Numerical Stability Of Two Classes Of Piecewise Continuous Differential Equations

This paper mainly deals with the numerical stability of two classes of piecewisecontinuous differential equations (i.e., impulsive differential equations and differen-tial equations with piecewise continuous arguments). These two classes of equationsappear in many fields of applied sciences, and the stability analysis of the numericalsolutions is of important theoretical and practical significance.The paper presents the background of some applications and research history ofimpulsive differential equations and differential equations with piecewise continuousarguments in detail, and surveys the development of the stability of the analytic andnumerical solutions of these two classes of differential equations.The asymptotic stability of the analytic solutions for linear impulsive differen-tial systems is investigated. The Runge-Kutta scheme with variable stepsize for thissystem is defined, which can preserve the original order of the method. In the case of2-norm, the conditions under which the numerical solutions of Runge-Kutta methodsandθ-methods preserve the asymptotic stability of the analytic solutions are given forthe coefficient L withμ[L]≤0,μ[?L]≤0 and arbitrary L respectively. Especially,such conditions are analyzed for the Runge-Kutta method whose stability function isgiven by the (r,s)-Pade′approximation to ex by using the theory of Order Stars andPade′approximation.The Lyapunov theorems are given for the impulsive differential and discrete sys-tems respectively, which are improvement of the previous Lyapunov theorems. Theasymptotic stability conditions for some kinds of impulsive differential equations andthe linear discrete equation are discussed. The conditions under which the numericalsolutions produced byθ-methods applying to the linear impulsive differential equationwith variable coefficient preserve the asymptotic stability of the analytic solutions areanalyzed.For d-dimension complex valued linear differential systems with piecewise con-tinuous arguments, the asymptotic stability of the analytic solutions is investigated.The Runge-Kutta scheme for this system is defined, which can preserve the originalorder of the method. In the case ofμ[L] < 0, the conditions under which the numerical solutions preserve the asymptotic stability of the analytic solutions are discussed forthe coefficient L being a complex, a normal and a real symmetric matrix respectively.Especially, such conditions are analyzed for the Runge-Kutta method whose stabilityfunction is given by the (r,s)-Pade′approximation to ex by using the theory of OrderStars and Pade′approximation for L being a real symmetric matrix.At last, a partial differential equation with piecewise continuous arguments isstudied. An example is given and shows a very interesting phenomenon that thenumerical solutions produced by the well known Crank-Nicolson formula applyingto such equations and original PDEs exhibit different properties. Hence, it is worthto study the numerical solutions of such equation. The finite difference method:θ-schemes for such kind of equations are defined and the conditions under which thenumerical solutions of the scheme preserve the asymptotic stability of the analyticsolutions are studied in detail.