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中文核心期刊

一类指数矩阵函数及其应用

An exponential matrix function and its application

  • 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.

     

    Abstract: A new exponential matrix function is introduced to obtainthe special solutions, based on Duhamel convolution integration for thenon-homogenous polynomial terms of the first order constant coefficientordinary equation, exponential functions or their products., Based on thenew exponential matrix, an efficient recursive algorithm is proposed in thispaper with a precise integration method (PTM) applied to the specialsolution. The processes of PTM for general solution and the proposed methodfor special solution are closely integrated. This combination is confirmedto have a high computational efficiency.. The proposed method is alsovalidated to have more convenient and more applied range. A general formulais given to greatly save computation time. Numerical examples are alsodemonstrated the validity and efficiency of the method.

     

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