Runge-Kutta方法求解多延迟积分微分方程的稳定性(英文)

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

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ddut=Lu(t)+M1u(t-τ1)+M2u(t-τ2)+K1∫t-tτ1u(θ)dθ+K2∫t-tτ2u(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.

This paper deals with the sufficient conditions of the asymptotical stability of Runge-Kutta （RK） method du for multi-delay integro-differential equations（DIDEs） with two constant delays on the basis of the linear equation du/dt = Lu（t） ＋ M1u（t - τ1 ） ＋ M2u（t - τ2） ＋ K1∫t-τ1^t u（θ）dθ ＋ K2∫t-τ2^t u（θ）dθ, where L, M1, M2, K1, K2 are constant complex matrices. In particular, we show that the same result as in the case 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 = τ/m is used, where m is a positive integer.