Application of Variational Iteration Method to Nonlinear Differential Equations of Fractional Order

Odibat, Zaid; Momani, Shaher

doi:10.1515/ijnsns.2006.7.1.27

Cited by 595 publications

(311 citation statements)

References 21 publications

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“…[9][10][11][12][13][14][15][16][17][18][19][20]. To solve the fractional KdV equation by means of variational iteration method, rewrite equation (1.2) in the form…”

Section: Analysis Of the Variational Iteration Methodsmentioning

confidence: 99%

“…[14], to nonlinear wave equations [15], to Helmholtz equation [16], to generalized Burger-Fisher and Burger equations [17], and other fields. Recently, the application of the variational iteration method is successfully extended to obtain analytical approximate solutions to linear and nonlinear differential equations of fractional order [18][19][20]. A comparison between the variational iteration method and Adomian decomposition method for solving fractional differential equations is given in Refs.…”

Section: Introductionmentioning

confidence: 99%

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Variational iteration method for solving the space‐ and time‐fractional KdV equation

Momani

Odibat

Alawneh

2007

Numerical Methods Partial

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This paper presents numerical solutions for the space-and time-fractional Korteweg-de Vries equation (KdV for short) using the variational iteration method. The space-and time-fractional derivatives are described in the Caputo sense. In this method, general Lagrange multipliers are introduced to construct correction functionals for the problems. The multipliers in the functionals can be identified optimally via variational theory. The iteration method, which produces the solutions in terms of convergent series with easily computable components, requiring no linearization or small perturbation. The numerical results show that the approach is easy to implement and accurate when applied to space-and time-fractional KdV equations. The method introduces a promising tool for solving many space-time fractional partial differential equations.

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“…[9][10][11][12][13][14][15][16][17][18][19][20]. To solve the fractional KdV equation by means of variational iteration method, rewrite equation (1.2) in the form…”

Section: Analysis Of the Variational Iteration Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Variational iteration method for solving the space‐ and time‐fractional KdV equation

Momani

Odibat

Alawneh

2007

Numerical Methods Partial

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“…For this, several solution methods have been developed to obtaining the approximate solutions of fractional differential equations. Some well-known methods for approximating solutions of FBVP are summarized as follows, but not limited to: Homotopy perturbation method [1,2], Differential transform method [3,4], Adomian decomposition method [5][6][7], Variational iteration method [8,9], Cubic spline method [10], Haar wavelet method [11] and Homotopy analysis method [12].…”

Section: Introductionmentioning

confidence: 99%

An efficient algorithm for solving fractional differential equations with boundary conditions

2016

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Abstract:In this paper, a sinc-collocation method is described to determine the approximate solution of fractional order boundary value problem (FBVP). The results obtained are presented as two new theorems. The fractional derivatives are defined in the Caputo sense, which is often used in fractional calculus. In order to demonstrate the efficiency and capacity of the present method, it is applied to some FBVP with variable coefficients. Obtained results are compared to exact solutions as well as Cubic Spline solutions. The comparisons can be used to conclude that sinc-collocation method is powerful and promising method for determining the approximate solutions of FBVPs in different types of scenarios.

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“…As a result, the solution of such equations can be approximated with a large variety of analytical numerical methods. Among these, homotopy perturbation method Odibat and Momani, 2008), variational iteration method (Odibat and Momani, 2006;Abbasbandy, 2007;Molliq et al, 2009), homotopy analysis method (Song and Zhang, 2007;Hashim et al, 2009;El-Ajou et al, 2010;Arqup and El-Ajou, 2013), transform methods (Arıkoğlu and Özkol, 2007;Oturanç et al, 2008;Ertürk and Momani, 2008;Momani et al, 2014) and other methods (Schneider, 1996;Arqub et al, 2012;Al-Smadi et al, 2013;Arqub et al, 2013;Arqub et al, 2015).…”

Section: Introductionmentioning

confidence: 99%

An Expansion Iterative Technique for Handling Fractional Differential Equations Using Fractional Power Series Scheme

Abu-Gdairi¹,

Al‐Smadi²,

Gumah³

2015

Journal of Mathematics and Statistics

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In this study, we present a new analytical numerical technique for solving a class of time Fractional Differential Equations (FDEs) with variable coefficients based on the generalized Taylor series formula in the Caputo sense. This method provided the solution in the form of a rapidly convergent power series under a multiple fractional differentiability with easily computable components. An efficacious experiment is given to guarantee the procedure, to illustrate the theoretical statements of the present technique and to show its potentiality, generality and superiority for solving wide range of FDEs. The results reveal that the method is easy to implement, very effective, fully compatible with the complexity of such problems, straightforward and simple.

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Application of Variational Iteration Method to Nonlinear Differential Equations of Fractional Order

Cited by 595 publications

References 21 publications

Variational iteration method for solving the space‐ and time‐fractional KdV equation

Variational iteration method for solving the space‐ and time‐fractional KdV equation

An efficient algorithm for solving fractional differential equations with boundary conditions

An Expansion Iterative Technique for Handling Fractional Differential Equations Using Fractional Power Series Scheme

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