线性定常系统非齐次两点边值问题的扩展精细积分方法

提出了一种求解非齐次线性两点边值问题的高精度和高稳定的扩展精细积分方法（EPIM）.首先引入了区段量（即区段矩阵和区段向量）来离散非齐次线性微分方程，建立了非齐次两点边值问题基于区段量的求解框架.在该框架下，不同区段的区段量可以并行计算，整体代数方程组的集成不依赖于边界条件.然后引入区段响应矩阵来处理两点边值问题的非齐次项，导出了多项式函数、指数函数、正/余弦函数及其组合函数形式的非齐次项对应的区段响应矩阵的加法定理，结合增量存储技术提出了EPIM.对具有上述函数形式的非齐次项，该方法可以得到计算机上的精确解，一般形式的非齐次项则利用上述函数近似求解.最后通过两个具有刚性特征的数值算例验证了该方法的高精度和高稳定性.

An Extended Precise Integration Method for Solving Inhomogeneous Two-Point Boundary Value Problems of Linear Time-Invariant Systems

An extended precise integration method (EPIM) for solving inhomogeneous two-point boundary value problems (TPBVPs) of linear time-invariant systems was proposed. Firstly, the interval quantities of the interval matrices and vectors were introduced to describe the discretization of the differential equations for the TPBVPs. Thus a general framework for solving the TPBVPs was established, where the interval quantities for different intervals were computed in parallel, and the assembled algebraic equations for global analysis were independent of the boundary conditions. Secondly the interval response matrices corresponding to the interval vectors were used to deal with the inhomogeneous terms. The addition theorems for the interval response matrices were derived with the inhomogeneous terms in the forms of polynomial function, sine/cosine function, exponential function and their combinations. Then the extended precise integration method was proposed in combination with the incremental storage technique, of which the accuracy approached the machine precision for the inhomogeneous terms in the above forms. The general forms of the inhomogeneous terms can be approximated with the mentioned forms. In comparison with the analytical methods, two numerical examples of stiff problems give results showing the high accuracy and stability of the proposed method.