Stability Analysis Of Numerical Methods For Integro Differential Equations With A Proportional Delay

Delay differential equations (DDEs) have been used widely in the fields of technology, engineering and economic management, ecology, environment, population, traffic, and so on. Generally, it is difficult to obtain analytical solutions of DDEs, so many scholars pay their attention to numerical analysis for DDEs. This paper studies the boundness and asymptotic stability of numerical methods for integro-differential equations with a proportional delay.In the first part, we briefly introduce the results and development of analytical and numerical stability of delay differential equations, and propose our study by building on predecessors'conclusions. Our ideas are studying a class of special integro-differential equations with ordinary delay, which are obtained from integro-differential equations with a proportional delay by exponential transformation.In the second part, we investigate stability of general linear methods with a compound quadrature rule, and prove that they can keep analytical stability of a class of linear and nonlinear neutral equations if the method is strictly stable at infinity. Besides, we derive results about boundness stability of algebraically stable general linear methods for non-neutral equations.In the third part, we investigate stability of Runge-Kutta methods with a Pouzet type quadrature rule. For algebraically stable and strictly stable at infinity Runge-Kutta methods, similar results on boundness and asymptotic stability are given.We give some numerical experiments to validate our conclusions.