非线性动力系统精细积分下的显式级数解

Explicit Series Solution Under Precise Integration for Nonlinear Dynamics System

基于钟万勰等提出的指数矩阵精细算法 ,对n维未知向量v的一阶微分方程 v=Hv +f(v ,t)进行求解 ,其中Hv和f(v ,t)分别是右端项的线性齐次部分和非线性部分。将非线性部分在所论时刻tk处展成t-tk=τ的Taylor级数形式 ,并通过指数矩阵eHt 及其精细算法对状态方程直接积分 ,推导出状态方程的级数形式闭合解 ,此解的精度易于控制。算法不需对矩阵 [H]求逆 ,数值计算的稳定性及效率均可确保 ,对大型问题计算更为有利。

Based on the precise integration method of the exponential matrix developed by Zong Wanxie, we discuss a general dynamics system governed by the equation =Hv+f(v,t), in which v is an unknown n dimensional vector,H is a coefficient matrix,Hv and f(v,t) are the linear homogeneous part and nonlinear part in the right members of the equation respectively.The nonlinear part f(v,t) can be expressed with τ=t-t k in taylor series at the time t k.The authors suggest that the integral of the state equation is evaluated directly through exponential matrix and its precise algorithm, thus a closed series solution can be successively obtained and the precision of the solution can be controlled easily.This algorithm avoids calculating the inversion of the  matrix, meanwhile the stabilization and efficiency of the computation can be ensured, so this method is especially benefit to the large-scale question.The numerical example is presented to demonstrate the effectiveness of this algorithm.