几类脉冲微分方程数值方法的收敛性和稳定性

Impulsive differential equations are widely applied in many science and technology fields such as population ecology, epidemiological dynamics, control engineering, information technology. For impulsive ordinary differential equations, impulsive retarded differential equations, impulsive differential equations with piecewise continuous arguments and impulsive pantograph differential equations, of which the impulsive times are fixed, we will find suitable transformation which can transform the impulsive differential equations into differential equations without impulses, construct new numerical methods based on the transformation, and study convergence and stability of the numerical methods by the properties of numerical methods for the equations without impulses. For impulsive variable delay differential equations with fixed impulsive times, we will construct high order discrete numerical methods with variable stepsizes or piecewise continuous Runge-Kutta methods, and obtain the numerical methods which can preserve the stability of the exact solutions under some conditions. For impulsive differential equations with impulses at variable times, we will construct convergent numerical methods whose forms are concise, which can preserve stability of the exact solutions under some conditions. The results of this project will provide new methods for the scientific calculation in many fields, such as population ecology, epidemic dynamics and control. Therefore, they are significant to the theoretical research and practical applications.

脉冲微分方程广泛应用于种群生态学、流行病动力学、控制工程、信息技术等领域。本项目将分别针对固定时刻的脉冲常微分方程、脉冲常延迟微分方程、自变量分段连续型脉冲微分方程和脉冲比例方程，寻找适当的变换，把脉冲微分方程变成没有脉冲扰动的方程，利用没有脉冲扰动的方程的数值方法和变换构造原方程数值方法，并应用没有脉冲扰动的方程数值方法的性质来研究原方程的数值方法的收敛性和稳定性；针对固定时刻的脉冲变延迟微分方程，构造高阶收敛的变步长离散的数值方法或分段连续的Runge-Kutta方法，并在一定的条件下得到能够保持精确解稳定性的数值方法；针对依赖状态的脉冲微分方程，构造形式简洁的、收敛的且在一定的条件下能够保持精确解稳定性的数值方法。该研究可为种群生态学、流行病动力学和控制等诸多领域中的科学计算提供新的方法，因此具有理论意义和应用价值。

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。（1）针对脉冲变延迟微分方程，首先研究了其精确解的连续性与光滑性，进而构造了脉冲连续龙格库塔方法，并证明了当相应的龙格库塔方法是p阶的时该方法也是p阶收敛的；（2）针对半线性脉冲微分方程，给出了其精确解渐近稳定的充分条件，并在该条件下得到了保持其渐近稳定性的指数龙哥库塔方法；（3） 针对非线性脉冲微分方程，在利普希茨条件下研究了精确解得渐近稳定性，进一步应用帕德逼近方法得到了保持其稳定的龙格库塔方法。该研究可为种群生态学、流行病动力学和控制等诸多领域中的科学计算提供新的方法，因此具有理论意义和应用价值。