Heun Method For A Class Of Stochastic Distributed Delay Systems

Stochastic time-delay systems, because of considering the delay and the surrounding environment of the influence of random noise on the system, and compared with the deterministic differential equations and the stochastic differential equations, it can more accurately describe the phenomena of disturbance in the nature and the objective law of things development. Thus stochastic time delay system is widely applied in biochemistry, medicine, mechanical control, physical, financial, population dynamics, and many other research fields. However, only a handful of the analytical solution of the stochastic delay system can be found in usual. So the numerical solution of stochastic systems with time-delay and its property are particularly important. So far, there have been many scholars studied the numerical methods for solving stochastic distributed delay systems, but most of the methods are limited to the Euler- Maruyama, Milstein, split step method and so on. And for Heun method's research has not found in literatures. So this article will study the Heun method for solving a class of stochastic distributed delay systems.This article consists of four chapters. The first chapter is the introduction. This part briefly introduces the stochastic time delay model in many fields of application and the research status of the numerical method for solving stochastic distributed delay systems at home and abroad. Then on the basis of this, we put forward the main research work in this paper. The second chapter studies the convergence of the extension of Heun method in a class of stochastic system distributed delay, the method of mean- square convergence's order is 12, and the numerical experiments demonstrate this conclusion. The third chapter discusses the mean-square stability and T-stability of Heun method and sufficient conditions for mean-square stable and T-stable have been obtained respectively. At last, numerical examples are used to be illustrated the effectiveness of our theoretical results. The fourth chapter to the full text is summarized and prospect.