The Computational Stability Of Delay Integro-differential Equations

Delay integro-differential equations have many application in modeling population dynamics, the spread of infectious diseases, biological sciences, physics and control theory. Because of the complexity of the delay integro-differential equations, it is difficult to get exact analytical solution. Hence, the research of numerical methods becomes important. In fact, the stability consists of two aspects, the stability of differential equations and the stability of numerical methods. Only stable system has a good anti-interference performance, only stable numerical methods preserve the stability of system. Hence, it makes sense to study the stable property of system, also numerical methods.In order to compute the stability determining characteristic roots of delay integro-differential equations, we used a linear multistep method which based on Lagrange interpolation and Gauss-Legendre quadrature rule. We explore in what condition the proposed scheme maintain the stability properties of the original equations. We derive and prove a sufficient condition for stability of a delay integro-differential equation which we called RHP-stability. Conditions are obtained under which the proposed scheme maintain RHP-stability. We compare the obtained results with the ones using Newton-Cotes formulas.In this paper, we derive and prove a sufficient condition for the stability of numerical methods, but it’s not necessary. When the condition is fulfilled, then RHP-stability is recovered, the numerical methods which we proposed are stable. But when the condition is violated, numerical methods may also be stable. Finally, these two aspects are respectively confirmed by numerical experiments.