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.

Dual solutions for nonlinear boundary value problems by the variational iteration method

2017 ◽  
Vol 27 (1) ◽  
pp. 210-220 ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Boundary Value Problems ◽  
Iteration Method ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Variational Iteration Method ◽  
Dual Solutions ◽  
Nonlinear Boundary Value Problems ◽  
Content Type ◽  
Variational Iteration ◽  
Linear And Nonlinear

Purpose The purpose of this paper is to use the variational iteration method (VIM) for studying boundary value problems (BVPs) characterized with dual solutions. Design/methodology/approach The VIM proved to be practical for solving linear and nonlinear problems arising in scientific and engineering applications. In this work, the aim is to use the VIM for a reliable treatment of nonlinear boundary value problems characterized with dual solutions. Findings The VIM is shown to solve nonlinear BVPs, either linear or nonlinear. It is shown that the VIM solves these models without requiring restrictive assumptions and in a straightforward manner. The conclusions are justified by investigating many scientific models. Research limitations/implications The VIM provides convergent series solutions for linear and nonlinear equations in the same manner. Practical implications The VIM is practical and shows more power compared to existing techniques. Social implications The VIM handles linear and nonlinear models in the same manner. Originality/value This work highlights a reliable technique for solving nonlinear BVPs that possess dual solutions. This paper has shown the power of the VIM for handling BVPs.

scholarly journals An Approximate Solution for Boundary Value Problems in Structural Engineering and Fluid Mechanics

2008 ◽  
Vol 2008 ◽  
pp. 1-13 ◽  
Author(s):  
A. Barari ◽  
M. Omidvar ◽  
D. D. Ganji ◽  
Abbas Tahmasebi Poor
Keyword(s):  
Fluid Mechanics ◽  
Exact Solutions ◽  
Boundary Value Problems ◽  
Iteration Method ◽  
Structural Engineering ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Variational Iteration Method ◽  
Nonlinear Boundary Value Problems ◽  
Linear And Nonlinear

Variational iteration method (VIM) is applied to solve linear and nonlinear boundary value problems with particular significance in structural engineering and fluid mechanics. These problems are used as mathematical models in viscoelastic and inelastic flows, deformation of beams, and plate deflection theory. Comparison is made between the exact solutions and the results of the variational iteration method (VIM). The results reveal that this method is very effective and simple, and that it yields the exact solutions. It was shown that this method can be used effectively for solving linear and nonlinear boundary value problems.

scholarly journals Analytical approximate solution of higher order boundary value problems via variational iteration method

BIBECHANA ◽  
2017 ◽  
Vol 15 ◽  
pp. 37-42
Author(s):  
Jamshad Ahmad ◽  
Zobia Hamid
Keyword(s):  
Approximate Solution ◽  
Mathematical Models ◽  
Boundary Value Problems ◽  
Iteration Method ◽  
Boundary Value ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Higher Order ◽  
Analytical Approximate Solution ◽  
Variational Iteration

In this paper, application of variational iteration method has been successfully extended to obtain approximate solutions of some higher order boundary value problems. We emphasize the power of the method by testing three different mathematical models of distinct orders. The results are obtained by using only little iteration.  BIBECHANA 15 (2018) 37-42

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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