A new modification of variational iteration method for solving reaction–diffusion system with fast reversible reaction

This paper presents a new modification of He’s variational iteration method using Adomian’s polynomials (VIMAP) to solve reaction–diffusion system with fast reversible reaction. An auxiliary parameter is introduced into the VIMAP and optimally identified to adjust the convergence region of the approximate solution. The results reveal that the VIMAP is very accurate comparing with those obtained by the VIM but is not valid for large solution domain, while the new modification have a remarkable accuracy for large domains.     

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.     

Modified variational iteration method for solving fourth-order boundary value problems

In this paper, we apply the modified variational iteration method (MVIM) for solving the fourth-order boundary value problems. 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.     

An effective modification of He’s variational iteration method

It is well known that one of the advantages of He’s variational iteration method is the free choice of initial approximation. Therefore, in this paper, we use this advantage to propose a reliable modification of He’s variational iteration method. Indeed, this constructs an initial trial-function without unknown parameters, which is called the modified variational iteration method. Some of the nonlinear and linear equations are examined by the modified method to illustrate the effectiveness and convenience of this method, and in all cases, the modified technique performed excellently. The results reveal that the proposed method is very effective and simple and gives exact solutions. The modification could lead to a promising approach for many applications in applied sciences.     

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.     

Exact and numerical solutions for non-linear Burger’s equation by VIM

In this article variational iteration method (VIM), established by He in (1999), is considered to solve nonlinear Bergur’s equation. This method is a powerful tool for solving a large number of problems. Using variational iteration method, it is possible to find the exact solution or a closed approximate solution of a problem. Comparing the results with those of Adomian’s decomposition and finite difference methods reveals significant points. To illustrate the ability and reliability of the method, some examples are provided.     

An improved variational iteration method for solving coupled KdV and Boussinesq-like B(m, n) equations

In this article, we implement a new analytical technique; He’s variational iteration method for solving the coupled KdV and Boussinesq-like equations. In this method, first general Lagrange multipliers are introduced to construct correction functional for the problems. The multipliers in the functional can be identified optimally via the variational theory. Next the components of obtained iteration formulae defined by partial sum of other sequence, specially constructed according to Adomian’s decomposition method (ADM). Also according to ADM we used a partial sum of Adomian polynomials instead of nonlinear terms in iteration formulae. The initial approximations can be freely chosen with possible unknown constants, which can be determined by imposing the initial conditions. The results reveal that the proposed method is very effective and can be applied for other nonlinear problems.     

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.     

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.     

Modified Variational Iteration Method for Heat and Wave-Like Equations

In this paper, we apply the modified variational iteration method (MVIM) for solving the heat and wave-like 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 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.     

Efficient computational algorithms for solving one class of fractional boundary value problems

In this paper, we introduce a modification of He’s variational iteration, homotopy analysis and optimal homotopy analysis methods for solving fractional boundary value problems. It is illustrated that the proposed methods are powerful fast numerical tools to find accurate solutions. It is illustrated that efficiency of these methods is based on proper choosing of initial guess.     

Variational iteration method for solving twelfth-order boundary-value problems using He’s polynomials

In this paper, we apply the variational iteration method using He’s polynomials (VIMHP) for solving the twelfth-order boundary-value problems. The proposed method is an elegant combination of variational iteration and the homotopy perturbation methods. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The suggested 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.     

The application of the modified variational iteration method on the generalized Fisher’s equation

In a recent paper, Abassy et al. (J. Comput. Appl. Math. 207:137–147, 2007) proposed a modified variational iteration method (MVIM) for a special kind of nonlinear differential equations. In this paper, we consider variational iteration method (VIM) and MVIM (proposed in Abassy et al., J. Comput. Appl. Math. 207:137–147, 2007) to obtain an approximate series solution to the generalized Fisher’s equation which converges to the exact solution in the region of convergence. It is also shown that the application of VIM to the generalized Fisher’s equation leads to calculation of unneeded terms for series solution. Therefore, we use MVIM to overcome this disadvantage. Comparison of error between VIM and MVIM is made. The results show that the MVIM is more effective than the VIM.     

Application of variational iteration method for modified Camassa‐Holm and Degasperis‐Procesi equations

In this article, the variational iteration method (VIM) is used to obtain approximate analytical solutions of the modified Camassa‐Holm and Degasperis‐Procesi equations. The method is capable of reducing the size of calculation and easily overcomes the difficulty of the perturbation technique or Adomian polynomials. The results reveal that the VIM is very effective. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Solution for an anti-symmetric quadratic nonlinear oscillator by a modified He’s homotopy perturbation method

In this paper He’s homotopy perturbation method has been adapted to calculate higher-order approximate periodic solutions for a nonlinear oscillator with discontinuity for which the elastic force term is an anti-symmetric and quadratic term. We find that 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. Just one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate period of less than 0.73% for all values of oscillation amplitude, while this relative error is as low as 0.040% when the second iteration is considered. Comparison of the result obtained using this method with those obtained by the harmonic balance method reveals that the former is very effective and convenient.     

A new modification of He’s homotopy perturbation method for rapid convergence of nonlinear undamped oscillators

In this paper we present a new efficient modification of the homotopy perturbation method with x ³ force nonlinear undamped oscillators for the first time that will accurate and facilitate the calculations. The He’s homotopy perturbation method is modified by adding a term to linear operator depends on the equation and boundary conditions. We find that this modified homotopy perturbation method works very well for the wide range of time and boundary conditions for nonlinear oscillator. Only two or three iteration leads to high accuracy of the solutions. We then conduct a comparative study between the new modification and the homotopy perturbation method for strongly nonlinear oscillators. Numerical illustrations are investigated to show the accurate of the techniques. The new modified method accelerates the rapid convergence of the solution, reduces the error solution and increases the validity range. The new modification introduces a promising tool for many nonlinear problems.     

On using the modified variational iteration method for solving the nonlinear coupled equations in the mathematical physics

This paper applied the modified variational iteration method to the nonlinear coupled partial differential equations via the generalized nonlinear Hirota Satsuma coupled KdV equations, the nonlinear coupled Kortewge–de Vries KdV equations and the nonlinear shallow water equations together with the initial conditions. The proposed modification is made by introducing Adomian’s polynomials in the correct functional. The suggested algorithm is quite efficient and is practically well suited for use in such problems. The proposed iterative scheme finds the solution without any discritization, liberalization, perturbation, or restrictive assumptions.     

Notes on the optimal variational iteration method

Here, various variational iteration algorithms are compared. An auxiliary parameter can be introduced in the iteration procedure, and can be identified optimally, which results in Turkyilmazoglu’s optimal variational iteration method. Some unknown auxiliary parameters can be also included in the initial solution, and can be optimally determined, that is Heri?anu and Marinca’s optimal variational iteration method.     

Combination of the variational iteration method and numerical algorithms for nonlinear problems

A very simple and efficient local variational iteration method (LVIM), or variational iteration method with local property, for solving problems of nonlinear science is proposed in this paper. The analytical iteration formula of this method is derived first using a general form of first order nonlinear differential equations, followed by straightforward discretization using Chebyshev polynomials and collocation method. The resulting numerical algorithm is very concise and easy to use, only involving highly sparse matrix operations of addition and multiplication, and no inversion of the Jacobian in nonlinear problems. Apart from the simple yet efficient iteration formula, another extraordinary feature of LVIM is that in each local domain, all the collocation nodes participate in the calculation simultaneously, thus each local domain can be regarded as one “node” in calculation through GPU acceleration and parallel processing. For illustration, the proposed algorithm of LVIM is applied to various nonlinear problems including Blasius equations in fluid mechanics, buckled bar equations in solid mechanics, the Chandrasekhar equation in astrophysics, the low-Earth-orbit equation in orbital mechanics, etc. Using the built-in highly optimized ode45 function of MATLAB as a comparison, it is found that the LVIM is not only very accurate, but also much faster by an order of magnitude than ode45 in all the numerical examples, especially when the nonlinear terms are very complicated and difficult to evaluate.     

Construction of exact periodic wave and solitary wave solutions for the long–short wave resonance equations by VIM

In this paper, He’s variational iteration method is employed to construct periodic wave and solitary wave solutions for the long–short wave resonance equations. The chosen initial solution can be in soliton form with some unknown parameters, which can be determined in the solution procedure. Some examples are given. The results reveal that the method is very effective and convenient.