variational iteration method

In the present investigation, Variational Iteration Method (VIM) is applied to solve the dynamic oscillation of a current-carrying wire in a magnetic field which is generated by a fixed current-carrying conductor parallel to the wire. Two linear springs are considered to restrict the wire to a rigid wall. In a special case, the periodic solution of the problem is obtained by VIM and compared with numerical solutions for different parameters. Results indicate high accuracy of this method which can be easily extended to solve other non-linear vibration equations, therefore can be applicable in other engineering problems.

H. Jafari proposed a new integral transform recently, namely, the Jafari transform, which covered all classes of integral transforms in the class of Laplace transform, such as Laplace, Sumudu, Elzaki, Aboodh, natural, Shehu transformation, etc. In this paper, we utilize a semi-analytical technique, namely the Jafari variational iteration method, abbreviated JVIM, and we apply this technique to resolve one-dimensional diffusion equations with fractional order type using the Caputo fractional derivative. The results are compared with the homotopy analysis Shehu transform method (HASTM). Also, the results show the suggested algorithm is efficient, accurate and a powerful technique for solving a wide variety of linear and non-linear problems arising in various scientific areas.

The main object of this manuscript is to achieve the solutions of a time-fractional nonlinear system of equations describing the unsteady flow of a polytropic gas using two different approaches based on the combination of new general integral transform in the sense of Caputo fractional derivative and homotopy perturbation method and variational iteration method, respectively. The solutions are obtained in the form of rapidly convergent infinite series with easily computable terms.  Numerical results reveal that the proposed approaches are very effective and simple to obtain approximate and analytical solutions for nonlinear systems of fractional partial differential equations.

To solve fractional-order differential equations (FODEs) with Antagana-Baleanu fractional operator (ABFO), an efficient strategy based on variational iteration method (VIM) and natural transform(NT) is given. Natural variational iteration technique is the name of this method (NVIM). This work also investigates the convergence of the solution of general FODEs obtained by the suggested method given the theory of fixed point and Banach spaces. Furthermore, the error analysis of the NVIM solution is also discussed. Two problems are solved to validate and efficacy demonstrate the of the present. It is also demonstrated that the results obtained from the suggested technique are in excellent agreement with the exact solution. The NVIMs numerical results reveal that the technique is simple to implement and computationally appealing.

The value of an auxiliary parameter incorporated into the well-known variational iteration method (VIM) to obtain solutions of wave equations in unbounded domains is discussed in this article. The suggested method, namely the optimal variational iteration method, is investigated for convergence. Furthermore, the proposed method is tested on one-dimensional and two-dimensional wave equations in unbounded domains in order to better understand the solution mechanism and choose the best auxiliary parameter.Comparisons with results from the standard variational iteration procedure demonstrate that the auxiliary parameter is very useful in tracking the convergence field of the approximate solution.

Solitary waves are interesting phenomena arising in various fields of physics, chemistry, and biology. Nonlinear continuous and discrete models supporting wave solutions of solitary behaviour have received increasing attention in recent years. Some examples of such integrable systems include Korteweg de-Vries (KdV) equation, the nonlinear Schrödinger (NLS) equation, and the Manakov system (MS). In this paper, we propose a discrete nonlocal version of the nonlinear Manakov system which admits spatial and temporal PT-symmetry. PT-symmetry property gives relevance to various fields in physics and has received a lot of attention in the studies of integrable nonlinear equations. In this work, the time evolution of solitary and periodic wave solutions in the proposed system has been numerically investigated. Suitable initial conditions have been considered to construct bright and dark solitons. The variational iteration method (VIM) was used to simulate the solution of the system. The error measurement of the simulation demonstrates the efficiency of the numerical method in constructing the different types of wave solutions.

The main aim of this work is to find the exact solution in the form of the Mittag-Leller function for the nonlinear time-fractional reaction-diffusion-convection equation via a new coupling method namely, the Aboodh variational iteration method (AVIM). The proposed method is a coupling of the Aboodh transform method with the variational iteration method and the fractional derivative defined with the Liouville-Caputo operator. Three different numerical applications are given to demonstrate the validity and applicability of the proposed method and compare it to existing methods. The results shown through figures and tables demonstrate the accuracy of our method. It is concluded here that the proposed method is very efficient, simple and can be applied to other nonlinear problems arising in science and engineering.

In this work, we study the Variational Iteration Method (VIM) accompanied by He's polynomials for obtaining the solution of the temperature distribution in the casting-mould heterogeneous system with fractional order. The combined form of VIM and the Homotopy Perturbation Method (HPM) is called the Variational Iteration Homotopy Perturbation Method (VIHPM) which yields to implement the system of equations directly. The identification of the Lagrange multiplier is essentially more reliable and higher accurate for such type of problems. The approximate solution converges rapidly to the exact solution which confirms the accuracy of this approach. Some graphical representations are demonstrated to show the validity of this approach.

In this work, we employ a combination of variational iteration method (VIM) and Pad´e approximation method, called the VIM-Pad´e technique, to solve some nonlinear initial value problems and a delay differential equation (DDE). Some examples are provided to illustrate the capability and reliability of the technique. The obtained results by using the VIM are compared to the results of this technique. This comparison shows that VIM-Pad´e technique is more effective than VIM and yields faster convergence compared to the VIM.

To obtain the approximate solution to Riccati matrix differential equations, a new variational iteration approach was ‎proposed, which is suggested to improve the accuracy and increase the convergence rate of the approximate solutons to the ‎exact solution. This technique was found to give very accurate results in a few number of iterations. In this paper, the ‎modified approaches were derived to give modified solutions of proposed and used and the convergence analysis to the exact ‎solution of the derived sequence of approximate solutions is also stated and proved. Two examples were also solved, which ‎shows the reliability and applicability of the proposed approach. ‎

In this paper, the optimal control of transmission dynamics of hand, foot and mouth disease (HFMD), formulated by a compartmental deterministic SEIPR (Susceptible-Incubation (Exposed)- Infected - Post infection virus shedding - Recovered) model with vaccination and treatment as control parameters is considered. The objective function is based on the combination of minimizing the number of infected individuals and the cost involved in the interventions of vaccination given to the susceptible population and treatment given to the infected population. The existence for the optimal control pair is proved and the characterization of the optimal control pair is obtained by applying the Pontryagin's maximum principle. The variational iteration method is adopted to solve the non-linear Hamilton equations derived from the Pontryagin's maximum principle theory. These equations constitute a two-point boundary value problem. By considering the correction functionals of the Hamilton equations, the Lagrange multipliers are easily identified and practical iteration formulas are derived. An algorithm is developed, based on this formulas, to determine iteratively the solutions of the Hamilton equations with a desired accuracy. With the aid of solutions obtained, the optimal control law can be easily deduced. The results were analyzed and interpreted graphically using Maple.

In this paper, we study numerical method for hybrid fuzzy differential equations (HFDEs) by an application of the variational iteration method (VIM). We state a convergence result and give numerical examples to illustrate the theory. Numerical results are presented for some problems to demonstrate the usefulness and accuracy of this approach. The method is easy to apply and produces very accurate numerical results.

‎In this work‎, ‎we consider the parabolic equation‎: ‎$u_t-u_{xx}=0$‎. ‎The purpose of this paper is to introduce the method of‎ ‎variational iteration method and radial basis functions for‎ ‎solving this equation‎. ‎Also, the method is implemented to three‎ ‎numerical examples‎. ‎The results reveal‎ ‎that the technique is very effective and simple.

In this paper, we have proposed a new iterative method for finding the solution of ordinary differential equations of the first order. In this method we have extended the idea of variational iteration method by changing the general Lagrange multiplier which is defined in the context of the variational iteration method.This causes the convergent rate of the method increased compared with the variational iteration method. To prevent consuming large amount of the CPU time and computer memory and to control requires significant amounts of computations, the Taylor expansion of the iterative functions in each iteration are applied. Finally to extend the convergence region of the truncated series, also the Pade approximants are used. Error analysis and convergence of the method are studied. Some examples are given to illustrate the performance and efficiency of the proposed method. For comparison, the results obtained by the our method and the variational iteration method are ‎presented.

In this paper, the variational iteration method for solving nth-order fuzzy integro differential equations (nth-FIDE) is proposed. In fact the problem is changed to the system of ordinary fuzzy integro-differential equations and then fuzzy solution of nth-FIDE is obtained. Some examples show the efficiency of the proposed method.

In this paper a modification of He''s variational iteration method (VIM) has been employed to solve Dung and Riccati equations. Sometimes, it is not easy or even impossible, to obtain the first few iterations of VIM, therefore, we suggest to approximate the integrand by using suitable expansions such as Taylor or Chebyshev expansions.

We introduce and discuss the Homotopy perturbation method, the Adomian decomposition method and the variational iteration method for solving the stefan problem with kinetics. Then, we give an example of the stefan problem with  kinetics and solve it by these methods.

This paper presents a comparison between variational iteration method (VIM) and modfied variational iteration method (MVIM) for approximate solution a system of Volterra integral equation of the first kind. We convert a system of Volterra integral equations to a system of Volterra integro-di®erential equations that use VIM and MVIM to approximate solution of this system and hence obtain an approximation for system of Volterra integral equations. Some examples are given to show the pertinent features of this methods.

In this paper, He''s highly prolic variational iteration method is applied ef- fectively for showing the existence, uniqueness and solving a class of singular second order two point boundary value problems. The process of nding solu- tion involves generation of a sequence of appropriate and approximate iterative solution function equally likely to converge to the exact solution of the given problem which being processed out and improvised on its own at every step re- cursively. Moreover, Illustrative examples available to the context in literature when treated with, by application of such proposed method fetch encouraging results so as to justify and reveal its eciency and usefulness of the method.