Abstract

In the modern era, increasing numbers of cores per chip are applied for decentralized systems, but there is not any appropriate symbolic computation approach to construct multicore analytic approximation. Thus, it is essential to develop an efficient, simple and unified way for decentralized Adomian decomposition method to increase the potential speed of the multicore systems. In our paper, we present an innovative parallel algorithm of constructing analytic solutions for nonlinear differential system, which based on the Adomian–Rach double decomposition method and Rach’s Adomian polynomials. Based on our algorithm, we further developed a user-friendly Python software package to construct analytic approximations of initial or boundary value problems. Finally, the scope of validity of our Python software package is illustrated by several different types of nonlinear examples. The obtained results demonstrate the effectiveness of our package by compared with exact solution and numeric method, the characteristics of each class of Adomian polynomials and the efficiency of parallel algorithm with multicore processors. We emphasis that the super-linear speedup may happens for the duration of constructing approximate solutions. So, it can be considered as a promising alternative algorithm of decentralized Adomian decomposition method for solving nonlinear problems in science and engineering.

Introduction

A large number of enigmas in engineering, biology, economics and other disciplines, e.g. flow and heat transfer problem, the simulations of the immune system, control and optimization theory, bound price problem, are often modeled using a system of nonlinear problems [1–۵]. In particular, various kinds of decentralized systems, appeared in economics, medicine etc., are convenient and cost effective [6,7]. A wide range of analytic methods, like the Adomian decomposition method (ADM) [1,3], the perturbationincremental method [8], the variational iteration method [9], the homotopy perturbation method [10], etc., is a reliable and efficient technique to handle such problems. It should be mentioned that the ADM is among the most simple and effective analytic methods to construct approximations of nonlinear differential equations, and have been modified and improved by Adomian and his collaborator, like the Adomian–Rach double decomposition [1,11], etc. These modifications, in most cases, undoubtedly have provided higher accuracy and faster convergence in nonlinear differential equations [1,12]. Furthermore, the ADM, which have been proved that it works efficiently for a large number of nonlinear problems including fractional differential equations [13], even stochastic system [1], is easy to be implemented by various programming languages, such as Maple [13], Mathematics [14], etc.