Simple parametrization methods for generating Adomian polynomials

Kataria, K. K.; Vellaisamy, P.

doi:10.2298/aadm160123001k

Cited by 17 publications

(11 citation statements)

References 16 publications

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“…Recently, Kataria and Vellaisamy proposed a novel method for calculating the Adomian polynomials (21),…”

Section: Methods: the Development Of Multistep Modified Reduced Differential Transform Methodsmentioning

confidence: 99%

“…In this paper, the modification by implementing Adomian polynomials and the multistep approach are combined to execute the MMRDTM for solving NLSEs of power law nonlinearity. In addition, we apply parametrization methods for generating Adomian polynomials in which the algorithm doesn't require tedious calculations of high derivatives (21). The proposed technique has the ability to rapidly yielding a fastconvergent sequence of analytical approximation.…”

Section: Introductionmentioning

confidence: 99%

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Approximate Analytical Solutions of Bright Optical Soliton for Nonlinear Schrödinger Equation of Power Law Nonlinearity

et al. 2021

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This paper introduces the Multistep Modified Reduced Differential Transform Method (MMRDTM).It is applied to approximate the solution for Nonlinear Schrodinger Equations (NLSEs) of power law nonlinearity. The proposed method has some advantages. An analytical approximation can be generated in a fast converging series by applying the proposed approach. On top of that, the number of computed terms is also significantly reduced. Compared to the RDTM, the nonlinear term in this method is replaced by related Adomian polynomials prior to the implementation of a multistep approach. As a consequence, only a smaller number of NLSE computed terms are required in the attained approximation. Moreover, the approximation also converges rapidly over a wide time frame. Two examples are provided for showing the ability and advantages of the proposed method to approximate the solution of the power law nonlinearity of NLSEs. For pictorial representation, graphical inputs are included to represent the solution and show the precision as well as the validity of the MMRDTM.

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“…Recently, Kataria and Vellaisamy proposed a novel method for calculating the Adomian polynomials (21),…”

Section: Methods: the Development Of Multistep Modified Reduced Differential Transform Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Approximate Analytical Solutions of Bright Optical Soliton for Nonlinear Schrödinger Equation of Power Law Nonlinearity

et al. 2021

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show abstract

“…Recently, Kataria and Vellaisamy proposed a novel method for calculating the Adomian polynomials [30], It can be observed that the algorithm does not involve tedious calculations with high derivatives. By combining equations ( 5) and ( 6 Finally, MMRDTM proposes the following solutions…”

Section: The Development Of Multistep Modified Reduced Differential Transform Methodsmentioning

confidence: 99%

“…The modification by implementing Adomian polynomials and the multistep method are combined in this paper to perform the MMRDTM for solving KdV equations with compact support in application of pattern formation liquid drop. Furthermore, we use parametrization methods to generate Adomian polynomials without the need for time-consuming high-derivative calculations [30]. The proposed technique will quickly produce a fast-convergent sequence of analytical approximations.…”

Section: Introductionmentioning

confidence: 99%

Solitary Wave Solutions with Compact Support for The Nonlinear Dispersive K(m,n) Equations by Using Approximate Analytical Method

Hussin

Azmi

Kılıçman

2021

ARFMTS

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The study of solitons and compactons is important in nonlinear physics. In this paper we combined the Adomian polynomials with the multi-step approach to present a new technique called Multi-step Modified Reduced Differential Transform Method (MMRDTM). The proposed technique has the advantage of producing an analytical approximation in a fast converging sequence with a reduced number of calculated terms. The MMRDTM is presented with some modification of the Reduced Differential Transformation Method (RDTM) with multi-step approach and its nonlinear term is replaced by the Adomian polynomials. Therefore, the nonlinear initial value problem can easily be solved with less computational effort. Besides that, the multi-step approach produces a solution in fast converging series that converges the solution in a wide time area. Two examples are provided to demonstrate the capability and benefits of the proposed method for approximating the solution of NKdVEs with compactons. Graphical inputs are used to represent the solution and to demonstrate the precision and validity of the MMRDTM in graphic illustration. From the results, it was found that it is possible to obtain highly accurate results or exact solutions by using the MMRDTM.

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“…The van der Pol equation, otherwise called the van der Pol oscillator is a model which describes the behaviour of electrical circuits [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. It was formulated by an electrical engineer and physicist, Balthasar van der Pol, and no exact solution has been obtained for the second order differential equation since then [12].…”

Section: Introductionmentioning

confidence: 99%

Comparison of the Solution of the Van der Pol Equation Using the Modified Adomian Decomposition Method and Truncated Taylor Series Method

Ndam

Adedire

2020

J. Nig. Soc. Phys. Sci.

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In this paper, we compare the solution of the van der Pol equation obtained by using the truncated Taylor series method and the modified Adomian decomposition method with the solution obtained by the Poincare-Lindstedt (P-L) method. The approximating 4-component modified Adomian decomposition method behaves more like an approximate P-L analytic method than the tenth-order Taylor series. Also, with the addition of just one term, the approximating 5-component modified Adomian decomposition method produces a more convergent solution to that of P-L analytic method than the twenty second-order Taylor series approximation as the independent variable t representing time progressively increases. A general comparison of the two solutions revealed that the absolute errors generated by the approximating polynomial from the Taylor series are greater than the ones generated from the modified Adomian decomposition method. It was further revealed that very few components of the modified Adomian decomposition could yield a series of about 3 times the order of the one obtained by using the Taylor series method. Hence, we recommend the inclusion of the modified Adomian Decomposition Method in modern mathematical tools.

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Simple parametrization methods for generating Adomian polynomials

Cited by 17 publications

References 16 publications

Approximate Analytical Solutions of Bright Optical Soliton for Nonlinear Schrödinger Equation of Power Law Nonlinearity

Approximate Analytical Solutions of Bright Optical Soliton for Nonlinear Schrödinger Equation of Power Law Nonlinearity

Solitary Wave Solutions with Compact Support for The Nonlinear Dispersive K(m,n) Equations by Using Approximate Analytical Method

Comparison of the Solution of the Van der Pol Equation Using the Modified Adomian Decomposition Method and Truncated Taylor Series Method

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