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

Yingying Xie,Mancang Wang,Lingfei Li

doi:10.3390/math9151796

Yingying Xie, Mancang Wang + Show 1 more

Open Access

https://doi.org/10.3390/math9151796

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Journal: Mathematics	Publication Date: Jul 29, 2021
Citations: 3	License type: CC BY 4.0

Affiliation: Xi'an Jiaotong University, Northwest University

Abstract

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.

Highlights

The Adomian decomposition method (ADM) was first introduced by the American mathematician Adomian and has been widely used for a class of deterministic and stochastic problems in scientific research fields [1,2,3,4,5,6]
The modification of ADM itself has acquired a lot of remarkable results and it can be flexibly applied to kinds of complex higher order equations, even partial differential equations
This paper firstly employs Laguerre polynomials and the second kind of Chebyshev polynomials to modify the ADM, that is, at the beginning of implementation of ADM, Laguerre and the second kind of Chebyshev orthogonal polynomials are used to expand the right hand terms which fail to integrate with parameters

Summary

Introduction

The Adomian decomposition method (ADM) was first introduced by the American mathematician Adomian and has been widely used for a class of deterministic and stochastic problems in scientific research fields [1,2,3,4,5,6] It is based on looking for a solution in view of a series u( x, t) = ∑∞. This paper firstly employs Laguerre polynomials and the second kind of Chebyshev polynomials to modify the ADM, that is, at the beginning of implementation of ADM, Laguerre and the second kind of Chebyshev orthogonal polynomials are used to expand the right hand terms which fail to integrate with parameters. The modified ADM presented in this paper is compared to ones with the Taylor expansion, the first kind of Chebyshev polynomials and Legendre polynomials. The Taylor expansion at zero makes the error further and further away from the origin

Modification of Adomian Decomposition Method

Test Problem

Concluding Remarks