MODIFIED VARIATIONAL ITERATION METHOD FOR SCHRODINGER EQUATIONS

In this paper, we apply the modified variational iteration method (MVIM) for solving Schrödinger equations. The proposed modification is made by introducing He’s polynomials in the correction functional of variational iteration method (VIM). The suggested iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. The use of Lagrange multiplier coupled with He’s polynomials are the clear advantages of this technique over the decomposition method. Several examples are given to verify the reliability and efficiency of the proposed algorithm. Key wordsVariational iteration method, partial differential equations, Schrödinger equations, He’s polynomials.

Lu [28], Ma [29], Momani and Odibat [30], Noor and Mohyud-Din [31,32,[34][35][36][37], Rafi and Danili [38], Sweilman [39], Sadighi and Ganji [39]).Recently, Ghorbani et.al. introduced He's polynomials by splitting the nonlinear term and also proved that He's polynomials are fully compatible with Adomian's polynomials but are easier to calculate and are more user friendly (see Ghorbani et.al. [11,12]).More recently, Noor and Mohyud-Din combined He's polynomials and correction functional of the variational iteration method (VIM) and applied this reliable version to a number of physical problems; (see Noor and Mohyud-Din [34][35][36]).The paper is devoted to the study of an important type of partial differential equation which is called the Schrödinger equation, is of the form ( ) and arises in various areas of applied sciences including nonlinear optics, plasma physics, super conductivity and quantum mechanics, (see, Sadighi and Ganji.[39] and the reference therein).Several techniques including decomposition and homotopy perturbation have been employed for the solution of such problems, (see, Mohyud-Din and Noor [32], Sadighi and Ganji [39]).In this paper, we apply the modified variational iteration method (MVIM) which is formulated by the elegant coupling of variational iteration method (VIM) and He's polynomials for solving Schrödinger equations.It is shown that the MVIM provides the solution in a rapid convergent series.We write the correction functional for the Schrödinger equations and calculate Lagrange multiplier optimally via variational theory.The He's polynomials are introduced in the correction functional.The use of Lagrange multiplier reduces the successive application of the integral operator and minimizes the computational work.Moreover, the selection of the initial value is done very carefully because the approximants are heavily dependent on it.Several examples are given to illustrate the reliability and performance of the proposed method.It is to be highlighted that the modified variational iteration method (MVIM) has certain advantages as compare to the decomposition method.Firstly, the use of Lagrange multiplier reduces the successive applications of the integral operator and hence minimizes the computational work to a tangible level while still maintaining a very high level of accuracy.Moreover, He's polynomials are easier to calculate as compare to Adomian's polynomials and this gives it a clear edge over the traditional decomposition method.The MVIM is also independent of the small parameter assumption (which is either not there in the physical problems or difficult to locate) and hence is more convenient to apply as compare to the traditional perturbation method.It is worth mentioning that the MVIM is applied without any discretization, restrictive assumption or transformation and is free from round off errors.We apply the proposed MVIM for all the nonlinear terms in the problem without discretizing either by finite difference or spline techniques at the nodes, involves laborious calculations coupled with a strong possibility of the ill-conditioned resultant equations which is a complicated problem to solve.Moreover, unlike the method of separation of variables that requires initial and boundary conditions, the VIMHP provides the solution by using the initial conditions only, (see Noor and Mohyud-Din [34][35][36]).

MODIFIED VARIATIONAL ITERATION METHOD (MVIM)
The modified variational iteration method (MVIM) is obtained by the elegant coupling of correction functional (2) of variational iteration method (VIM) with He's polynomials and is given by .

NUMERCICAL APPLICATOIONS
In this section, we apply the modified variational iteration (MVIM) for solving Schrödinger equations.The results are very encouraging indicating the reliability and efficiency of the proposed method.The correction functional is given as Making the correction functional stationary, the Lagrange multipliers can be identified as Comparing the co-efficient of like powers of p, approximants are obtained ), The solution in a series form is given by The correction functional is given as Making the correction functional stationary, the Lagrange multipliers can be identified as Comparing the co-efficient of like powers of p, approximants are obtained , ) , ( : Comparing the co-efficient of like powers of p, approximants are obtained , ) , ( : 0  In this paper, we applied modified variational iteration method (MVIM) for solving Schrödinger equations.The method is applied in a direct way without using linearization, transformation, discretization or restrictive assumptions.It may be concluded that MVIM is very powerful and efficient in finding the analytical solutions for a wide class of boundary value problems.The method gives more realistic series solutions that converge very rapidly in physical problems.It is worth mentioning that the method is capable of reducing the volume of the computational work as compare to the classical methods while still maintaining the high accuracy of the numerical result.
the solution u will be readily obtained upon using the determined Lagrange multiplier and any selective function , 0 u consequently, the solution is given by 1, then (6)  corresponds to (4) and becomes the approximate solution of the form,

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The solution in a series form is given by