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 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.

scholarly journals Adomian Decomposition Method for a Class of Nonlinear Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
J. A. Sánchez Cano
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Nonlinear Problems ◽  
Coupled System ◽  
Nonlinear Ordinary Differential Equations ◽  
Adomian Decomposition

The Adomian decomposition method together with some properties of nested integrals is used to provide a solution to a class of nonlinear ordinary differential equations and a coupled system.

scholarly journals Application of Laplace–Adomian Decomposition Method for the Analytical Solution of Third-Order Dispersive Fractional Partial Differential Equations

Entropy ◽  
2019 ◽  
Vol 21 (4) ◽  
pp. 335 ◽  
Author(s):  
Rasool Shah ◽  
Hassan Khan ◽  
Muhammad Arif ◽  
Poom Kumam
Keyword(s):  
Partial Differential Equations ◽  
Analytical Solution ◽  
Differential Equations ◽  
Fractional Order ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Third Order ◽  
Partial Differential ◽  
Adomian Decomposition ◽  
In Series

In the present article, we related the analytical solution of the fractional-order dispersive partial differential equations, using the Laplace–Adomian decomposition method. The Caputo operator is used to define the derivative of fractional-order. Laplace–Adomian decomposition method solutions for both fractional and integer orders are obtained in series form, showing higher convergence of the proposed method. Illustrative examples are considered to confirm the validity of the present method. The fractional order solutions that are convergent to integer order solutions are also investigated.

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.

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