A Swarm-Intelligence Based Formulation for Solving Nonlinear ODEs: γβII-(2+3)P method

Abstract

The interconnectivity relationships between the weights of an important integration formula are discovered by swarm intelligence. These relationships make it possible to generate wide spectrum of ODEs-solving techniques in a continuous weighting space. So, against the conventional methods, new formulations would have variable weights which are tuned up by the weighting rules. Moreover, in an innovative attempt, Hermite interpolation is coupled with the integration formulas to rise up the accuracy and flexibility of the formulation. Hermite interpolation precisely evaluates the internal points of the step. This paper, in brief, develops the new idea of weighting rules in formulation of time-integrators. Four class of weighting rules are proposed: 1) Symmetry weighting rule (SWR) 2) Consistency weighting rule (CWR), 3) Fundamental weighting rule (FWR), and 4) Auxiliary weighting rule (FWR). The two first rules are adopted from common characteristics of available quadratures. The third is the most important one which is disclosed here for the first time. It formulates the correlation between the weights of the optimal integrators. Vibration equation under seismic loading is introduced here as the benchmark problem. Finding the optimal formulas to precisely solve this challenging problem guide us to the FWR formula. Notably, there are wide variants of FWRs which would be presented in near future. This research pioneers in application of swarm intelligence formulation of the FWRs for a given pattern of integration. Evolutionary grey wolf optimization (GWO) accompanied with statistics processes extracts the FWR. Finally, all the weighting rules and Hermite interpolation formulas are integrated in a unified algorithm so-called γβII−e+iP. It can effectively cope with various types of nonlinear ODEs.