An improved piecewise variational iteration method for. Application of the general variational iteration method to a. In this paper, we extend variational iteration method vim to find approximate solutions of linear and nonlinear thirteenth order differential equations in boundary value problems. Modified variational iteration method for the solution of a class of differential equations m. In this paper, we apply the modified variational iteration method mvim for solving the heat and wavelike equations.
The variational iteration method, proposed by he, was used to find the approximate solutions for the linear partial differential equations 116, 117. Subsequentworksreectthe exibility, consistency, and eectiveness of. Revised variational iteration method for solving systems. Two numerical examples are presented for the numerical illustration of the method and their results are compared with. The variational iteration method which obtains the analytical or numerical solutions of a wide spectrum of differential equations, as well as integral equations, was proposed in the late 90s by. This paper employs the variational iteration method to obtain analytical solutions of secondorder delay differential equations. Variational iteration method and sumudu transform for.
It provides a sequence of iterated solutions which is converge to the exact solution of the problem. In this paper, we propose a new variational inference method for deep exponentialfamily def models. The variational iteration method vim developed in 1999 by he in,,,,, will be used to study the linear wave equation, nonlinear wave equation, and wavelike equation in bounded and unbounded domains. A simple local variational iteration method and related. A novel modification of the variational iteration method vim is proposed by means of the laplace transform. The value of the multiplier is chosen using variational theory so thateach iteration. The purpose in this paper is to study the strong convergence of general iterative scheme to find a common element of the set of a finite family of nonexpansive mappings, the set of solutions of variational inequalities for a relaxed cocoercive mapping and the set of solutions of split mixed equilibrium problem. Variational iteration method for solving initial and boundary. In the same manner the rest of components can be obtained using the iteration formula 15 and by matlab software. Pdf variational iteration method for solving nonlinear.
The variational iteration method is a new method for solving linear and nonlinear problems and was introduced by a chinese mathematician, he. Pdf variational iteration method for solving some models of. Variational iteration method for special nonlinear partial. Keywords delay differentialalgebraic equations, variational iteration method, convergence 1. For variational iteration method, the key is the identification of the lagrangian multiplier. Revised variational iteration method for solving system of ordinary differential. The new approach is based on variational iteration theory and sumudu transform. Introduction the variational iteration method vim was first proposed by he 1, 2, and has been extensively discussed by many authors 310. Variational iteration method an overview sciencedirect topics. Research article revised variational iteration method for. Application of this method to the helmholtz equation is investigated in momani et al.
Pdf the variational iteration method for solving fuzzy. This however, relies on the merging procedure used within proposed method being allowed to store enough components to describe the smoothed distri. The basis for this method is the variational principle. W abstractthe variational iteration method vim has been shown to solve effectively, easily and accurately a large class of. Variational iteration method for solving coupled schrodinger. Special nonlinear partial differential equations, variational iteration method, lagrange multiplier 1 introduction in 1999, the variational iteration method vim was first proposed by jihuan he 1,2. It is shown in this paper that the application of vim to a special kind of nonlinear. The variational iteration method for analytic treatment of. Variational iteration method for solving volterra and. Modified variational iteration method, lagrange multiplier, taylors series, bratutype differential equation. The variational iteration method 811,15 has been extensively worked out over a.
The derivation of local variational iteration method lvim involves two stages. Variational iteration method for solving nonlinear wbk. Tahmina akter et al variational iteration method for solving coupled schrodingerkleingordon equation errors is avoided. In this paper, we use the variational iteration method vim to find the approximate analytical solution for an initial value problem involving the fuzzy duffing ordinary differential equation. Variational iteration methodsome recent results and new. The algorithm repeats these two steps until convergence, i. Applications of this method have been enlarged due to its flexibility, convenience and efficiency. This method is more efficient and easy to handle such nonlinear partial differential equations. Variational iteration method and sumudu transform for solving. Research article revised variational iteration method for solving systems of nonlinear fractionalorder differential equations c.
The variational iteration method vim 1821, which is a wellestablished technique with wide applications for ordinary differential equations, partial differential equations and delay differential equations, etc. Reduced basis for variational inequalities application to. Projectionbased model reduction for contact problems. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The variational iteration method vim was proposed by he in 1997 1921. The famous peronamalik pm equation which was at first introduced for image restoration has been solved via various numerical methods. Variational iteration method for fractional calculus a. A numerical implementation of the variational iteration. Baleanu 4 department of mathematics, faculty of science, istanbul university, vezneciler, istanbul, turkey. Deriving the iteration formula, proving its convergence to the exact solution and studying the approximate solution of volterra and fredholm integrodifferential equation by using the variational iteration method vim are given in 1.
Variational iteration method for initial and boundary value problems. The variational iteration method is used for solving autonomous ordinary differential system in he 2000. Oct 18, 2011 the purpose of this paper is to provide a comparison of the adomian decomposition method adm with the variational iteration method vim for solving the lane. In quantum mechanics, the variational method is one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states. Application of hes variational iteration method for solution. Variational iteration method for solving telegraph equations. Pdf n this paper, the variational iteration method is implemented for solving nonlinear initial value problems. A reduced basis method for parametrized variational inequalities. Convergence of variational iteration method for secondorder. For linear problems, their exact solutions can be obtained by only one iteration step due to the fact that the lagrangian multiplier can be identified exactly. Inertial iterative schemes with variable step sizes for.
Variational iteration method for solving nonlinear boundary. Fractional variational iteration method for fractional. Variational iteration method a kind of nonlinear analytical technique. Variational iteration method and homotopy perturbation method are introduced to solve the kawahara equation. In 4 he modified the general lagrange multiplier method 5 and constructed an iterative. Pdf the variational iteration method for solving linear and.
Variational iteration method for solving nonlinear wbk equations m. The variational iteration method for solving linear and nonlinear. Comparison of the obtained results with the numerical solution shows that both methods lead to remarkably accurate solutions. Variational iteration method for a class of nonlinear di. Modified variational iteration method for the solution of a. A study on the convergence of variational iteration method article in mathematical and computer modelling 51910. In this paper, a new kind of analytical technique for a nonlinear problem called the variational iteration method is described and used to give approximate solutions for some wellknown nonlinear problems. The main feature of the method is that the solution of a mathematical problem with linearization. The vim was developed by he 1719 for solving linear, nonlinear, initial and boundary value problems. Only one iteration leads to high accuracy of the solutions and. This method is a very powerful method for solving a large amount of problems. Variational inequalities with linear constraints reduced basis procedures haasdonk, salomon, wohlmuth. Pdf toward a modified variational iteration method. The variational iteration method vim attracted much attention in the past few years as a promising method for solving nonlinear differential equations.
Basic idea of variational iteration method variational iteration method is a powerful device to solve the various kinds of linear and nonlinear functional equations. In this paper we will solve it for the first time via applying a new numerical method called the variational. The variational iteration method for solving linear and nonlinear odes and scientific models with variable coefficients article pdf available in open engineering 41 february 2014 with 2,3. Variational iteration method for solving system of. In this paper, hes variational iteration method is implemented to give approximate and analytical solutions for a class of boundary value problems. Variational iteration method for solving nonlinear wbk equations. The proposed iterative scheme finds the solution without any discretization.
But in fact, the first ideas of the method was issued by 20. Variational iteration method is uniquely qualified to address this challenge. According to variational iteration method, see abbasbandy 2007, abdou and soliman 2005. Pdf variational iteration method for image restoration. Variational iteration method for solving system of fractional. Modified variational iteration method for heat and wave.
It has been used to solve effectively, easily and accurately a large class of nonlinear problems with approximation 6. The idea of the vim is to construct an iteration method based on a correction functional that includes a generalized lagrange multiplier. In order to overcome the demerit, in this paper we will apply hes variational iteration method for solving semidifferential equations of th order. The proposed modification is made by introducing hes polynomials in the correction functional. It had been proved by many authors 23 to be a powerful mathematical tool for solving various types of nonlinear problems, which represent a plenty of modern science branches. The variational iteration method vim proposed by jihuan he is a new analytical method to solve nonlinear equations. The results are compared with those obtained by the numerical methods available in the literature to establish the efficiency of the method. Modified variational iteration method for analytical. The nonclassical calculi such as qcalculus, fractional calculus and qfractional calculus have been hot topics in both applied and pure sciences. In this study, we investigate the application of a new modification of the piecewise variational iteration method for simulating the solution of the strongly nonlinear oscillators.
Variational iteration method for solving some models of. Oscillator problems are frequently encountered in many major fields of science and engineering. The variational iteration method for analytic treatment of homogeneous and inhomogeneous partial differential equations. The main goal of the present study is to find the analytic. A few numerical cases were solved to demonstrate methodology of this new. Solution of linear and nonlinear pdes by the hes variational. Exact solutions of some nonlinear partial differential equations using the variational iteration method linked with laplace transforms and the pade technique. Variational iteration method for image restoration. Inertial iterative schemes with variable step sizes for variational inequality problem involving pseudomonotone operator author jamilu abubakar, poom kumam, habib ur. Pdf variational iteration method for delay differential. Then the method is successfully extended to fractional differential equations. A numerical implementation of the variational iteration method and so on. The purpose of this study is to implement a new modification of the variational iteration method hsvim, which is a combination of spectral method and variational iteration method for heat.
Variational iteration method for a class of nonlinear. Modified variational iteration method for second order. In this scheme the solution takes the form of a convergent series with easily. Among all these methods applied to solve dierential equations, one of them is the variational iteration method vim which was rstly initiated by jihuan he which could be seenthroughout. Abstract in this paper, the variational iteration method is proposed to solve system of nonlinear volterras integrodifferential equations.
This paper outlines a detailed study of the coupling of hes polynomials with correction functional of variational iteration method vim for solving. Hybrid hyperreduced modeling for contact mechanics. The method converges to the exact solution after an iteration. Modified variational iteration method for the solution of. A variational expectationmaximisation algorithm for. Variational iteration method vim to illustrate the basic concept of the hes vim, we consider the following general differential equation lu nu gx, 1 where l is a linear operator, n a nonlinear operator and gx is the inhomogeneous term.
The volterra integral equation of the second kind is an integral equation of the. This study mainly concentrates on the analytical aspects, and the variational iteration method is extended in a new way to solve an initial value problem. New applications of the modified variational iteration method. In this method, a correction functional is constructed by a general lagrange multiplier, which can be identified via variational theory. The fact that variational iteration technique solves nonlinear problems without using adomian polynomials can be considered as a clear advantage of this method over the decomposition method. Hes variational iteration method in this section, we brie. Solution of variational problems using new iterative method.
In this paper, a modified variational iteration method mvim for the solution of a differential equation of bratutype is presented. Pdf toward a modified variational iteration method magdy. Solution of nonlinear partial differential equations by new. The proposed method, which called the piecewise spectral variational iteration method, is based on a combination of spectral method and variational iteration method, and applying it on smaller subintervals and. In this research, a new approach is presented for solving delay differential equations ddes which is a blend of sumudu transform and variational iteration method vim. Pdf we apply the variational iteration method vim for solving linear and nonlinear ordinary differential equations with variable coefficients. Pdf on the convergence of variational iteration method. Variational iteration method for solving nonlinear. We find the solutions of variational problems by using the new iterative method. A general lagrange multiplier is used to construct a correction functional. This allows calculating approximate wavefunctions such as molecular orbitals.
Application of hes variational iteration method to solve. Variational iteration method is implemented to construct solitary solutions for nonlinear dispersive equations. General iterative scheme for split mixed equilibrium. Recently, fractional differential equations have been investigated via the famous variational iteration method. Variational iteration method solutions for certain. Development of variational iteration method the variational iteration method 19 has been shown to solve effectively, easily and accurately a large class of nonlinear problems with approximations converging rapidly to accurate solutions. This method is used for solving burgers and coupled burgers equations in abdou et al.
The main advantage of the method is that it can be applied directly to all types of nonlinear differential and integral equations, homogeneous. In this work, the variational iteration method vim is used for analytic treatment of the linear and nonlinear systems of partial. Optimal variational iteration method for nonlinear problems. Modified variational iteration method for heat and wavelike. The decomposition method and the variational iteration method vim 4, 17 are comparatively new approaches to provides an analytic approximate solution to linear and nonlinear problems, and they are in particular valuable as tools for applied mathematicians and scientists 3. Then some new linear and nonlinear models have appeared. Several linear fractional differential equations are analytically solved as examples and the methodology is demonstrated. Application of variational iteration method to a general.
This will make the solution procedure easier, more effective, and more straightforward. The flexibility and adaptation provided by the method have made the method a strong candidate for approximate analytical solutions. Subsequently, in 1999, the variational iteration method vim was first proposed by jihuan he 17, 18. Several techniques including the adomian decomposition method, the variational iteration method, the weighted finite difference techniques and the laplace decomposition method have been used to solve nonlinear. A comparative study of variational iteration method and he. Pdf application of variational iteration method to linear partial. He 1999, 2000, 2006 developed the variational iteration method for solving linear, nonlinear, initial and boundary value problems. The analytical iteration formula of this method is derived first using a general form of first order nonlinear differential equations, followed by straightforward. A study on the convergence of variational iteration method. Ghanbari 1department of mathematics, university of mazandaran babolsar 47416. A very simple and efficient local variational iteration method for solving problems of nonlinear science is proposedin this paper. The main property of the method is in its flexibility and ability to solve nonlinear equations accurately and conveniently.
Variational iteration method for solving differential. The idea of the vim is to construct an iteration method based on a. New applications of the variational iteration method from. The variational iteration method vim is applied to the numerical simulation of linear thirdorder dispersive partial differential equations pdes. Comparison of the adomian decomposition method and the. This is done with an uncommon sumudu transform alongside variational theory. Variational iteration method for solving nonlinear boundary value problems. Vim allows for the solution of the differential equation. Our method converts nonconjugate factors in a def model. Variational iteration method 1,2 was proposed to find analytical solutions for nonlinear equations, i. The method converge s to the exact solution after an iteration. Starting from the pioneer ideas of the inokutisekinemura method, jihuan he 3 developed the variational iteration method vim in 1999.
The main goal of this paper is to extend the variational iteration method to. Variational iteration method has been favourably applied to various kinds of nonlinear problems. In this section, we combined laplace transform and variational iteration method to solve the nonlinear partial differential equations. Variational iteration method an overview sciencedirect. A new modified variational iteration method was found, inspired and driven by wus thoughts and combining with the sumudu transform 20. Abstract in this paper, the author used the variational iteration method vim to find the analytical. Using the hes variational iteration method, it is possible to find the exact solutions or better approximate solutions of the partial differential equations. The variational iteration method of the local fractional operator was employed to solve the local fractional partial differential equations 75, 104, 118125. Variational inference on deep exponential family by using. The variational iteration method, which produces the solutions in terms of convergent series, requiring no linearization or small perturbation.