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.     

Variational iteration method for solving integro-differential equations

In this Letter, the variational iteration method is applied to solve integro-differential equations. Some examples are given to illustrate the effectiveness of the method, the results show that the method provides a straightforward and powerful mathematical tool for solving various integro-differential equations.     

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).     

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.     

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.     

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.     

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.     

Solution of the Bethe-Goldstone equation for the reaction matrix in finite nuclei

A method for solving the BG equation for the reaction matrixt in finite nuclei is presented. The application of this method is demonstrated for a one-dimensional case, which is similar to the problem where the internucleon potential acts only in the relatives-state. The single particle potential has a harmonic oscillator form and the phenomenological internucleon potentialv(r) contains a hard core and an attractive part of the Yukawa type. By taking the exclusion principle into account exactly an infinite system of integral equations is obtained. It is proved that the solution of the corresponding finite system converges to the exact solution. An iteration method for solving such a finite system with an arbitrary number of equations is developed. Its main feature consists in the exclusion of the dependence on the hard core part ofv(r) (which is treated as the limit case of a rectangular repulsive potential with a variable heightv ₀). This exclusion transforms the original system to a system of integral equations depending only on the attractive part ofv(r) and to a linear algebraic system. Both these systems can be solved by iteration for all values ofv ₀ as well as for v₀= +. The numerical results confirm the rapid convergence of the proposed iteration method and demonstrate that the solution of the finite system with a sufficiently large number of equations approximates the exact solution very precisely.     

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.     

The application of He's variational iteration method to nonlinear equations arising in heat transfer

In this Letter, we purpose to solve nonlinear equations arising in heat transfer through the He's variational iteration method (VIM) and show that it is strongly and simply capable of solving a large class of linear or nonlinear differential equations without the tangible restriction of sensitivity to the degree of the nonlinear term and also is very user friend because it reduces the size of calculations besides, its iterations are direct and straightforward. 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$c_n$, as used in this Letter.     

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.     

一类扰动洛伦兹系统的解法

研究了一个大气物理中洛伦兹系统的求解问题.首先利用广义变分原理构造一组变分迭代,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.广义变分迭代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 变分原理 近似解     

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.     

Variational iteration method for solving an inverse parabolic equation

The variational iteration method is applied to solving an inverse problem of determining an unknown parameter in a linear parabolic equation. This method is based on the use of Lagrange multipliers for identification of optimal values of parameters in a functional. We get a rapid convergent sequence tending to the exact solution of the inverse problem. To show the efficiency of the present method, some interesting examples are presented.     

Solutions of Heat-Like and Wave-Like Equations with Variable Coefficients by Means of the Homotopy Analysis Method

We employ the homotopy analysis method （HAM） to obtain approximate analytical solutions to the heat-like and wave-like equations. The HAM contains the auxiliary parameter h, which provides a convenient way of controlling the convergence region of series solutions. The analysis is accompanied by several linear and nonlinear heat-like and wave-like equations with initial boundary value problems. The results obtained prove that HAM is very effective and simple with less error than the Adomian decomposition method and the variational iteration method.     

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.     

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.     

一类强非线性发展方程孤波变分迭代解法

研究了一类强非线性发展方程. 利用变分迭代方法,首先构造了相应的变分;其次选取了适当的初始近似;再用迭代方法得到了孤波的任意次精度的近似解. 关键词： 发展方程 非线性 孤波 近似方法.     

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.     

Beyond Adomian method: The variational iteration method for solving heat-like and wave-like equations with variable coefficients

Recently Adomian method was used to solve various kinds of heat-like and wave-like equations. In this Letter, an alternative approach called the variational iteration method is presented to overcome the demerit of complex calculation of Adomian polynomial. Some examples are given to show the reliability and the efficiency of the variational iteration method.