Probabilistic interpretations of nonclassic Adomian polynomials

Abstract

The Adomian decomposition method (ADM) is a powerful tool for solving numerous nonlinear functional equations and a large class of initial/boundary value problems. The main task in the application of ADM is the computation of Adomian polynomials (APs). In addition to classic APs, Rach’s nonclassic APs, called class I-IV polynomials, are used to solve a wide range of nonlinear functional equations. In this paper, we present probabilistic interpretations for the nonclassic APs, including class V polynomials studied by Duan. We derive the recurrence relations for the computation of nonclassic APs using our approach. Some numerical examples are discussed to show that the probabilistic approach to compute the nonclassic APs is attractive and is also simple. Finally, a probabilistic proof is given for the known fact that the class IV APs are the classic APs. The probabilistic approach offers an alternative method to the existing analytical or combinatorial approach.

Keywords:

• Nonclassic Adomian polynomials
• Poisson distribution of order n
• probabilistic method
• recurrence relations

AMS CLASSIFICATION:

• 65L99
• 93E25
• 60E05

Acknowledgments

The authors are grateful to Dr. Rach for bringing out the nonclassic Adomian polynomials to the attention of the authors and his encouragements for exploring the probabilistic connections to them.