变延迟微分方程数值解稳定性

NUMERICAL STABILITY OF DELAY DIFFERENTIAL EQUATIONS WITH VARIABLE DELAYS

考虑了变延迟微分方程初值问题，我们研究用于求解这类问题的数值解法线性θ－方法的稳定性。在一定条件下求解常延迟微分方程的θ－方法是渐近稳定的。通过对比常延迟微分方程与变延迟微分方程的数值解，我们给出了交延迟微分方程线性θ－方法惭近稳定的充分条件。与变延迟微分方程精确解的稳定性相似，在一定条件下，变延迟微分方程线性θ－方法数值解的渐近稳定性不依赖于延迟项随时间的变化。而且我们证明了，任何数值方法，只要将其用于常延迟微分方程是稳定的，那么对变延迟问题它也是稳定的。

Abstract This paper deals with the numerical solution of initial value problems for differential e-quations with variable delays.We investigate the stability properties of linear θ-methods inthe numerical solution of these problems.It is well known that numerical solution of θ-meth-ods is stable under certain conditions for delay differential equations with constant delays. Bycomparision of numerical solution for variable delay differential equations with that for con-stant delay differntial equations,we present sufficient conditions for linear θ-methods to beasymptotic stable for retarded differntial equations with variable delays. Similar to what wehave known on the stability properties of the exact solution of variable delay differential e-quations,it turns out to be that numerical stability is independent of the variation of the de-lay term with time under certain conditions.Moreover,our theorems imply that numericalsolution of any method for variable delay differential equations is asymptotically stable if it isasymptotically stable for constant delay problems.SUBJECT TERMS:delay differential equations, numerical solution,θ-methods,asymptotic stability