The Use of Variational Iteration Method, Differential Transform Method and Adomian Decomposition Method for Solving Different Types of Nonlinear Partial Differential Equations

Bıldık, Necdet; Konuralp, Ali

doi:10.1515/ijnsns.2006.7.1.65

Cited by 238 publications

(109 citation statements)

References 13 publications

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“…In addition, it is not guaranteed that a perturbation result is valid in the whole region of all physical parameters. In order to overcome the restrictions of perturbation techniques, some non perturbation methods are developed, such as Lyapunov's artificial small parameter method (Liao, 2003;Abbasbandy, 2006) the Adomian decomposition method (Wazwaz, 2005Bildik & Konuralp, 2006 and so on. In general, all these methods are based on a so-called artificial parameter, and the approximation of solutions are due to series with the artificial small parameter.…”

Section: Homotopy Analysis Methods (Ham)mentioning

confidence: 99%

Mathematical Modelling of Spray Combustion-Numerical and Analytical Analysis with application to Engineering Science

Nave¹,

Ajadi²,

Gol’dshtein³

2016

APR

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In the present paper we applied two well-known analytical method to the problem of thermal explosion of monodisperse and polydisperse fuel spray. The methods are the method of integral manifold (MIM) and the homotopy analysis method (HAM). The MIM method used as a basic tools for the analysis of SPS system of ordinary differential equations which means that the physical/ mathematical model should contain a small parameter in the governing equations. The HAM is always valid no matter whether there exist small physical parameters or not in contrast to the classical perturbation methods which requires the existence of a small parameter in the system (in general this is not the case). According to the theory of HAM, the convergence and the rate of solution series are dependent on the convergent control parameter ℏ. This means that this parameter gives one a convenient way to adjust and to control the convergent region of the solutions. γ (conventional parameter of Semenov's theory of thermal explosion), the final dimensionless adiabatic temperature of the thermally insulated system after explosion. This parameter is small compared with unity for most gaseous mixture due to the high exothermicity and activation energy of the chemical reactionwhere ϵ d is the emissivity of the droplet surface ϵ 1,2 dimensionless parameters, introduced for the first time (Gol'dshtein, Goldfarb, Shreiber, & Zinoviev, 1996) and describe the relations between the thermo physical properties of the gas and liquid phases

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Section: Homotopy Analysis Methods (Ham)mentioning

confidence: 99%

Mathematical Modelling of Spray Combustion-Numerical and Analytical Analysis with application to Engineering Science

Nave¹,

Ajadi²,

Gol’dshtein³

2016

APR

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“…There are many effective methods to solve this problem, like Adomian decomposition method [8][9][10], variation iteration method [11], differential transform method [12], residual power series method [13,14], iteration method [15], homotopy perturbation method [16], homotopy analysis method [17], and so on. Furthermore, for the nonlinear problem, the multiple exp-function method [18,19], the transformed rational function method [20][21][22], and invariant subspace method [23,24] are three systematical approaches to handle the nonlinear terms.…”

Section: Discrete Dynamics In Nature and Societymentioning

confidence: 99%

Homotopy Series Solutions to Time-Space Fractional Coupled Systems

Zhang

Cai

Chen

et al. 2017

Discrete Dynamics in Nature and Society

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We apply the homotopy perturbation Sumudu transform method (HPSTM) to the time-space fractional coupled systems in the sense of Riemann-Liouville fractional integral and Caputo derivative. The HPSTM is a combination of Sumudu transform and homotopy perturbation method, which can be easily handled with nonlinear coupled system. We apply the method to the coupled Burgers system, the coupled KdV system, the generalized Hirota-Satsuma coupled KdV system, the coupled WBK system, and the coupled shallow water system. The simplicity and validity of the method can be shown by the applications and the numerical results.

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“…Perturbation method [6] provides the most versatile tools available in nonlinear analysis of engineering problem, but its limitations restrict its application [7] [8]: Perturbation method [9] is based on assuming a small parameter. The majority of nonlinear problems, especially those having strong nonlinearity, have no small parameters at all.…”

Section: Introductionmentioning

confidence: 99%

Implementation of Homotopy Analysis Method on circular permeable slider containing of incompressible Newtonian fluid

Rahimi

Rahimi-Esbo

Ganji

et al. 2016

bspm

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The aim of this paper is to examine the classical problem of an incompressible Newtonian fluid through the porous of a circular slider which is moving laterally on a horizontal plan. Employing the similarity variables, the governing differential equations have been reduced to ordinary ones and solved via Homotopy Analysis Method (HAM). The analytical solution for the coupled Nonlinear Ordinary Differential Equations resulting from the momentum equation is obtained and Velocity fields have been computed and discussed for different values of the Reynolds number. Also the fourth-order Runge-Kutta numerical method (NUM) is used for the validity of these analytical methods and excellent agreement are observed between the solutions obtained from HAM and numerical results.

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The Use of Variational Iteration Method, Differential Transform Method and Adomian Decomposition Method for Solving Different Types of Nonlinear Partial Differential Equations

Cited by 238 publications

References 13 publications

Mathematical Modelling of Spray Combustion-Numerical and Analytical Analysis with application to Engineering Science

Mathematical Modelling of Spray Combustion-Numerical and Analytical Analysis with application to Engineering Science

Homotopy Series Solutions to Time-Space Fractional Coupled Systems

Implementation of Homotopy Analysis Method on circular permeable slider containing of incompressible Newtonian fluid

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