[Paper Review] A developed new algorithm for evaluating Adomian polynomials

Mustapha Azreg‐Aïnou|ArXiv.org|Jun 1, 2009

Fuzzy Systems and Optimization8 references19 citations

TL;DR

This paper introduces a novel algorithm for evaluating Adomian polynomials (APs) by defining reduced polynomials (RPs) that depend only on the difference of their subscripts, enabling a systematic, compact, and efficient computation. The method simplifies hand calculations and leads to a highly optimized Mathematica program, with explicit general forms for APs up to $ A_{10} $, significantly reducing computational complexity in the Adomian decomposition method.

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.

Motivation & Objective

• To address the computational complexity of evaluating Adomian polynomials (APs) in the Adomian decomposition method (ADM), which is a major bottleneck in solving nonlinear differential and integral equations.
• To introduce a new class of objects called reduced polynomials (RPs), which are independent of the nonlinear operator and depend only on the difference of their subscripts.
• To derive general analytical expressions for RPs for small differences in subscripts, revealing that each RP depends on $ m-k+1 $ functions, where $ m $ and $ k $ are the subscripts.
• To develop a simple, compact, and efficient Mathematica program for computing individual RPs and APs in their general forms.
• To provide explicit, previously unpublished analytical expressions for $ A_5 $ through $ A_{10} $, enhancing the utility of ADM in analytical and computational studies.

Proposed method

• The paper defines reduced polynomials (RPs), denoted $ Z_{m,k} $, which are independent of the nonlinear operator $ N $, and depend only on the difference $ m-k $.
• It establishes that $ Z_{m,k} $ depends on $ m-k+1 $ functions $ (u_1, u_2, ..., u_{m-k+1}) $ when $ k > 1 $, and $ Z_{m,1} $ depends only on $ u_m $.
• The Adomian polynomial $ A_m $ is decomposed as a sum over $ k $, where each term is the product of $ Z_{m,k} $ and the $ k $-th derivative of the nonlinear function $ F $, i.e., $ A_m = igoplus_{k=1}^m Z_{m,k} F^{(k)}(u_0) $.
• The authors derive general expressions for $ Z_{m,k} $ for small values of $ m-k $, enabling systematic construction of APs.
• A Mathematica program is implemented based on the derived properties, allowing automatic generation of RPs and APs in symbolic form.
• The method is validated by deriving and presenting the first explicit general forms of $ A_5 $ through $ A_{10} $, which are not available in the literature.

Experimental results

Research questions

• RQ1 How can the evaluation of Adomian polynomials be systematized and simplified to reduce computational burden in the Adomian decomposition method?
• RQ2 What intrinsic structural properties of reduced polynomials (RPs) govern their dependence on the components $ u_1, u_2, ..., u_m $?
• RQ3 Can a general analytical expression for RPs be derived for small differences in their subscripts, and what is the functional dependence on the $ u_i $ components?
• RQ4 To what extent can the new RP-based algorithm be implemented in a compact and efficient symbolic computation program?
• RQ5 What are the explicit general forms of Adomian polynomials $ A_5 $ through $ A_{10} $, and how do they compare with existing literature?

Key findings

• The reduced polynomial $ Z_{m,k} $ depends on exactly $ m-k+1 $ functions $ (u_1, ..., u_{m-k+1}) $ when $ k > 1 $, and $ Z_{m,1} $ depends only on $ u_m $, revealing a clear structural pattern.
• The general form of $ A_m $ is expressed as a sum of $ m $ terms, each being the product of $ Z_{m,k} $ and $ F^{(k)}(u_0) $, which separates the operator-independent and operator-dependent parts.
• The derived expressions for $ A_5 $ through $ A_{10} $ are the first complete, explicit, and general forms of these polynomials published in the literature, filling a critical gap.
• The algorithm enables a dramatically simplified and compact Mathematica program for computing RPs and APs, reducing code complexity and execution time.
• The method provides clear, systematic hints for elegant hand calculations of APs, significantly reducing the effort required for manual derivation.
• The approach is general and applicable to any nonlinear operator depending on a single variable, making it broadly useful across diverse applications in nonlinear science and engineering.