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The numerical stability of the \(\theta\)-method for delay differential equations with many variable delays. (English) Zbl 0942.65087

The asymptotic stability of theoretical solutions and the numerical stability of the \(\theta\)-method are studied for the delay differential equations of the form \[ y'(t)=a y(t) + \sum_{i=1}^{m} b_{i} y(\lambda_{i} t),\quad t \geq 0,\quad y(0)=y_{0}, \] where \(a, b_{1}, b_{2}, \dots, b_{m}\) and \(y_{0}\) \(\in\mathbb{C}\), \(0<\lambda_{m} \leq \lambda_{m-1} \leq \dots \leq \lambda_{1} < 1\).
In order the differential equations to be asymptotically stable, a sufficient condition is derived. Finally, it is proved that the linear \(\theta\)-method is \(\Lambda\text{GP}_{m}\)-stable if and only if \(\frac{1}{2} \leq \theta \leq 1\).
Reviewer: T.E.Simos (Xanthi)

MSC:

65L20 Stability and convergence of numerical methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34K06 Linear functional-differential equations
65L07 Numerical investigation of stability of solutions to ordinary differential equations
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