A developed new algorithm for evaluating Adomian polynomials

TL;DR

This paper introduces a new algorithm for evaluating Adomian polynomials by expressing them through reduced polynomials, providing detailed properties, and implementing a simple Mathematica program for their derivation.

Contribution

The paper develops a novel, general algorithm for calculating Adomian polynomials using reduced polynomials, enhancing computational efficiency and analytical understanding.

Findings

Derived properties of reduced polynomials and Adomian polynomials

Constructed general analytical expressions for small subscript differences

Implemented a simple Mathematica program for polynomial derivation

Abstract

Adomian polynomials (AP's) are expressed in terms of new objects called reduced polynomials (RP's). These new objects, which carry two subscripts, are independent of the form of the nonlinear operator. Apart from the well-known two properties of AP's, curiously enough no further properties are discussed in the literature. We derive and discuss in full detail the properties of the RP's and AP's. We focus on the case where the nonlinear operator depends on one variable and construct the most general analytical expressions of the RP's for small values of the difference of their subscripts. It is shown that each RP depends on a number of functions equal to the difference of its subscripts plus one. These new properties lead to implement a dramatically simple and compact Mathematica program for the derivation of individual RP's and AP's in their general forms and provide useful hints for elegant hand calculations of AP's. Application of the program is considered.