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The purpose of this paper is to employ an alternative approach to reconstruct the standard variational iteration algorithm II proposed by He, including Lagrange multiplier, and to give a simpler formulation of Adomian decomposition and modified Adomian decomposition method in terms of newly proposed variational iteration method-II (VIM). Through careful investigation of the earlier variational iteration algorithm and Adomian decomposition method, we find unnecessary calculations for Lagrange multiplier and also repeated calculations involved in each iteration, respectively. Several examples are given to verify the reliability and efficiency of the method.

Over the last few decades several analytical/approximate methods have been developed to solve nonlinear ordinary and partial differential equations. For initial and boundary-value problems in ordinary and partial differential equations, some of these techniques include the perturbation method [

The Adomian decomposition method [

He’s variational iteration method [

In the present study, we have linked up variational iteration method and Adomian decomposition method through Lagrange multiplier, which shows that VIM is another form of expressing ADM and vice versa. This study reveals that there is no need to integrate the differential equation again and again as we do in Adomian decomposition method. Advantage of new iterative scheme over the variational iteration method is that it avoids the unnecessary calculations and we can construct Lagrange multiplier very easily without construction of the correctional functional.

In order to elucidate the solution procedure, we consider the following

In (

In order to illustrate the solution procedure, we consider the following examples for ordinary and partial differential equations.

Consider the Blasius equation

Consider the nonhomogeneous wave equation

This paper helps us to gain insight into the idea of Adomian decomposition method and variational iteration method. By keeping in view both methods, we propose more simplified forms to calculate Lagrange multipliers. By introducing this Lagrange multiplier in ADM and VIM following the observations that have been made,

there is no need to do integration process again and again like we do in Adomian decomposition method and one can get the same results of Adomian method.

It is easy to calculate the Lagrange multiplier of He’s variational iteration method.

This new approach avoids the unnecessary calculations like we did in He’s variational iteration method and Adomian decomposition method.

This study shows that VIM is another form of expressing ADM and vice versa.

So we can say that the present method is parallel form of ADM and can give good results of VIM with less effort.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors have made the same contribution. All authors read and approved the final paper.

The authors are grateful to the reviewers for their comments and useful suggestions and the third author was supported by Project no. FEKT-S-14-2200 of Faculty of Electrical Engineering and Communication, Brno University of Technology, Czech Republic.