An Analysis Of Delay-Dependent Stability Of Numerical Methods For Delay Differential Equations

Delay differential equations (DDEs) play an important role in describing the phenomenon in the fields of Physics, Engineering, Biology, Medical Science, Economics and so on. In the case of long time integration, the stability of numerical methods is important. Therefore, the stability analysis of numerical methods for solving DDEs has received scholar's widespread attention in recent years. One of the important problems is to study the delay-dependent stability of numerical methods for DDEs. In this paper, we also concern with this problem.At the beginning of this paper, we give a brief introduction to some applications of DDEs in different fields and collect some results of analytical stability theory for DDEs with constant coefficients. Then we review specially the theory development on delay-dependent stability of numerical methods for DDEs.In the second chapter, we deal with the delay-dependent stability of BDF methods for a class of second-order DDEs. Firstly, by citing the work from the literature, we plot the analytical region in the parameter plane. And then, by using boundary locus method, we obtain a necessary condition which guarantees that the analytical region is a sub set of numerical stability region. At last, the results ofτ(0 )-compatibility of BDF methods are obtained.In the third chapter, we discuss the delay-dependent stability of numerical methods for distributed delay differential equations. For a class of two-step methods, one order BDF method and two order BDF method applied to model equations, we investigate their Iτ(0 )- stability respectively, and plot the stability region of the numerical methods. Finally, we give some numerical examples to confirm the results in our paper.