Delay-dependent stability analysis of trapezium rule for second order delay differential equations with three parameters
The paper is concerned with the study of the conservation of stability properties by applying the trapezoidal rule to the second order delay differential equations with three parameters: \[ y''=ay'(t)+by(t)+cy(t-\tau), \quad t\geq 0. \] This problem which in general cannot be solved exactly is replaced by using the trapezoidal rule by the numerical method \[ \begin{aligned} y_{n+1}-y_{n}&=\tfrac{1}{2}\, hY_{n+1}+\tfrac{1}{2}\, hY_{n},\\ Y_{n+1}-Y_{n}&=\tfrac{1}{2}\, haY_{n+1}+\tfrac{1}{2}\, haY_{n}+\tfrac{1}{2}\, hby_{n+1}+\tfrac{1}{2}\, hby_{n}+\tfrac{1}{2}\, hcy_{n+1-m}+\tfrac{1}{2}\,hc y_{n-m},\end{aligned} \] where \(h=\tau /m\) is a consistent step size and \(Y_{n}\) an approximation to \(y(x_{n})\). The main result is given in the following theorem: The difference equation is asymptotically stable if: \[ a<-b\tau \quad \text{and}\quad b\tau^{2}<-c\tau^{}<\frac{4m^{2}(1-\cos\eta)+h^{2}b(1+\cos\eta)}{\cos m\eta(1+\cos\eta)} \] where \(\eta\) is the root of \[ a=-((4m(1-\cos\eta)+h^{2}b(1+\cos\eta ))/(2\tau\sin\eta)\tan m\eta \] with \(\eta\in(0,\pi/2m)\).
second order delay differential equations
scientific article; zbMATH DE number 5828189