Runge-Kutta方法求解多延迟积分微分方程的稳定性

讨论了用Runge.Kutta方法求解带有两个延迟常量的多延迟积分微分方程du/dt=Lu(t)+M1u(t-T1)+M2u(t-T2)+K1∫5t-T1u(θ)dθ+K2∫5t-T2u(θ)dθ的数值稳定性,并给出了其渐进稳定的充分条件.这里的L,M1,M2,K1,K2都是复矩阵.特别当K1,K2=0时,亦可以得到相同的结论,即每一个A稳定的RK方法都可以证明其解的延迟独立稳定性.

Stability of Runge-Kutta methods for multi-delay integro-differential equations

This paper deals with the sufficient conditions of the asymptotical stability of Runge-Kutta (PK) method for multi-delay integro-differenfial equations(DIDEs) with two constant delays on the basis of the linear equation du/dt = Lu(t) + M1u(t-T1) + M2u(t-T2)+ K1∫tut-T1u(θ)dθ +K2∫tt-T2 u(θ)dθ, where L, M1,M2, K1,K2 are constant complex matrices. In patticular, we show that the same result as in the caso K1 ,K2 = 0 holds for this test equation,i.e.,every A-stable RK method preserves the delay-independent stability of the exact solution whenever a step-size of the form h = T/m is used, where m is a positive integer.