Reliability of Adomian decomposition method for high order nonlinear differential equations

2013 ◽  
Vol 7 ◽  
pp. 2735-2743
Author(s):  
Ayad R. Khudair
Keyword(s):  
Differential Equations ◽  
Decomposition Method ◽  
Nonlinear Differential Equations ◽  
Adomian Decomposition Method ◽  
High Order ◽  
Adomian Decomposition

Numerical solution of Lane-Emden type equations using Adomian decomposition method with unequal step-size partitions

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Umesh Umesh ◽  
Manoj Kumar
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Decomposition Method ◽  
Nonlinear Differential Equations ◽  
Adomian Decomposition Method ◽  
The Other ◽  
Step Size ◽  
Adomian Polynomials ◽  
Content Type ◽  
Adomian Decomposition

Purpose The purpose of this paper is to obtain the highly accurate numerical solution of Lane–Emden-type equations using modified Adomian decomposition method (MADM) for unequal step-size partitions. Design/methodology/approach First, the authors describe the standard Adomian decomposition scheme and the Adomian polynomials for solving nonlinear differential equations. After that, for the fast calculation of the Adomian polynomials, an algorithm is presented based on Duan’s corollary and Rach’s rule. Then, MADM is discussed for the unequal step-size partitions of the domain, to obtain the numerical solution of Lane–Emden-type equations. Moreover, convergence analysis and an error bound for the approximate solution are discussed. Findings The proposed method removes the singular behaviour of the problems and provides the high precision numerical solution in the large effective region of convergence in comparison to the other existing methods, as shown in the tested examples. Originality/value Unlike the other methods, the proposed method does not require linearization or perturbation to obtain an analytical and numerical solution of singular differential equations, and the obtained results are more physically realistic.

scholarly journals Treatment for third-order nonlinear differential equations based on the Adomian decomposition method

2017 ◽  
Vol 20 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Xueqin Lv ◽  
Jianfang Gao
Keyword(s):  
Differential Equations ◽  
Decomposition Method ◽  
Nonlinear Differential Equations ◽  
Adomian Decomposition Method ◽  
Differential Algebraic Equations ◽  
Nonlinear Ordinary Differential Equations ◽  
Algebraic Equations ◽  
Third Order ◽  
Adomian Decomposition ◽  
Linear And Nonlinear

The Adomian decomposition method (ADM) is an efficient method for solving linear and nonlinear ordinary differential equations, differential algebraic equations, partial differential equations, stochastic differential equations, and integral equations. Based on the ADM, a new analytical and numerical treatment is introduced in this research for third-order boundary-value problems. The effectiveness of the proposed approach is verified by numerical examples.

scholarly journals Natural Decomposition Approximation Solution for First Order Nonlinear Differential Equations

2018 ◽  
Vol 7 (4.5) ◽  
pp. 442
Author(s):  
A. Patra ◽  
T. T. Shone ◽  
B. B. Mishra
Keyword(s):  
Differential Equations ◽  
Decomposition Method ◽  
Nonlinear Differential Equations ◽  
Adomian Decomposition Method ◽  
Approximation Solution ◽  
Transform Method ◽  
First Order ◽  
Adomian Decomposition ◽  
Natural Decomposition ◽  
First Order Differential Equations

In this research paper, we propose the Natural decomposition method (NDM) to solve nonlinear first order differential equations. We compare the results obtained by NDM with the exact solutions. This method is a combination of the natural transform method and adomian decomposition method. By showing the less error one can be concluded that the NDM is easy to use and efficient.  

scholarly journals Adomian Decomposition Method with Orthogonal Polynomials: Laguerre Polynomials and the Second Kind of Chebyshev Polynomials

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1796
Author(s):  
Yingying Xie ◽  
Lingfei Li ◽  
Mancang Wang
Keyword(s):  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Chebyshev Polynomials ◽  
Decomposition Method ◽  
Nonlinear Differential Equations ◽  
Laguerre Polynomials ◽  
Adomian Decomposition Method ◽  
Numerical Results ◽  
Adomian Decomposition ◽  
Linear And Nonlinear

In this paper, a new efficient and practical modification of the Adomian decomposition method is proposed with Laguerre polynomials and the second kind of Chebyshev polynomials which has not been introduced in other articles to the best of our knowledge. This approach can be utilized to approximately solve linear and nonlinear differential equations. The proposed formulations are examined by a representative example and the numerical results confirm their efficiency and accuracy.

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