Variational iteration method for elastodynamic Green’s functions

Akmaz, Hakan K.

doi:10.1016/j.na.2008.10.057

Cited by 7 publications

(3 citation statements)

References 17 publications

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“…The classical variational iteration method was used in many papers for a variety of problems arising from mathematical physics, see for example, Laplace equation by Wazwaz (2007), quadratic Riccati differential equation by Abbasbandy (2007), a coupled MKdV system by Inc and Cavlak (2008), differential-difference equation by Yildirim (2008), singular equations of Lane-Emden type by Yildirim and Ozis (2009), nth-order integro-differential equations by Abbasbandy and Shivanian (2009), a class of singular boundary value problems by Geng (2009), elastodynamic Greens functions by Akmaz (2009), singular fourth-order parabolic partial differential equations by Noor et al (2009), biochemical reaction model by Goh et al (2010) and fractional equations by Wu and Lee (2010), among many others. However, for some physical models the classical variational iteration method unfortunately fails to converge to the real solution, see for instance the sample works on the nonlinear evolution equations by Ganji and Sadighi (2007) and by Liang and Jeffrey (2009).…”

Section: Introductionmentioning

confidence: 99%

Convergent optimal variational iteration method and applications to heat and fluid flow problems

Türkyılmazoğlu

2016

International Journal of Numerical Methods for Heat &Amp; Fluid Flow

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Purpose – In an earlier paper (Turkyilmazoglu, 2011a), the author introduced a new optimal variational iteration method. The idea was to insert a parameter into the classical variational iteration formula in an aim to prevent divergence or to accelerate the slow convergence property of the classical approach. The purpose of this paper is to approve the superiority of the proposed method over the traditional one on several physical problems treated before by the classical variational iteration method. Design/methodology/approach – A sufficient condition theorem with an upper bound for the error is also presented to further justify the convergence of the new variational iteration method. Findings – The optimal variational iteration method is found to be useful for heat and fluid flow problems. Originality/value – The optimal variational iteration method is shown to be convergent under sufficient conditions. A novel approach to obtain the optimal convergence parameter is introduced.

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Section: Introductionmentioning

confidence: 99%

Convergent optimal variational iteration method and applications to heat and fluid flow problems

Türkyılmazoğlu

2016

International Journal of Numerical Methods for Heat &Amp; Fluid Flow

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“…This method, which was first proposed by He [7,8] in 1998, has been proved by many authors to be a powerful mathematical tool for various kinds of non-linear problems [9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

An inverse problem of identifying the coefficient of semilinear parabolic equation

Parzilvand¹,

Shahrezaee²

2016

Mathematical Researches

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In this paper, a variational iteration method (VIM), which is a well-known method for solving nonlinear equations, has been employed to solve an inverse parabolic partial differential equation. Inverse problems in partial differential equations can be used to model many real problems in engineering and other physical sciences. The VIM is to construct correction functional using general Lagrange multipliers identified optimally via the variational theory.This method provides a sequence of function which converges to the exact solution of the problem. This technique does not require any discretization, linearization or small perturbations and therefore reduces the numerical computations a lot. Numerical examples are examined to show the efficiency of the technique.

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“…So far, many types of analytical and numerical solutions have been proposed for many problems arising in the fields covering chemistry, biology, physics and branches of all engineering fields [17][18][19][20][21][22][23][24][25][26][27][28]. In many of those methods particularly numerical methods, various techniques were setup by using various basis functions like B-spline functions of different degrees.…”

Section: Introductionmentioning

confidence: 99%

Motion of Patterns Modeled by the Gray-Scott Autocatalysis System in One Dimension

Korkmaz¹,

Ersoy²,

Dağ³

2016

Preprint

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Occupation of an interval by self-replicating initial pulses is studied numerically. Two different approximates in different categories are proposed for the numerical solutions of some initial-boundary value problems. The sinc differential quadrature combined with third-fourth order implicit Rosenbrock and exponential B-spline collocation methods are setup to obtain the numerical solutions of the mentioned problems. The numerical simulations containing occupation of single initial pulse, non or slow occupation model and covering the domain with two initial pulses are demonstrated by using both proposed methods.

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Variational iteration method for elastodynamic Green’s functions

Cited by 7 publications

References 17 publications

Convergent optimal variational iteration method and applications to heat and fluid flow problems

Convergent optimal variational iteration method and applications to heat and fluid flow problems

An inverse problem of identifying the coefficient of semilinear parabolic equation

Motion of Patterns Modeled by the Gray-Scott Autocatalysis System in One Dimension

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