An aftertreatment technique for improving the accuracy of Adomian's decomposition method

Jiao, Yong‐Chang; Yamamoto, Y.; Dang, Chuangyin; Hao, Yue

doi:10.1016/s0898-1221(01)00321-2

Cited by 102 publications

(61 citation statements)

References 8 publications

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“…Most of these equations have no analytical solution and so, numerical methods may be required to obtain their approximate solutions [9][10][11][12][13]. For this reason, the Laguerre collocation method has been presented.…”

Section: Resultsmentioning

confidence: 99%

Laguerre Polynomial Solutions of a Class of Initial and Boundary Value Problems Arising in Science and Engineering Fields

Gürbüz

Sezer

2016

Acta Phys. Pol. A

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In this study, we consider high-order nonlinear ordinary differential equations with the initial and boundary conditions. These kinds of differential equations are essential tools for modelling problems in physics, biology, neurology, engineering, ecology, economy, astrophysics, physiology and so forth. Each of the mentioned problems are described by one of the following equations with the specific physical conditions: Riccati, Duffing, EmdenFowler, Lane Emden type equations. We seek the approximate solution of these special differential equations by means of a operational matrix technique, called the Laguerre collocation method. The proposed method is based on the Laguerre series expansion and the collocation points. By using the method, the mentioned special differential equations together with conditions are transformed into a matrix form which corresponds to a system of nonlinear algebraic equations with unknown Laguerre coefficients, and thereby the problem is approximately solved in terms of Laguerre polynomials. In addition, some numerical examples are presented to demonstrate the efficiency of the proposed method and the obtained results are compared with the existing results in literature.

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Section: Resultsmentioning

confidence: 99%

Laguerre Polynomial Solutions of a Class of Initial and Boundary Value Problems Arising in Science and Engineering Fields

Gürbüz

Sezer

2016

Acta Phys. Pol. A

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“…Generally ADM yields the Taylor series of the true solution. Jiao et al [14] has observed that although the series can be rapidly convergent in a very small region, it has very slow convergence rate in the wider region and the truncated series solution is an inaccurate solution in that region, which will greatly restrict the application area of the method. In order to improve the convergence rate and accuracy of ADM, we have applied the following techniques.…”

Section: Techniques For Improving Accuracymentioning

confidence: 99%

Solution of a class of singular boundary value problems

Mittal

Nigam

2008

Numer Algor

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In this work a class of singular ordinary differential equations is considered. These problems arise from many engineering and physics applications such as electro-hydrodynamics and some thermal explosions. Adomian decomposition method is applied to solve these singular boundary value problems. The approximate solution is calculated in the form of series with easily computable components. The method is tested for its efficiency by considering four examples and results are compared with previous known results. Techniques that can be applied to obtain higher accuracy of the present method has also been discussed.

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“…The situation can be improved if one is able to perform a (partial) summation of the series. In the literature [18] Padé approximation has been used with success to improve the convergence of the series. We are not going to follow this path here, since we are interested in the short-time behavior of the solutions.…”

Section: Explicit Solutionsmentioning

confidence: 99%

Studying nonlinear effects on the early stage of phase ordering using a decomposition method

et al. 2005

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Nonlinear effects on the early stage of phase ordering are studied using Adomian's decomposition method for the Ginzburg-Landau equation for a nonconserved order parameter. While the long-time regime and the linear behavior at short times of the theory are well understood, the onset of nonlinearities at short times and the breaking of the linear theory at different length scales are less understood. In the Adomians decomposition method, the solution is systematically calculated in the form of a polynomial expansion for the order parameter, with a time dependence given as a series expansion. The method is very accurate for short times, which allows to incorporate the short-time dynamics of the nonlinear terms in a analytical and controllable way.

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An aftertreatment technique for improving the accuracy of Adomian's decomposition method

Cited by 102 publications

References 8 publications

Laguerre Polynomial Solutions of a Class of Initial and Boundary Value Problems Arising in Science and Engineering Fields

Laguerre Polynomial Solutions of a Class of Initial and Boundary Value Problems Arising in Science and Engineering Fields

Solution of a class of singular boundary value problems

Studying nonlinear effects on the early stage of phase ordering using a decomposition method

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