Variational iteration technique for solving a system of nonlinear equations

In this paper, we use the variational iteration technique to suggest and analyze some new iterative methods for solving a system of nonlinear equations. We prove that the new method has fourth-order convergence. Several numerical examples are given to illustrate the efficiency and performance of the new iterative methods. Our results can be viewed as a refinement and improvement of the previously known results.     

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.     

Construction of Newton-like iteration methods for solving nonlinear equations

In this paper, we present a simple, and yet powerful and easily applicable scheme in constructing the Newton-like iteration formulae for the computation of the solutions of nonlinear equations. The new scheme is based on the homotopy analysis method applied to equations in general form equivalent to the nonlinear equations. It provides a tool to develop new Newton-like iteration methods or to improve the existing iteration methods which contains the well-known Newton iteration formula in logic; those all improve the Newton method. The orders of convergence and corresponding error equations of the obtained iteration formulae are derived analytically or with the help of Maple. Some numerical tests are given to support the theory developed in this paper.     

A remark on He’s variational iteration technique for solving nonlinear equations

The variational iteration method (VIM) can be usefully applied for solving many linear and nonlinear scientific and engineering problems. In this note we show that He’s approach for solving nonlinear equations, arising from the VIM, is, actually, Schröder’s method presented in his classical work from 1870.     

Variational iteration method for solving cubic nonlinear Schrödinger equation

The variational iteration method is applied to solve the cubic nonlinear Schrödinger (CNLS) equation in one and two space variables. In both cases, we will reduce the CNLS equation to a coupled system of nonlinear equations. Numerical experiments are made to verify the efficiency of the method. Comparison with the theoretical solution shows that the variational iteration method is of high accuracy.     

Variational iteration method for solving generalized Burger–Fisher and Burger equations

We consider variational iteration method to investigate generalized Burger–Fisher and Burger equations. In this method, general Lagrange multipliers are introduced to construct correction functionals for the problems. The multipliers in the functionals can be identified optimally via variational theory. Comparison with Adomian decomposition method reveals that the approximate solutions obtained by the proposed method converge to its exact solution faster than those of Adomian’s method. Its remarkable accuracy is finally demonstrated in the study of some values of constants in generalized Burger–Fisher and Burger equations.     

Variational Iteration Method for Delay Differential Equations

Since1930'sand40's,theexamplesofdelaydifferentialequationsarisinginpracticalapplicationshavebeenescalatedrapidly,andhavebeenstudiedextensively(fordetails,see[1]).Inthispaperwewillproposeanovelmethodcalledvariationaliterationmethod[2]tosolvesuchproblems.Considerfollowingpopulationgrowthmodel[1]x′(t)+cθ(t-1)x(t)+cθ(t-1)=0(1a)x(0)=θ(0),-1≤t≤0(1b)　　Accordingtovariationaliterationmethod[2],thecorrectionfunctionalcanbeconstructedasfollowsxn+1(t)=xn(t)+∫t0λ[x′nτ+cθ(τ-1)xn(τ)+cθ(τ-1…     

Variational iteration method for solving sixth-order boundary value problems

In this paper, we have shown that sixth-order boundary value problems can be transformed into a system of integral equations, which can be solved by using variational iteration method. The analytical results of the equations have been obtained in terms of convergent series with easily computable components. Several examples are given to illustrate the efficiency and implementation of the method. It is observed that the proposed technique is more useful and is easier to implement because one does not need to calculate the Adomian’s polynomials which is itself a difficult task.     

Variational iteration method for solving two-point boundary value problems

Variational iteration method is introduced to solve two-point boundary value problems. Numerical results demonstrate that the method is promising and may overcome the difficulty arising in Adomian decomposition method.     

A nonlinear iteration method for solving a two-dimensional nonlinear coupled system of parabolic and hyperbolic equations

A nonlinear iteration method for solving a class of two-dimensional nonlinear coupled systems of parabolic and hyperbolic equations is studied. A simple iterative finite difference scheme is designed; the calculation complexity is reduced by decoupling the nonlinear system, and the precision is assured by timely evaluation updating. A strict theoretical analysis is carried out as regards the convergence and approximation properties of the iterative scheme, and the related stability and approximation properties of the nonlinear fully implicit finite difference (FIFD) scheme. The iterative algorithm has a linear constringent ratio; its solution gives a second-order spatial approximation and first-order temporal approximation to the real solution. The corresponding nonlinear FIFD scheme is stable and gives the same order of approximation. Numerical tests verify the results of the theoretical analysis. The discrete functional analysis and inductive hypothesis reasoning techniques used in this paper are helpful for overcoming difficulties arising from the nonlinearity and coupling and lead to a related theoretical analysis for nonlinear FI schemes.     

Modified variational iteration method for solving Helmholtz equations

In this paper, we apply the modified variational iteration method (MVIM) for solving the Helmholtz equations. The proposed modification is made by introducing He's polynomials in the correction functional. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed iterative scheme finds the solution without any discretization, linearization, or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that the proposed technique solves nonlinear problems without using the Adomian's polynomials can be considered as a clear advantage of this algorithm over the decomposition method.     

Variational iteration method for fractional heat- and wave-like equations

This paper applies the variational iteration method to obtaining analytical solutions of fractional heat- and wave-like equations with variable coefficients. Comparison with the Adomian decomposition method shows that the VIM is a powerful method for the solution of linear and nonlinear fractional differential equations.     

Some geometrical iteration methods for nonlinear equations

This paper describes geometrical essentials of some iteration methods （e.g. Newton iteration, secant line method, etc.） for solving nonlinear equations and advances some geometrical methods of iteration that are flexible and efficient.     

求解Sylvester矩阵方程的一种改进的梯度方法

提出了一种改进的梯度迭代算法来求解Sylvester矩阵方程和Lyapunov矩阵方程.该梯度算法是通过构造一种特殊的矩阵分裂,综合利用Jaucobi迭代算法和梯度迭代算法的求解思路.与已知的梯度算法相比,提高了算法的迭代效率.同时研究了该算法在满足初始条件下的收敛性.数值算例验证了该算法的有效性.     

Two CSCS-based iteration methods for solving absolute value equations

Recently, two families of HSS-based iteration methods are constructed for solving the system of absolute value equations (AVEs), which is a class of non-differentiable NP-hard problems. In this study, we establish the Picard-CSCS iteration method and the nonlinear CSCS-like iteration method for AVEs involving the Toeplitz matrix. Then, we analyze the convergence of the Picard-CSCS iteration method for solving AVEs. By using the theory about nonsmooth analysis, we particularly prove the convergence of the nonlinear CSCS-like iteration solver for AVEs. The advantage of these methods is that they do not require the storage of coefficient matrices at all, and the sub-system of linear equations can be solved efficiently via the fast Fourier transforms (FFTs). Therefore, computational cost and storage can be saved in practical implementations. Numerical examples including numerical solutions of nonlinear fractional diffusion equations are reported to show the effectiveness of the proposed methods in comparison with some existing methods.     

Numerical stability for solving nonlinear equations

Summary The concepts of the condition number, numerical stability and well-behavior for solving systems of nonlinear equationsF(x)=0 are introduced. Necessary and sufficient conditions for numerical stability and well-behavior of a stationary are given. We prove numerical stability and well-behavior of the Newton iteration for solving systems of equations and of some variants of secant iteration for solving a single equation under a natural assumption on the computed evaluation ofF. Furthermore we show that the Steffensen iteration is unstable and show how to modify it to have well-behavior and hence stability.This work was supported in part by the Office of Naval Research under Contract N 00014-67-0314-0010 NR 044-422 and by the National Science Foundation under Grant GJ 32111     

An effective variational iteration algorithm for solving Riccati differential equations

The piecewise variational iteration method (VIM) for solving Riccati differential equations (RDEs) provides a solution as a sequence of iterates. Therefore, its application to RDEs leads to the calculation of terms that are not needed and more time is consumed in repeated calculations for series solutions. In order to overcome these shortcomings, we propose an easy-to-use piecewise-truncated VIM algorithm for solving the RDEs. Some examples are given to demonstrate the simplicity and efficiency of the proposed method. Comparisons with the classical fourth-order Runge–Kutta method (RK4) verify that the new method is very effective and convenient for solving Riccati differential equations.     

Modified variational iteration method for solving Fisher’s equations

In this paper, we apply the modified variational iteration method (MVIM) for solving Fisher’s equations. The proposed modification is made by introducing He’s polynomials in the correction functional. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that the proposed technique solves nonlinear problems without using Adomian’s polynomials can be considered as a clear advantage of this algorithm over the decomposition method.