Adomian decomposition method is a special case of Lyapunov’s artificial small parameter method

Zhang, Xiaolong; Liang, Shuxiu

doi:10.1016/j.aml.2015.04.011

Cited by 11 publications

(5 citation statements)

References 10 publications

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“…Despite their success, these methods are constrained by the necessity of small physical parameters and limited flexibility, making them primarily applicable to weakly nonlinear problems. Non--perturbation methods, like Lyapunov's Artificial Small Parameter Method and the Adomian Decomposition Method [9], offer alternatives but come with limitations, such as the lack of freedom to choose nonlinear operators and uncertainty regarding convergence. Consequently, both perturbation and traditional non-perturbation methods are typically valid for weakly nonlinear problems.…”

Section: Introductionmentioning

confidence: 99%

Homotopy analysis of explicit series solutions in a modified Nosé-Hoover oscillator

Deng,

Duarte,

Januário

et al. 2024

J. Appl. Math. Comput. Mech.

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Section: Introductionmentioning

confidence: 99%

Homotopy analysis of explicit series solutions in a modified Nosé-Hoover oscillator

Deng,

Duarte,

Januário

et al. 2024

J. Appl. Math. Comput. Mech.

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“…A variety of iterative techniques such as the homotopy perturbation method (HPM) [7][8], the Adomian decomposition method (ADM) [9][10], and the variational iteration method (VIM) [11][12][13] have been suggested to solve PDEs. By using terms from an infinite series, these methods generate either an approximate or an exact solution that quickly converges to an accurate solution.…”

Section: Introductionmentioning

confidence: 99%

Mixed Transform Iterative Method for Solving Wave Like Equations

S.Ismael,

H. Al-Fayadh

2023

ANJS

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In this article, a hybrid method that combines Aboodh transform, variational iteration method, and the homotopyperturbation method is presented for approximate the solution of important partial differential equations that describe wave like differential equations in one, two and three dimensions. The suggested method utilizes only the initial conditions to providean analytical solution, in contrast to the method of separation of variables, that also requires boundary conditions. This method makes use of the Aboodh's transform advantageous on the Lagrange multiplier computation, hence there is no need to use the convolution theorem or any integration in a recurrence relation for the process of finding it. The obtained exact solutions are found as a convergent series with components that are simple to compute. In addition, the method needsno any assumptions that change the problem's physical nature, such as those that involve discretization, linearization, or minor factors compared to certain other techniques. Some examples are given to show how efficient and useful the method is.

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“…It has been successfully applied to solve many problems in physical sciences such as the nonlinear Klein-Gordon equation [5], the Lane-Emden problem [6], the hydromagnetic peristaltic flow [7], the nonlinear fin problem [8] and the plate flow [9]. To make the method more effective, many researchers have done lots of improvements, for example, giving the noise terms [10], comparing with other methods [11,12], discussing the convergence rate [13], analysing the error [14], modifying the recursive scheme [15,16]and indicating the essence [17]. Recently, Duan and his colleagues introduced a convergence parameter into the ADM firstly [18,19,20].…”

Section: Introductionmentioning

confidence: 99%

A novel analytic approximation method with a convergence acceleration parameter for solving nonlinear problems

Zhang

Zou

Liang

et al. 2018

Communications in Nonlinear Science and Numerical Simulation

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In this paper, a new analytic method with a convergencecontrol parameter c is first proposed. The parameter c is used to adjust and control the convergence region and rate of the resulting series solution. It turns out that the convergence region and rate can be greatly enlarged by choosing a proper value of c. Furthermore, a numerical approach for finding the optimal value of the convergence-control parameter is given. At the same time, it is found that the traditional Adomian decomposition method is only a special case of the new method. The effectiveness and applicability of the new technique are demonstrated by several physical models including nonlinear heat transfer problems, nano-electromechanical systems, diffusion and dissipation phenomena, and dispersive waves. Moreover, the ideas proposed in this paper may offer us possibilities to greatly improve current analytic and numerical techniques.Mathematics Subject Classification (2010). 34E10; 35B20; 76M45.

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Adomian decomposition method is a special case of Lyapunov’s artificial small parameter method

Cited by 11 publications

References 10 publications

Homotopy analysis of explicit series solutions in a modified Nosé-Hoover oscillator

Homotopy analysis of explicit series solutions in a modified Nosé-Hoover oscillator

Mixed Transform Iterative Method for Solving Wave Like Equations

A novel analytic approximation method with a convergence acceleration parameter for solving nonlinear problems

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