Numerical Andanalytic Dissipativity For Functional Differential Equations With Distributed Delay On Unbounded Interval

The models of distributed delay integro-differential equations widely appear in thecensus, the spread of disease, neural networks, power engineering and other scientificfields. These systems not only care about the current state, but also relate to the previousinformation. The studies in science and engineering often pay close attention to the longtime behaviors of the system. The dissipativity is a kind of the long time behaviorcharacteristics of the system, in other words, the system has a bounded attractor,meanwhile, the solution of the system enters the absorbing set and then keeps inside aftera certain period. The dissipative research is an important topic in the study of the system,and has great significance in grasping the characteristics of the solution.This paper studies the numerical and analytic dissipativity for several classes offunctional differential equations with distributed delay on unbounded interval. Firstly, thearticle introduces the development of the theoretical and numerical dissipativity of delaydifferential equations. Secondly it discusses the analytic dissipativity for two classes offunctional differential equations with distributed delay on unbounded interval, and provesthe theoretical dissipativity under certain conditions by using the definition of dissipativity,Cauchy inequality, Halanay inequality and other technique, and gives the attractors of thesystems. Then we further study the dissipativity of Runge-Kutta method and linearmultistep method, the discrete schemes of the two methods are also given in this paper.The integral terms are discretized by different methods, and through further theoreticalproof, sufficient conditions of Runge-Kutta method and linear multistep method forkeeping dissipativity of corresponding system are given respectively. In the articlenumerical experiments verify the theoretical results. Finally, the paper makes a summaryabout the whole article, and discusses the direction for further researches.