一类指数矩阵函数及其应用  

AN EXPONENTIAL MATRIX FUNCTION AND ITS APPLICATION

作　　者：富明慧[1] 林敬华[1] 

机构地区：[1]中山大学应用力学与工程系,广东省近岸海洋工程重点实验室,广州510275

出　　处：《力学学报》2009年第5期808-814,共7页Chinese Journal of Theoretical and Applied Mechanics

基　　金：国家自然科学基金(10672194);广东省自然科学基金(031552)资助项目~~

摘　　要：研究了一阶常微分方程组特解的精细积分方法.针对非齐次项为多项式、指数函数以及二者的乘积的情况,在Duhamel积分形式特解的基础上,引入了一类指数矩阵函数.通过该类函数的线性组合即可表达出非齐次方程的特解.建立了该类指数矩阵函数的一种高效递推算法,并在此基础上实现了特解的精细积分.由于特解的积分过程能充分利用通解精细积分过程的中间量,因此两个精细积分过程能有机地结合起来,形成了一种高效、统一的广义精细积分法.对上述递推算法做了进一步优化,并给出了通用的计算公式.算例结果证明了该方法的有效性.A new exponential matrix function is introduced to obtain the special solutions, based on Duhamel convolution integration for the non-homogenous polynomial terms of the first order constant coefficient ordinary equation, exponential functions or their products., Based on the new exponential matrix, an efficient recursive algorithm is proposed in this paper with a precise integration method （PTM） applied to the special solution. The processes of PTM for general solution and the proposed method for special solution are closely integrated. This combination is confirmed to have a high computational efficiency.. The proposed method is also validated to have more convenient and more applied range. A general formula is given to greatly save computation time. Numerical examples are also demonstrated the validity and efficiency of the method.

关 键 词：一阶常微分方程组 特解 精细积分法 指数矩阵运算 递推算法 

分 类 号：O321[理学—一般力学与力学基础]