Variational Iteration algorithm-II Research Articles

Modified Variational Iteration Algorithm-II: Convergence and Applications to Diffusion Models

Variational iteration method has been extensively employed to deal with linear and nonlinear differential equations of integer and fractional order. The key property of the technique is its ability and flexibility to investigate linear and nonlinear models conveniently and accurately. The current study presents an improved algorithm to the variational iteration algorithm-II (VIA-II) for the numerical treatment of diffusion as well as convection-diffusion equations. This newly introduced modification is termed as the modified variational iteration algorithm-II (MVIA-II). The convergence of the MVIA-II is studied in the case of solving nonlinear equations. The main advantage of the MVIA-II improvement is an auxiliary parameter which makes sure a fast convergence of the standard VIA-II iteration algorithm. In order to verify the stability, accuracy, and computational speed of the method, the obtained solutions are compared numerically and graphically with the exact ones as well as with the results obtained by the previously proposed compact finite difference method and second kind Chebyshev wavelets. The comparison revealed that the modified version yields accurate results, converges rapidly, and offers better robustness in comparison with other methods used in the literature. Moreover, the basic idea depicted in this study is relied upon the possibility of the MVIA-II being utilized to handle nonlinear differential equations that arise in different fields of physical and biological sciences. A strong motivation for such applications is the fact that any discretization, transformation, or any assumptions are not required for this proposed algorithm in finding appropriate numerical solutions.

An efficient approach for the numerical solution of fifth-order KdV equations

Abstract The main aim of this article is to use a new and simple algorithm namely the modified variational iteration algorithm-II (MVIA-II) to obtain numerical solutions of different types of fifth-order Korteweg-de Vries (KdV) equations. In order to assess the precision, stability and accuracy of the solutions, five test problems are offered for different types of fifth-order KdV equations. Numerical results are compared with the Adomian decomposition method, Laplace decomposition method, modified Adomian decomposition method and the homotopy perturbation transform method, which reveals that the MVIA-II exceptionally productive, computationally attractive and has more accuracy than the others.

Study on numerical solution of dispersive water wave phenomena by using a reliable modification of variational iteration algorithm

In this paper, we propose an incredible modification of variational iteration algorithm-II that will accelerate the fast convergence of the series solution. We utilize the proposed algorithm on KdV, mKdV and combined KdV–mKdV equations to illustrate its accuracy and reliability. The consequences of modified variational iteration algorithm-II (MVIA-II) show that the proposed technique minimizes the size of calculations, converges rapidly, computationally efficient and yields accurate results. This modification demonstrated the high efficiency and attained very good agreement in illustrated models and might be promptly reached out to other nonlinear PDEs.

A general numerical algorithm for nonlinear differential equations by the variational iteration method

Purpose The purpose of this paper is to suggest a general numerical algorithm for nonlinear problems by the variational iteration method (VIM). Design/methodology/approach Firstly, the Laplace transform technique is used to reconstruct the variational iteration algorithm-II. Secondly, its convergence is strictly proved. Thirdly, the numerical steps for the algorithm is given. Finally, some examples are given to show the solution process and the effectiveness of the method. Findings No variational theory is needed to construct the numerical algorithm, and the incorporation of the Laplace method into the VIM makes the solution process much simpler. Originality/value A universal iteration formulation is suggested for nonlinear problems. The VIM cleans up the numerical road to differential equations.

Numerical solution of Korteweg–de Vries-Burgers equation by the modified variational iteration algorithm-II arising in shallow water waves

In this paper, an effective modification of variational iteration algorithm-II is presented for the numerical solution of the Korteweg–de Vries-Burgers equation, Burgers equation and Kortewege–de Vries equation. In this modification, an auxiliary parameter is introduced which make sure the convergence of the standard algorithm-II. In order to assess the precision of the solutions, numerical computations obtained from the time evaluation of the solutions of the Kortewege–de Vries-Burgers equation with different values for dispersion and diffusion coefficients, show that the proposed algorithm converges rapidly, yields accurate results and offers better accuracy and robustness in comparison with other previous numerical methods. Furthermore, the method can be readily implemented for illuminating viably an enormous number of nonlinear differential equations with better accuracy. Furthermore, any transformation, weak nonlinearity assumption to find an explicit solution or any discretization to find the numerical solution is not necessary for this proposed algorithm.

Analytic approximate solutions for some nonlinear Parabolic dynamical wave equations

In this paper, modified variational iteration algorithm-II is investigated for finding approximate solutions of nonlinear Parabolic equations. Comparisons of the MVIA-II with trigonometric B-spline collocation method, variational iteration method, homotopy perturbation transform method, Adomian decomposition method, and modified variational iteration method are carried out, which show that the proposed algorithm (MVIA-II) is robust one. Some nonlinear Parabolic equations are given to demonstrate the implementation and accuracy of the MVIA-II.

Numerical solution of coupled Schrödinger–KdV equation via modified variational iteration algorithm-II

This paper applied the modified variational iteration algorithm-II to obtain the numerical solution of coupled Schrodinger–KdV equation. For this purpose, we use the Jacobi periodic solution and Soliton solution as initial guess for a better approximation. Numerical results and comparison exact and approximate solution have demonstrated that modified variational iteration algorithm-II is very effective and convenient for nonlinear equations, especially for the coupled Schrodinger–KdV equation.

Convergence of variational iteration method applied to two-point diffusion problems

A new approach is introduced to construct the variational iteration algorithm-II for diffusion problems. Based on this algorithm, the sufficient conditions for convergence of variational iteration method applied to this type of two-point boundary value problems are established. An error analysis is also presented. Several examples are given to illustrate the convergence concept and error estimate.

Higher order explicit solutions for nonlinear dynamic model of column buckling using variational approach and variational iteration algorithm-II

This paper deals with the determination of approximate solutions for a model of column buckling using two efficient and powerful methods called He’s variational approach and variational iteration algorithm-II. These methods are used to find analytical approximate solution of nonlinear dynamic equation of a model for the column buckling. First and second order approximate solutions of the equation of the system are achieved. To validate the solutions, the analytical results have been compared with those resulted from Runge-Kutta 4th order method. A good agreement of the approximate frequencies and periodic solutions with the numerical results and the exact solution shows that the present methods can be easily extended to other nonlinear oscillation problems in engineering. The accuracy and convenience of the proposed methods are also revealed in comparisons with the other solution techniques.

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.

Variational iteration algorithm-II for solving the system of third order non-linear integro-differential equations

Variational iteration method is widely used for solving nonlinear, integro-differential equations. Through careful investigation of the iteration formulas of the earlier variational iteration algorithm (VIM), we find unnecessary repeated calculations in each iteration. To overcome this shortcoming, the variational iteration algorithm-II (He et al., 2010) will be used in this paper to solve the system of non linear integro-differential equations. Beside this, the comparison of the exact solution with approximated solution by VIM-II is illustrated by the graphs. Several examples are given to verify the reliability and efficiency of the method. Key words: Variational iteration algorithm-II (VIM), system of nonlinear integro-differential equations.

Variational iteration algorithm-II for solving linear and non-linear ODEs

Variational iteration method is widely used for solving linear and non-linear ODE. In this paper, the variational iteration algorithm-II is constructed and some examples are given to show its simple solution procedure and high accuracy of the obtained results. Key words: Variational iteration method, series solution, ordinary differential equations.

An Aproximation to Solution of Space and Time Fractional Telegraph Equations by the Variational Iteration Method

Sevimlican suggested an effective algorithm for space and time fractional telegraph equations by the variational iteration method. This paper shows that algorithm can be updated by either variational iteration algorithm‐II or the fractional variational iteration method.

Second-order two-point boundary value problems using the variational iteration algorithm-II

In this paper, we introduce an improved variational iteration method (VIM) for nonlinear second-order boundary value problems. The main advantage of this modification is that it can avoid additional computation in determining the unknown parameters in initial approximation when solving boundary value problems using the conventional VIM. Also, iterative sequences obtained using the improved VIM do satisfy the boundary conditions while iterative sequences obtained using conventional VIM may not, in general, satisfy the boundary conditions. Numerical results reveal that the improved method is accurate and efficient.

An Alternative Approach to Differential-Difference Equations Using the Variational Iteration Method

Although a variational iteration algorithm was proposed by Yildirim (Math. Prob. Eng. 2008 (2008), Article ID 869614) that successfully solves differential-difference equations, the method involves some repeated and unnecessary iterations in each step. An alternative iteration algorithm (variational iteration algorithm-II) is constructed in this paper that overcomes this shortcoming and promises to provide a universal mathematical tool for many differential-difference equations.

Study of the effects of the Reynolds number on circular porous slider via variational iteration algorithm-II

In this paper the problem of the porous slider where the fluid is injected through the porous bottom is considered. The similarity transformation reduces the governing equations into coupled nonlinear ordinary differential equations. The resulting equations are solved by using He’s variational iteration algorithm-II. The resulting series solution contains the well known Reynolds number. The influence of the Reynolds number on the velocity field has been discussed graphically.

Analytical approach to two-dimensional viscous flow with a shrinking sheet via variational iteration algorithm-II

The purpose of this paper is to employ an analytical approach to a two-dimensional viscous flow with a shrinking sheet. A comparative study of the variational iteration algorithm-II (VIM-II) and the Adomian decomposition method (ADM) are discussed. Both approaches have been applied to obtain the solution of a two-dimensional viscous flow due to a shrinking sheet. This study outlines the significant features of the two methods. Comparison is made with the ADM to highlight the significant features of the VIM-II and its capability of handling completely integrable equations. Through careful investigation of the iteration formulas of the earlier variational iteration algorithm (VIM), we find unnecessary repeated calculations in each iteration. To overcome this shortcoming, we suggest the VIM-II, which has advantages over other iteration formulas, such as the VIM, and the ADM. Further iterations can produce more accurate results and decrease the error.