两个相同部件并联可修系统解研究

Study of solutions of two parallel repairable systems with same components

中文关键词: 半离散化  逼近  C0半群  算子

英文关键词: semi-discretization  approximation  semi-group of class C0  arithmetic

中文摘要:

      研究了两个相同部件并联可修系统解的问题．利用半离散化逼近方法将抛物型偏微分方程组化为矩阵常微分方程组，即用初等阶梯函数对并联可修系统的修复率μ(x)进行逼近，使该系统转化为半离散化系统．并对该系统的动态解用C0半群理论中的Trotter定理加以证明，得到该解的收敛性．最后假设该并联系统的修复率为常数，利用Matlab软件进行数值实验，从实验图形中发现该可修系统的数值解和理论证明的结论是一致的．结果表明，离散后的常微分方程组的解收敛于原抛物型偏微分方程组的解，从而为该模型的进一步数值计算打下了基础．

英文摘要:

      The problem of solving two parallel repairable systems with the same components is studied. Parabolic partial differential equations are transformed into matrix ordinary differential equations by using semi-discrete approximation method, that is, the repairability rate μ(x)of parallel repairable systems is approximated by elementary step functions. The system is transformed into a semi-discrete system. The dynamic solution of the system is proved by the Trotter theorem of semi-group theory C0, and the convergence of the solution is obtained. Finally, assuming that the repairability rate of the parallel system is constant, the numerical solution of the repairable system is found to be consistent with the theoretical proof from experimental graph by using the Matlab software to carry out numerical experiments. The results show that the solution of the discrete ordinary differential equations converges to the solution of the original parabolic partial differential equation, thus lay foundation for further numerical calculation of the model