method called the variational iteration method or VIM. The results obtained by the variational iteration method are compared with the classical Runge–Kutta method (fourth-order) to gauge its effectiveness. Numerical values from these analyses provide us with some useful observation on the behaviour of the infection subjected to certain conditions.     

Variational iteration method for solving non-linear partial differential equations

In this paper, we shall use the variational iteration method to solve some problems of non-linear partial differential equations (PDEs) such as the combined KdV–MKdV equation and Camassa–Holm equation. The variational iteration method is superior than the other non-linear methods, such as the perturbation methods where this method does not depend on small parameters, such that it can fined wide application in non-linear problems without linearization or small perturbation. In this method, the problems are initially approximated with possible unknowns, then a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory.     

On solving the chaotic Chen system: a new time marching design for the variational iteration method using Adomian’s polynomial

This paper centres on the effectiveness of the variational iteration method and its modifications for numerically solving the chaotic Chen system, which is a three-dimensional system of ODEs with quadratic nonlinearities. This research implements the multistage variational iteration method with an emphasis on the new multistage hybrid of variational iteration method with Adomian polynomials. Numerical comparisons are made between the multistage variational iteration method, the multistage variational iteration method using the Adomian’s polynomials and the classic fourth-order Runge-Kutta method. Our work shows that the new multistage hybrid provides good accuracy and efficiency with a performance that surpasses that of the multistage variational iteration method.     

Harmonic wave generation in non linear thermoelasticity by variational iteration method and Adomian's method

This paper applies the variational iteration method and Adomian's decomposition method to solve numerically the harmonic wave generation in a nonlinear, one-dimensional elastic half-space model subjected initially to a prescribed harmonic displacement. The results show that the variational iteration method is much easier, more convenient, and more stable and efficient than Adomian decomposition method.     

A procedure for determining a spacewise dependent heat source and the initial temperature

The inverse problem of determining a spacewise dependent heat source, together with the initial temperature for the parabolic heat equation, using the usual conditions of the direct problem and information from two supplementary temperature measurements at different instants of time is studied. These spacewise dependent temperature measurements ensure that this inverse problem has a unique solution, despite the solution being unstable, hence the problem is ill-posed. We propose an iterative algorithm for the stable reconstruction of both the initial data and the source based on a sequence of well-posed direct problems for the parabolic heat equation, which are solved at each iteration step using the boundary element method. The instability is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented for a typical benchmark test example, which has the input measured data perturbed by increasing amounts of random noise. The numerical results show that the proposed procedure gives accurate numerical approximations in relatively few iterations.     

A variational method for identifying a spacewise-dependent heat source

The inverse problem of determining a spacewise-dependent heatsource for the parabolic heat equation using the usual conditionsof the direct problem and information from one supplementarytemperature measurement at a given instant of time is studied.This spacewise-dependent temperature measurement ensures thatthis inverse problem has a unique solution, but the solutionis unstable and hence the problem is ill-posed. We propose avariational conjugate gradient-type iterative algorithm forthe stable reconstruction of the heat source based on a sequenceof well-posed direct problems for the parabolic heat equationwhich are solved at each iteration step using the boundary elementmethod. The instability is overcome by stopping the iterativeprocedure at the first iteration for which the discrepancy principleis satisfied. Numerical results are presented which have theinput measured data perturbed by increasing amounts of randomnoise. The numerical results show that the proposed procedureyields stable and accurate numerical approximations after onlya few iterations.     

An approximation to the solution of telegraph equation by variational iteration method

The variational iteration method (VIM) has been applied to solve many functional equations. In this article, this method is applied to obtain an approximate solution for the Telegraph equation. Some examples are presented to show the ability of the proposed method. The results of applying VIM are exactly the same as those obtained by Adomian decomposition method. It seems less computation is needed in proposed method.© 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Solitary wave solutions of the improved KdV equation by VIM

The variational iteration method (VIM) is applied to solve numerically the improved Korteweg-de Vries equation (IKdV). A correction function is constructed with a general Lagrange multiplier that can be identified optimally via the variational theory. This technique provides a sequence of functions with easily computable components that converge rapidly to the exact solution of the IKdV equation. Propagation of single, interaction of two, and three solitary waves, and also birth of solitons have been discussed. Three invariants of motion have been evaluated to determine the conservation properties of the problem. This procedure is promising for solving other nonlinear equations.     

Constructing approximate Green’s function for a vector equation for the electric field using the variational iteration method

In this work, we use He’s variational iteration method (VIM) to find approximate Green’s functions for a vector equation for the electric field with anisotropic dielectric permittivity and magnetic permeability. We present numerical examples which show that an approximate solution of an initial value problem (IVP) for a vector equation can be obtained by using these approximate Green’s functions.     

He's variational iteration method for the solution of nonlinear Newell-Whitehead-Segel equation

In this paper, we apply He''s Variational iteration method (VIM) for solving nonlinear Newell-Whitehead-Segel equation. By using this method three different cases of Newell-Whitehead-Segel equation have been discussed. Comparison of the obtained result with exact solutions shows that the method used is an effective and highly promising method for solving different cases of nonlinear Newell-Whitehead-Segel equation.