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 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.

Application of Variational Iteration Method for nth-Order Integro-Differential Equations

2009 ◽  
Vol 64 (7-8) ◽  
pp. 439-444 ◽  
Author(s):  
Said Abbasbandy ◽  
Elyas Shivanian
Keyword(s):  
Differential Equations ◽  
Iteration Method ◽  
Initial Conditions ◽  
Initial Approximation ◽  
Variational Iteration Method ◽  
Integrodifferential Equations ◽  
Variational Iteration

AbstractIn this paper, the variational iteration method is proposed to solve Fredholm’s nth-order integrodifferential equations. The initial approximation is selected wisely which satisfies the initial conditions. The results reveal that this method is very effective and convenient in comparison with other methods.

scholarly journals Solitary and compacton solutions of fractional KdV-like equations

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 328-336 ◽  
Author(s):  
Bo Tang ◽  
Yingzhe Fan ◽  
Jianping Zhao ◽  
Xuemin Wang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Reliable Tool ◽  
Variational Iteration ◽  
He’S Polynomials ◽  
Liouville Derivative

AbstractIn this paper, based on Jumarie’s modified Riemann-Liouville derivative, we apply the fractional variational iteration method using He’s polynomials to obtain solitary and compacton solutions of fractional KdV-like equations. The results show that the proposed method provides a very effective and reliable tool for solving fractional KdV-like equations, and the method can also be extended to many other fractional partial differential equations.

scholarly journals Comment on “Variational Iteration Method for Fractional Calculus Using He’s Polynomials”

2012 ◽  
Vol 2012 ◽  
pp. 1-2 ◽  
Author(s):  
Ji-Huan He
Keyword(s):  
Iteration Method ◽  
Variational Iteration Method ◽  
Initial Boundary ◽  
Iteration Algorithm ◽  
Initial Boundary Value Problems ◽  
Variational Iteration ◽  
He’S Polynomials ◽  
Fractional Differential ◽  
Initial Boundary Value ◽  
Modified Variational Iteration Method

Recently Liu applied the variational homotopy perturbation method for fractional initial boundary value problems. This note concludes that the method is a modified variational iteration method using He’s polynomials. A standard variational iteration algorithm for fractional differential equations is suggested.

Variational Iteration Method for Delay Differential Equations Using He’s Polynomials

2010 ◽  
Vol 65 (12) ◽  
pp. 1045-1048 ◽  
Author(s):  
Syed Tauseef Mohyud-Din ◽  
Ahmet Yildirim
Keyword(s):  
Applied Sciences ◽  
Signal Processing ◽  
Differential Equations ◽  
Iteration Method ◽  
Delay Differential Equations ◽  
Digital Images ◽  
Variational Iteration Method ◽  
Variational Iteration ◽  
He’S Polynomials ◽  
Delay Differential

January 21, 2010 In this paper, we apply the variational iteration method using He’s polynomials (VIMHP) for solving delay differential equations which are otherwise too difficult to solve. These equations arise very frequently in signal processing, digital images, physics, and applied sciences. Numerical results reveal the complete reliability and efficiency of the proposed combination.

Variational Iteration Method for Burgers’ and Coupled Burgers’ Equations Using He’s Polynomials

2010 ◽  
Vol 65 (4) ◽  
pp. 263-267 ◽  
Author(s):  
Syed Tauseef Mohyud-Din ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Lagrange Multiplier ◽  
Iteration Method ◽  
Decomposition Method ◽  
Variational Iteration Method ◽  
Burgers Equations ◽  
Variational Iteration ◽  
He’S Polynomials ◽  
Proposed Modification

In this paper, we apply a modified version of the variational iteration method (MVIM) for solving Burgers’ and coupled Burgers’ equations. The proposed modification is made by introducing He’s polynomials in the correction functional of the variational iteration method (VIM). The use of Lagrange multiplier coupled with He’s polynomials are the clear advantages of this technique over the decomposition method.

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 Solution of Nonlinear Partial Differential Equations by New Laplace Variational Iteration Method

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Eman M. A. Hilal ◽  
Tarig M. Elzaki
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Iteration Method ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Variational Iteration Method ◽  
Convolution Integral ◽  
Partial Differential ◽  
Variational Iteration ◽  
Modified Variational Iteration Method

The aim of this study is to give a good strategy for solving some linear and nonlinear partial differential equations in engineering and physics fields, by combining Laplace transform and the modified variational iteration method. This method is based on the variational iteration method, Laplace transforms, and convolution integral, introducing an alternative Laplace correction functional and expressing the integral as a convolution. Some examples in physical engineering are provided to illustrate the simplicity and reliability of this method. The solutions of these examples are contingent only on the initial conditions.

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