Application of He’s variational iteration method to Helmholtz equation

Momani, Shaher; Abuasad, Salah

doi:10.1016/j.chaos.2005.04.113

Cited by 392 publications

(174 citation statements)

References 18 publications

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“…The subscript n indicates the nth approximation and u'n is considered as a restricted variation [21][22][23], i.e., δ un = 0.…”

Section: Variation Iteration Methodsmentioning

confidence: 99%

Analytical approximate solutions for investigating non-damped system of cantilever beams

Jafari¹,

Ahmadi²,

Ghasempour³

et al. 2018

ntmsci

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In the current study, a comprehensive analysis of nonlinear free vibrations of a flexible rotating beam attached with a setting angle to a rigid hub is perused. The system is supposed to be subjected to an external torque and a rotation dissipation load which collectively referred to as perturbations. A set of two strongly coupled second-order ordinary nonlinear differential equations with an ignorable coordinate is used to depict the dynamics of the individual mode in plane flexural deflection of the arm and its overall rotational motion. For the unperturbed system which is our especial case in this work, the system angular momentum parameter, not the hub rotational speed, becomes a specified constant of motion. Furthermore, in order to consider the effect of beam deflections on the overall motion and vice versa, utilizing torque-controlled approach is necessary. It is noteworthy that here three powerful analytical methods for instance Modified Differential Transformation Method (MDTM), Variation Iteration Method (VIM) and finally Homotopy Perturbation Method (HPM) have been introduced to investigate the comportment and frequency of cantilever rotating beams with a setting angle. These methods are useful and practical for solving a wide variety of nonlinear equations. Comparisons amongst the obtained results by all the presented methods and Numerical solution reveal that all the afore-mentioned methods are significantly effective and efficient especially MDTM.

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“…The subscript n indicates the nth approximation and u'n is considered as a restricted variation [21][22][23], i.e., δ un = 0.…”

Section: Variation Iteration Methodsmentioning

confidence: 99%

Analytical approximate solutions for investigating non-damped system of cantilever beams

Jafari¹,

Ahmadi²,

Ghasempour³

et al. 2018

ntmsci

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show abstract

“…[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%

“…The variational iteration method was first proposed by He [9][10][11][12][13][14][15], and was successfully applied to autonomous ordinary differential equations in Ref. [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].…”

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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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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“…where λ is a general Lagrangian multiplier [19,20] which can be identified optimally via the variational theory. The subscript indicates the -th approximation and˜ considered as a restricted variation [19,20].…”

Section: Variational Iteration Methodsmentioning

confidence: 99%

Analytical accuracy of the one dimensional heat transfer in geometry with logarithmic various surfaces

et al. 2014

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Abstract:In this study, heat transfer and temperature distribution equations for logarithmic surface are investigated analytically and numerically. Employing the similarity variables, the governing differential equations have been reduced to ordinary ones and solved via Homotopy perturbation method (HPM), Variational iteration method (VIM), Adomian decomposition method (ADM). The influence of the some physical parameters such as rate of effectiveness of temperature on non-dimensional temperature profiles is considered. Also the fourth-order RungeKutta numerical method (NUM) is used for the validity of these analytical methods and excellent agreement are observed between the solutions obtained from HPM, VIM, ADM and numerical results.

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Application of He’s variational iteration method to Helmholtz equation

Cited by 392 publications

References 18 publications

Analytical approximate solutions for investigating non-damped system of cantilever beams

Analytical approximate solutions for investigating non-damped system of cantilever beams

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

Analytical accuracy of the one dimensional heat transfer in geometry with logarithmic various surfaces

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