An efficient algorithm for the multivariable Adomian polynomials

In this article the sum of the series of multivariable Adomian polynomials is demonstrated to be identical to a rearrangement of the multivariable Taylor expansion of an analytic function of the decomposition series of solutions u 1, u 2, … , u m about the initial solution components u 1,0, u 2,0, …...

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Bibliographic Details
Published in:Applied mathematics and computation Vol. 217; no. 6; pp. 2456 - 2467
Main Author: Duan, Jun-Sheng
Format: Journal Article
Language:English
Published: Amsterdam Elsevier Inc 15.11.2010
Elsevier
Subjects:
Acceleration of convergence
Adomian decomposition method
Adomian polynomials
Algorithms
Derivatives
Difference and functional equations, recurrence relations
Differential equation
Differential equations
Exact sciences and technology
Functions of a complex variable
Mathematical analysis
Mathematical models
Mathematics
Matrices
Matrix methods
Multivariable
Multivariable function
Nonlinear operator
Numerical analysis
Numerical analysis. Scientific computation
Ordinary differential equations
Sciences and techniques of general use
Multivariable function
Adomian polynomials
Adomian decomposition method
Nonlinear operator
Differential equation
Polynomial
Transcendental equation
Convergence acceleration
Recurrence relation
Decomposition method
Difference equation
Parameterization
Algorithm
Non linear operator
Non linear equation
Series summation
Numerical analysis
Applied mathematics
Analytical function
ISSN:0096-3003, 1873-5649
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Summary:In this article the sum of the series of multivariable Adomian polynomials is demonstrated to be identical to a rearrangement of the multivariable Taylor expansion of an analytic function of the decomposition series of solutions u 1, u 2, … , u m about the initial solution components u 1,0, u 2,0, … , u m,0 ; of course the multivariable Adomian polynomials were developed and are eminently practical for the solution of coupled nonlinear differential equations. The index matrices and their simplified forms of the multivariable Adomian polynomials are introduced. We obtain the recurrence relations for the simplified index matrices, which provide a convenient algorithm for rapid generation of the multivariable Adomian polynomials. Another alternative algorithm for term recurrence is established. In these algorithms recurrence processes do not require complicated operations such as parametrization, expanding and regrouping, derivatives, etc. as practiced in prior art. The MATHEMATICA program generating the Adomian polynomials based on the algorithm in this article is designed.
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ISSN:0096-3003
1873-5649
DOI:10.1016/j.amc.2010.07.046