Optimal variational iteration method using Adomian’s polynomials for physical problems on finite and semi-infinite intervals

2017 ◽  
Vol 132 (5) ◽  
Author(s):  
Syed Tauseef Mohyud-Din ◽  
Waseem Sikander ◽  
Umar Khan ◽  
Naveed Ahmed
Keyword(s):  
Iteration Method ◽  
Variational Iteration Method ◽  
Variational Iteration ◽  
Physical Problems ◽  
Infinite Intervals ◽  
Adomian’S Polynomials

TRAVELING WAVE SOLUTIONS OF SEVENTH-ORDER GENERALIZED KdV EQUATIONS BY VARIATIONAL ITERATION METHOD USING ADOMIAN'S POLYNOMIALS

2009 ◽  
Vol 23 (15) ◽  
pp. 3265-3277 ◽  
Author(s):  
SYED TAUSEEF MOHYUD-DIN ◽  
MUHAMMAD ASLAM NOOR ◽  
KHALIDA INAYAT NOOR
Keyword(s):  
Applied Sciences ◽  
Iteration Method ◽  
Traveling Wave Solutions ◽  
Variational Iteration Method ◽  
Variational Iteration ◽  
Kdv Equations ◽  
Wave Solutions ◽  
Seventh Order ◽  
Generalized Kdv Equations ◽  
Adomian’S Polynomials

In this paper, we apply the variational iteration method using Adomian's polynomials (VIMAP) to investigate propagating traveling solitary wave solutions of seventh-order generalized KdV (SOG-KdV) equations, which play a very important role in mathematical physics, engineering, and applied sciences. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed iterative scheme finds the solution without any discretization, perturbation, linearization, or restrictive assumptions and is formulated by the elegant coupling of variational iteration method and Adomian's polynomials.

Convergent optimal variational iteration method and applications to heat and fluid flow problems

2016 ◽  
Vol 26 (3/4) ◽  
pp. 790-804 ◽  
Author(s):  
Mustafa Turkyilmazoglu
Keyword(s):  
Fluid Flow ◽  
Iteration Method ◽  
Sufficient Conditions ◽  
Variational Iteration Method ◽  
Content Type ◽  
Flow Problems ◽  
Variational Iteration ◽  
Physical Problems ◽  
Novel Approach ◽  
Heat And Fluid Flow

Purpose – In an earlier paper (Turkyilmazoglu, 2011a), the author introduced a new optimal variational iteration method. The idea was to insert a parameter into the classical variational iteration formula in an aim to prevent divergence or to accelerate the slow convergence property of the classical approach. The purpose of this paper is to approve the superiority of the proposed method over the traditional one on several physical problems treated before by the classical variational iteration method. Design/methodology/approach – A sufficient condition theorem with an upper bound for the error is also presented to further justify the convergence of the new variational iteration method. Findings – The optimal variational iteration method is found to be useful for heat and fluid flow problems. Originality/value – The optimal variational iteration method is shown to be convergent under sufficient conditions. A novel approach to obtain the optimal convergence parameter is introduced.

VARIATIONAL ITERATION METHOD FOR SOLVING NONLINEAR HIGHER-ORDER INITIAL AND BOUNDARY VALUE PROBLEMS

2009 ◽  
Vol 06 (04) ◽  
pp. 521-555 ◽  
Author(s):  
SYED TAUSEEF MOHYUD-DIN ◽  
MUHAMMAD ASLAM NOOR ◽  
KHALIDA INAYAT NOOR
Keyword(s):  
Boundary Value Problems ◽  
Iteration Method ◽  
Nonlinear Boundary ◽  
Iterative Scheme ◽  
Boundary Value ◽  
Variational Iteration Method ◽  
Higher Order ◽  
Nonlinear Boundary Value Problems ◽  
Variational Iteration ◽  
Adomian’S Polynomials

In this paper, we apply variational iteration method (VIM) and variational iteration method using Adomian's polynomials for solving nonlinear boundary value problems. The proposed iterative scheme finds the solution without any discretization, linearization, perturbation, or restrictive assumptions. Several examples are given to verify the accuracy and efficiency of the method. We have also considered an example where the proposed VIM is not reliable.

Modified Variational Iteration Method for Integro-Differential Equations and Coupled Systems

2010 ◽  
Vol 65 (4) ◽  
pp. 277-284 ◽  
Author(s):  
Syed Tauseef Mohyud-Din
Keyword(s):  
Differential Equations ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Integrodifferential Equations ◽  
Coupled Systems ◽  
Variational Iteration ◽  
He’S Polynomials ◽  
Proposed Modification ◽  
Adomian’S Polynomials ◽  
Modified Variational Iteration Method

In this paper, we apply the modified variational iteration method (mVIM) for solving integrodifferential equations and coupled systems of integro-differential equations. The proposed modification is made by the elegant coupling of He’s polynomials and the correction functional of variational iteration method. The proposed mVIM is applied without any discretization, transformation or restrictive assumptions and is free from round off errors and calculation of the so-called Adomian’s polynomials.

scholarly journals Fractional Variational Iteration Method versus Adomian’s Decomposition Method in Some Fractional Partial Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Junqiang Song ◽  
Fukang Yin ◽  
Xiaoqun Cao ◽  
Fengshun Lu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Iteration Method ◽  
Decomposition Method ◽  
Variational Iteration Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Variational Iteration ◽  
Adomian’S Decomposition Method ◽  
Adomian’S Polynomials

A comparative study is presented about the Adomian’s decomposition method (ADM), variational iteration method (VIM), and fractional variational iteration method (FVIM) in dealing with fractional partial differential equations (FPDEs). The study outlines the significant features of the ADM and FVIM methods. It is found that FVIM is identical to ADM in certain scenarios. Numerical results from three examples demonstrate that FVIM has similar efficiency, convenience, and accuracy like ADM. Moreover, the approximate series are also part of the exact solution while not requiring the evaluation of the Adomian’s polynomials.

scholarly journals Variational Iteration Method for Fifth-Order Boundary Value Problems Using He's Polynomials

2008 ◽  
Vol 2008 ◽  
pp. 1-12 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Syed Tauseef Mohyud-Din
Keyword(s):  
Boundary Value Problems ◽  
Iteration Method ◽  
Boundary Value ◽  
Nonlinear Problems ◽  
Variational Iteration Method ◽  
Variational Iteration ◽  
Homotopy Perturbation ◽  
He’S Polynomials ◽  
Fifth Order ◽  
Adomian’S Polynomials

We apply the variational iteration method using He's polynomials (VIMHP) for solving the fifth-order boundary value problems. The proposed method is an elegant combination of variational iteration and the homotopy perturbation methods and is mainly due to Ghorbani (2007). The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed iterative scheme finds the solution without any discritization, linearization, or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that the proposed technique solves nonlinear problems without using Adomian's polynomials can be considered as a clear advantage of this algorithm over the decomposition method.

scholarly journals Solution of Singular and Nonsingular Initial and Boundary Value Problems by Modified Variational Iteration Method

2008 ◽  
Vol 2008 ◽  
pp. 1-23 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Syed Tauseef Mohyud-Din
Keyword(s):  
Boundary Value Problems ◽  
Iteration Method ◽  
Boundary Value ◽  
Variational Iteration Method ◽  
Suggested Algorithm ◽  
Variational Iteration ◽  
Proposed Modification ◽  
Efficiency And Reliability ◽  
Adomian’S Polynomials ◽  
Modified Variational Iteration Method

We apply the modified variational iteration method (MVIM) for solving the singular and nonsingular initial and boundary value problems in this paper. The proposed modification is made by introducing Adomian's polynomials in the correct functional. The suggested algorithm is quite efficient and is practically well suited for use in such problems. The proposed iterative scheme finds the solution without any discretization, linearization, perturbation, or restrictive assumptions. Several examples are given to verify the efficiency and reliability of the suggested algorithm.

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