Stability And Oscillation Analysis For Differential Equation With Piecewise Continuous Arguments Of Mixed Type

This paper deals with stability and oscillation of analytical solution and numerical solution of delay differential equation with piecewise continuous argument (EPCA) of mixed type. The EPCA has considerable applications to population dynamics, control theory, environmental science, commercial sales and so on. It contains arguments that are constant on certain intervals, so it can be considered as a piecewise ODE.In this paper, we investigate the EPCA with delay[t] and 2 and obtain the condition of stability and oscillation of analytical solution.Euler method, linear θ-method and Runge-Kutta method are applied to the equation.The stability and oscillation of numerical solutions in these methods are studied.In second chapter, according to the character of EPCA that arguments are constant on certain intervals and the theory of solution of differential equation, we get the expression of analytic solution. From the expression of analytic solution, we obtain the condition of stability and oscillation of analytical solution.In third chapter, we investigate the numerical solution of Euler method. It is proved that when the stepsize is sufficiently small, the numerical solution conserves stability and oscillation of analytic solution.In fourth chapter, we investigate the numerical solution of linear θ-method and obtain the condition of stability and oscillation of numerical solution. It is proved that when the parameter θ satisfies some conditions and the stepsize is sufficiently small, the numerical solution conserves stability and oscillation of analytic solution.In fifth chapter, the numerical solution in the Runge-Kutta method is considered. The stability function of the Runge-Kutta considered here is given by the (r,s)-Pade approximation of e. Using Pade approximation theory and the Order Star theory, we prove that when the parameters satisfy some conditions and the stepsize is sufficiently small, the numerical solution conserves stability and oscillation of analytic solution.