Stability Of Numerical Methods For Two Classes Of Nonlinear Differential Equations With Piecewise Constant Delay

Differential equations with piecewise constant delay have been widely applied to sci-ence and engineering, such as biology, economics, electrostatics, hydrodynamic, electro-magnetic field theory, chemistry and control theory and so on. It is important for studying the theory and numerical methods for differential equations with piecewise constant delay. The stability, which reflects the influence of the initial value for the solution, is a very im-portant property of differential equation. Therefore, it is necessary to study the stability of the solutions and numerical methods for differential equation with piecewise constant delay. The work of this paper includes two parts.First, on the base of existing research results in the literature, we further consider the following initial value problem of differential equation with two piecewise constant delays: The sufficient conditions are given respectively, which guarantee system itself to be stable or asymptotically stable. At the same time, the stability of Runge-Kutta methods for solving this system is studied. We prove that the BNf-stable Runge-Kutta methods is stable or asymptotically stable under some assumptions about the problem class.Secondly, we consider the neutral delay differential equations with piecewise constant delay as follows The stability of this problem class itself and Runge-Kutta methods for solving this system are given in literature. In this paper, we further study the numerical stability of a kind of linear multistep methods to solve this systes and the sufficient condition is given in order to guarantee the methods to be stability.Finally, some numerical experiments are present, the results further verify the theoret-ical results.