Numerical Stability Of Delay Differential Equations

As we know, many problems in physics, engineering, biology and economics can be modeled by systems of ordinary differential equations(ODEs). Actually, in many realistic models, we should know some past states of these systems. So it is converted into delay differential equa-tions(DDEs), which are widely used in bionomics, environment science and electrodynamics. And many problems in network of electric power can be modeled by system of neutral delay differential equations(NDDEs).Linear multi-step method, θ-method, and Runge-Kutta method are effectively numerical methods for solving DDEs and NDDEs. Block θ-method has good stability, and doesn't use the high order derivatives, so it is a kind of potential methods.The purpose of this paper is to study the stability behavior of numerical solution of systems of NDDEs and DDEs. Based on Lagrange interpolation, for two different models, we will show the sufficient and necessary conditions of asymptotic stability of linear multi-step method for NDDEs with many delays; we will give and prove the condition of GP-stability of block θ-method for DDE with complex coefficients, and also we will show that block θ-method for DDE with complex coefficients is GPL-stable if and only if θ = 1.