Standard Variational Iteration Method Research Articles

Modified Variational Iteration Method Approach for Fuzzy Initial Value Problem

In this paper, an application of  modified variational iteration method (MVIM) for solving homogeneous and nonhomogeneous ordinary differential equations with fuzzy initial value conditions (FDEs) are examined. This modified form is about a procedure some changes in the general formula of the variational iteration method (VIM) . MVIM showed a qualitative and excellent improvement over the standard VIM. To illustrate how this modified form is works we picked some numerical examples and solved them. The efficacy and validity of MVIM have been confirmed through results obtained.  

Novel approximate solution for fractional differential equations by the optimal variational iteration method

In this work, the optimal variational iteration method (OVIM) is used to solve partial and ordinary fractional differential equations. The standard variational iteration method (VIM) is reassessed by introducing a parameter that accelerates convergence. The value of the convergence acceleration coefficient is determined by calculating the residual of the parameter within the L2 norm. The numerical results proved that the proposed method gives much better and more accurate results than the standard VIM method, as the effect of adding the parameter to the VIM method and its calculation method was clear in improving the method and increasing the convergence of the solution, where we solved several examples of ordinary and partial fractional differential equations. In addition, the fractional Kawahara and foam drainage equations have also been solved. All results were obtained using the Mathematica®12 program. The results were also compared with the optimal Adomian decomposition method (ADM), and the proposed method showed higher efficiency and better accuracy.

Modified Variational Iteration Method for Solving Nonlinear Partial Differential Equation Using Adomian Polynomials

The aim of this paper is to solve numerically the Cauchy problems of nonlinear partial differential equation (PDE) in a modified variational iteration approach. The standard variational iteration method (VIM) is first studied before modifying it using the standard Adomian polynomials in decomposing the nonlinear terms of the PDE to attain the new iterative scheme modified variational iteration method (MVIM). The VIM was used to iteratively determine the nonlinear parabolic partial differential equation to obtain some results. Also, the modified VIM was used to solve the nonlinear PDEs with the aid of Maple 18 software. The results show that the new scheme MVIM encourages rapid convergence for the problem under consideration. From the results, it is observed that for the values the MVIM converges faster to exact result than the VIM though both of them attained a maximum error of order 10^-9. The resulting numerical evidences were competing with the standard VIM as to the convergence, accuracy and effectiveness. The results obtained show that the modified VIM is a better approximant of the above nonlinear equation than the traditional VIM. On the basis of the analysis and computation we strongly advocate that the modified with finite Adomian polynomials as decomposer of nonlinear terms in partial differential equations and any other mathematical equation be encouraged as a numerical method.

New Perspective on the Conventional Solutions of the Nonlinear Time-Fractional Partial Differential Equations

The role of integer and noninteger order partial differential equations (PDE) is essential in applied sciences and engineering. Exact solutions of these equations are sometimes difficult to find. Therefore, it takes time to develop some numerical techniques to find accurate numerical solutions of these types of differential equations. This work aims to present a novel approach termed as fractional iteration algorithm-I for finding the numerical solution of nonlinear noninteger order partial differential equations. The proposed approach is developed and tested on nonlinear fractional-order Fornberg–Whitham equation and employed without using any transformation, Adomian polynomials, small perturbation, discretization, or linearization. The fractional derivatives are taken in the Caputo sense. To assess the efficiency and precision of the suggested method, the tabulated numerical results are compared with the standard variational iteration method and the exact solution as well. In addition, numerical results for different cases of the fractional-order α are presented graphically, which show the effectiveness of the proposed procedure and revealed that the proposed scheme is very effective, suitable for fractional PDEs, and may be viewed as a generalization of the existing methods for solving integer and noninteger order differential equations.

A computational method based on the modification of the variational iteration method for determining the solution of the optimal control problems

AbstractThis article presents a modification of the variational iteration method (VIM) for solving Hamilton‐Jacobi‐Bellman equations of linear and nonlinear optimal control problems. In this method, the Lagrange multiplier is chosen in such a way that a sequence of value functions that are produced by this presented method (PM) converges to exact solution faster than the standard VIM. This fast convergence is due to choosing an exponentially decaying Lagrange multiplier for the first time. Finally, some examples are presented to demonstrate the effectiveness of the PM for finding the solution of the optimal control problems.

An efficient iterative approach for three-dimensional modified anomalous fractional sub-diffusion equations on a large domain

The fractional sub-diffusion equation, which is obtained by replacing the time derivative in ordinary diffusion by a fractional derivative of order ϑ with 0 < vartheta < 1, has appeared in numerous complex system. In this paper, we suggest an efficient and accurate iterative method based on coupling the variational iteration method (VIM) with an auxiliary parameter for solving three-dimensional FDEs described in the Riemann–Liouville sense. However, though the standard VIM is often invalid on large domains, the VIM with an auxiliary parameter is highly efficient in approximating the solution of complex systems even on large domains. The procedure of obtaining an optimal auxiliary parameter is illustrated through some examples, while the theoretical analysis confirms the convergence of the proposed method. Comparing the results of standard VIM and modified VIM by the auxiliary parameter confirms the effectiveness of using the new technique on the magnitude of the convergence region.

A New Variational Iteration Method for a Class of Fractional Convection-Diffusion Equations in Large Domains

In this paper, we introduced a new generalization method to solve fractional convection–diffusion equations based on the well-known variational iteration method (VIM) improved by an auxiliary parameter. The suggested method was highly effective in controlling the convergence region of the approximate solution. By solving some fractional convection–diffusion equations with a propounded method and comparing it with standard VIM, it was concluded that complete reliability, efficiency, and accuracy of this method are guaranteed. Additionally, we studied and investigated the convergence of the proposed method, namely the VIM with an auxiliary parameter. We also offered the optimal choice of the auxiliary parameter in the proposed method. It was noticed that the approach could be applied to other models of physics.

A numerical approach for a class of astrophysics equations using piecewise spectral-variational iteration method

PurposeThe main purpose of this paper is to implement the piecewise spectral-variational iteration method (PSVIM) to solve the nonlinear Lane-Emden equations arising in mathematical physics and astrophysics.Design/methodology/approachThis method is based on a combination of Chebyshev interpolation and standard variational iteration method (VIM) and matching it to a sequence of subintervals. Unlike the spectral method and the VIM, the proposed PSVIM does not require the solution of any linear or nonlinear system of equations and analytical integration.FindingsSome well-known classes of Lane-Emden type equations are solved as examples to demonstrate the accuracy and easy implementation of this technique.Originality/valueIn this paper, a new and efficient technique is proposed to solve the nonlinear Lane-Emden equations. The proposed method overcomes the difficulties arising in calculating complicated and time-consuming integrals and terms that are not needed in the standard VIM.

Optimal variational iteration method for nonlinear problems

This paper focuses on the study of boundary value problems using well-known He’s variational iteration method which is coupled with an auxiliary parameter. Three examples are given to show the efficiency and importance of the proposed algorithm. The reliability and accuracy has been proved by comparing our results with the solution obtained by standard variational iteration method.

Solving differential-algebraic equations through variational iteration method with an auxiliary parameter

In this study, we consider the importance of an auxiliary parameter, which is introduced into the well-known variational iteration method to obtain solutions for differential-algebraic equations. We study the convergence of the proposed method, which is called the variational iteration method with an auxiliary parameter. In addition, the proposed method is applied to two differential-algebraic equations to elucidate the solution procedure and to select the optimal auxiliary parameter. Comparisons with the results obtained by the standard variational iteration method demonstrate that the auxiliary parameter is highly effective in controlling the convergence region of the approximate solution.

Variational iteration method with an auxiliary parameter for solving wave-like and heat-like equations in large domains

The present study is an attempt to obtain solutions of wave-like and heat-like equations with high-dimensional in large domains through the well-known variational iteration method (VIM) which is coupled with an auxiliary parameter. The convergence of the proposed method, namely variational iteration method with an auxiliary parameter, is under investigation. In addition, four examples are provided to elucidate the solution procedure and to optimally choose the auxiliary parameter. Comparing the results of the standard variational iteration method, the auxiliary parameter proves very effective in controlling the convergence region of approximate solution.

Controlled Variational Iteration Method for Bratu Equation Arising in Electro-Spun Organic Nanofibers Elaboration

In this paper, an effective formulation of the variational iteration method is suggested for solving Bratu equation arising in electro-spinning. The suggested formulation depends on embedding a nonzero auxiliary parameter that controls the solution convergence region. An alternative formulation of the Bratu equation is suggested as well. The proposed formula eliminates the complexity that appears when solving using the standard variational iteration algorithms illustrated in [1,2] without approximating the exponential term. A suitable choice of the auxiliary parameter results in an accurate approximation compared with the approximation of the standard variational iteration method.

Tri-prong scheme for regularized long wave equation

This paper witnesses the application of a tri-prong scheme comprising the well-known Variational Iteration (VIM), Adomian’s polynomials and an auxiliary parameter to obtain solutions of regularized long wave (RLW) equation in large domain. Computational work elucidates the solution procedure appropriately and comparison with results by the standard variational iteration method shows that the auxiliary parameter proves very effective to control the convergence region of approximate solutions.

Optimal Variational Asymptotic Method for Nonlinear Fractional Partial Differential Equations.

We propose optimal variational asymptotic method to solve time fractional nonlinear partial differential equations. In the proposed method, an arbitrary number of auxiliary parameters γ 0, γ 1, γ 2,… and auxiliary functions H 0(x), H 1(x), H 2(x),… are introduced in the correction functional of the standard variational iteration method. The optimal values of these parameters are obtained by minimizing the square residual error. To test the method, we apply it to solve two important classes of nonlinear partial differential equations: (1) the fractional advection-diffusion equation with nonlinear source term and (2) the fractional Swift-Hohenberg equation. Only few iterations are required to achieve fairly accurate solutions of both the first and second problems.

Variational iteration method for Bratu-like equation arising in electrospinning

This paper points out that the so called enhanced variational iteration method (Colantoni & Boubaker, 2014) for a nonlinear equation arising in electrospinning and vibration-electrospinning process is the standard variational iteration method. An effective algorithm using the variational iteration algorithm-II is suggested for Bratu-like equation arising in electrospinning. A suitable choice of initial guess results in a relatively accurate solution by one or few iteration.

Modified variational iteration method (non-homogeneous initial value problem)

A new application of modified variational iteration method (MVIM) for non-linear non-homogeneous differential equations is introduced. The modified form introduces a change in the formulation of variational iteration relation and provides a qualitative improvement over standard variational iteration method (VIM). This paper is an extension of the previous work in Abassy (2010) [33], Abassy et al. (2007) [31,29,32,30]. Some illustrative numerical examples and Mathematica program codes are introduced.

A piecewise variational iteration method for treating a nonlinear oscillator of a mass attached to a stretched elastic wire

In this paper, we introduce a piecewise variational iteration method for treating a nonlinear oscillator of a mass attached to a stretched elastic wire. For the nonlinear oscillator, the present method can produce a good approximation to the exact solution in a very large region, while the standard variational iteration method (VIM) gives a good approximation only in a small region. The numerical results obtained show that the present method does not share the drawback of the standard VIM and can provide very accurate analytical approximate solutions for both small and large values of the oscillation amplitude and parameter.

Numerical solutions of Duffing equations involving both integral and non-integral forcing terms

In this paper, an improved variational iteration method is presented for solving Duffing equations involving both integral and non-integral forcing terms. The main advantage of this modification over the standard variational iteration method (VIM) is that it can avoid unnecessary repeated computation in determining the unknown parameters in the initial solution. Numerical results reveal that the improved method is simple and efficient.

The multistage variational iteration method for a class of nonlinear system of ODEs

This paper implements the multistage variational iteration method (MVIM) to solve a class of nonlinear system of first-order ordinary differential equations (ODEs). The domain of validity of the solutions via the standard variational iteration method (VIM) is extended by the simple multistage strategy. Comparisons with the exact solution and the fourth-order Runge–Kutta (RK4) method show that the MVIM is a reliable method for nonlinear equations.