脉冲微分方程数值分析与应用

Impulsive differential equations (IDEs) are widely applied in the fields of science and technology, such as control systems, physics, biology, medical science, economics and so on. Their outstanding characteristic is fully taken into account the effect of instantaneous mutations on the state, and more profound, more accurately reflect the changes of things. It is of great significance to research the theory and application of numerical methods for IDEs. The project is devote to investigate the stability, convergence of numerical methods for impulsive ordinary differential equations and impulsive functional differential equations. The theory of numerical methods for IDEs is innovative, which can be regarded as improvements and extension of the algorithm theory for ordinary differential equations and has significance in theory and practice. As application of the aforementioned theory, highly efficient numerical methods with good stability and high convergent order for solving IDEs will be constructed and to serve the large-scale scientific computing in practical problems of modern science and engineering.

脉冲微分方程在控制系统、物理学、生物学、医学、经济学等众多科学技术领域有广泛应用，其突出的特点是充分考虑到瞬时突变现象对状态的影响，更深刻、更准确地反映事物的变化规律。由于脉冲微分方程的真解一般很难获得，因此对其算法理论进行深入研究具有毋庸置疑的重要性与广阔应用前景。本项目重点研究求解脉冲常微分方程与脉冲泛函微分方程的数值方法的稳定性和收敛性，建立具有普遍指导意义的算法理论。以所获理论结果为指针，构造具有良好稳定性和高收敛阶的高效数值方法，为现代科学技术及工程实际问题中大规模科学计算服务。

本项目研究了非线性脉冲常微分方程与脉冲延迟微分方程Runge-Kutta方法的稳定性和散逸性，获得了方法的稳定性和散逸性条件；同时研究了求解刚性和非刚性问题的线性多步法、单支方法、Runge-Kutta方法、多步Runge-Kutta方法的收敛性，结果表明：求解刚性常微分方程的p阶B-收敛的数值方法用于求解刚性脉冲微分方程也是p阶B-收敛的，非刚性问题也有类似结论。由于脉冲微分方程充分考虑了瞬时突变现象对状态的影响，更深刻、更准确地反映了事物的变化规律，在控制系统、物理学、生物学、医学、经济学等众多领域有广泛应用，因此本项目得到的脉冲微分方程数值方法的相关结果可为构造具有良好稳定性和高精度的数值方法提供理论依据，具有较大的理论意义和实际应用价值。