Oscillation Analysis Of Numerical Solution For Two Kinds Of Special Delay Differential Equations

This paper deals with the oscillation analysis of numerical solution for two kinds of special delay differential equation (that is differential equations with piecewise continuous arguments of advanced type and linear constant coefficient delay differ-ential equations). The two kinds of equations as important mathematical models are widely used in many fields such as physics, biology theory. Hence, the devel-opment of appropriate numerical methods and the discussion of their properties become a research project with theoretical significance and practical value.The paper describes in detail the application background of two kinds of equa-tions, and reviews the current situation of research on the oscillation of numerical solutions for the two kinds of equations at domestic and abroad.For differential equations with piecewise continuous arguments using the equiv-alent conditions for oscillation of the equation, the paper discusses the necessary and sufficient conditions for oscillation of analytical solutions. When solving the equation by applying θ-method, we transform the oscillation of the equation into that of a difference equation according to the characteristic of the equation itself, discuss the necessary and sufficient conditions for oscillation of numerical solutions in the θ-method by using of judging theorem that the oscillation of difference equa-tion is equivalent to its characteristic equation without positive root, and study the condition of the θ-method that preserve the oscillation of the original equation.Currently, the research on the oscillation of numerical solutions for delay differ-ential equations is confined to a few kinds of special equations, and for the general delay differential equations, some related research findings have not been found. The last chapter begins with the linear constant coefficient equation and discusses the oscillation of numerical solutions in the explicit Euler, the implicit Euler, the trapezoid method and the θ-method respectively by using of the conclusion that the oscillation of difference equation is equivalent to its characteristic equation without positive root. Through the application of several important inequality, the pape studies the conditions under which these numerical methods preserve oscillation of the original equations and discusses the conditions that the oscillation of equations must not be kept.