Generalisation of the Lagrange multipliers for variational iterations applied to systems of differential equations

Altintan, Derya; Uğur, Ömür

doi:10.1016/j.mcm.2011.05.013

Cited by 5 publications

(5 citation statements)

References 30 publications

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“…Therefore (3) become (5) If in (5), then we may write (5) as (6) Integrating the integral sign in (6) using integration by part and thereafter setting since the extremum condition of requires that is stationary we get the system (7) Solving the above system of equations, it is easy to see that (8) Similarly, if in (5), the following system of equation is easily derived (9) And the Lagrange multiplier solved for as (10) …and so on.…”

Section: Overview Of Variational Iteration Methods (Vim)mentioning

confidence: 99%

“…Authors in [3] and [5] suggested a universal way to identify the Lagrange multiplier for VIM by implementing Laplace transform. Altintan and Ugor [6] proposed a method that defines the Lagrange multipliers for VIM as Matrix-valued functions. Building on existing methods and variational theories for the determination of the Lagrange multiplier for VIM, in this paper we apply the operator D-Method and integrating factor to certain aspect of the determination of accurate Lagrange multiplier for VIM.…”

Section: Original Research Articlementioning

confidence: 99%

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On the Computation of the Lagrange Multiplier for the Variational Iteration Method (VIM) for Solving Differential Equations

Okiotor

Ogunfiditimi

Durojaye

2020

JAMCS

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In this study, the Variational Iteration Method (VIM) is applied in finding the solution of differential equations with emphasis laid on the choice of the Lagrange multiplier used while employing VIM. Building on existing methods and variational theories, the operator D-Method and integrating factor are employed in certain aspects in the determination of exact Lagrange multiplier for VIM. When results of the computed exact Lagrange multiplier were compared with results of approximate Lagrange multiplier, it was observed that the computed exact Lagrange multiplier reduced significantly the number of iterations required to get a good approximate result, and in some cases the result converged to the exact solution after a single iteration. Evaluations are carried out using MAPLE Software.

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Section: Overview Of Variational Iteration Methods (Vim)mentioning

confidence: 99%

Section: Original Research Articlementioning

confidence: 99%

On the Computation of the Lagrange Multiplier for the Variational Iteration Method (VIM) for Solving Differential Equations

Okiotor

Ogunfiditimi

Durojaye

2020

JAMCS

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“…A kind of VIM with a suitably modified Lagrange Multipliers for system of differential equations proposed by Altintan and Ugur [32] will be followed here. Even if the essence of the method is to tackle the nonlinear problems, here we will apply restricted variation to the part of matrix A which makes an exact solution to Eq.…”

Section: Variational Iteration Methodsmentioning

confidence: 99%

“…Reviews and more detailed explanations about the method can be seen in [26,27]. Since the initiation of the method, there have been many modifications and improvements introduced to it [28][29][30][31][32], for the reviews of which we refer to the note by He [33]. VIM is recently used for solutions of many different structural problems, see for example [34][35][36][37][38].…”

Section: Introductionmentioning

confidence: 99%

Some closed-form solutions for buckling of straight beams with varying cross-section by Variational Iteration Method with Generalized Lagrange Multipliers

Eroğlu

Tüfekçi

2018

International Journal of Engineering and Applied Sciences

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This study aims to derive approximate closed-form solutions for critical loads of straight beams with variable cross-section. The governing equations are derived for purely flexible beam for small displacements and rotation and turned into non-dimensional form. Approximate solutions to the set of equations for stability problems are searched by Variational Iteration Method with Generalized Lagrange Multipliers. It turns out that highly accurate approximate buckling loads for cantilever beams with constant or variable section can be obtained in closed-form. Many novel closed-form solutions for critical load of such structures, which may serve as benchmark solutions, are presented.

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“…But accurate determination of the Lagrange multiplier for VIM can sometimes be challenging [5,6,4]. Over the years, a number of authors have proposed various methods for the determination of the Lagrange multiplier for VIM [7,8,4,9]. In [6], we proposed the use of Integrating Factor method and Operator D-Method in certain aspects of the determination of the Lagrange Multiplier for VIM.…”

Section: Introductionmentioning

confidence: 99%

Generalization of the Expansion of the Lagrange Multiplier Identifier Integrals for the Variational Iteration Method

Okiotor

Ogunfiditimi²

2023

JAMCS

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This study presents a theorem for resolving certain integrals. These integrals usually arise in the Correction Functional of the Variational Iteration Method (VIM). Resolving these integrals in the Correction Functional of VIM enables the accurate determination of the Lagrange multiplier for the method, which in turn leads to faster convergence rate for the solution of VIM. The study begins with a brief overview of the Lagrange Multiplier and VIM. This is followed by the theorem and a few experimental problems considered to demonstrate the usefulness of the theorem in the implementation of VIM. Comparison of results obtained shows that accurately determined Lagrange Multiplier for VIM improves the convergence rate of the method compared to using approximately determined Lagrange multiplier for the scheme. Evaluations are carried out using Maple 18 software.

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Generalisation of the Lagrange multipliers for variational iterations applied to systems of differential equations

Cited by 5 publications

References 30 publications

On the Computation of the Lagrange Multiplier for the Variational Iteration Method (VIM) for Solving Differential Equations

On the Computation of the Lagrange Multiplier for the Variational Iteration Method (VIM) for Solving Differential Equations

Some closed-form solutions for buckling of straight beams with varying cross-section by Variational Iteration Method with Generalized Lagrange Multipliers

Generalization of the Expansion of the Lagrange Multiplier Identifier Integrals for the Variational Iteration Method

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