The Dissipativity Of Theoretical Solution And Numerical Solution For A Class Of Implicit Neutral Functional Integro-differential Equations

Functional integro-differential equations(FIDEs) exist widely in elastic mechanics,fluid mechanics, the spread of diseases, population migration, electromagnetic fieldtheory and other areas. These areas not only care about the current state, but also focuson the long time behavior of the system. Dissipativity is a kind of the behavior, itcharacterized by the property of possessing a bounded absorbing set which all trajectoriesenter in a finite time and thereafter remain inside. There is no doubt that it is important todepict the property of solution by dissipativity.This article is concerned with the dissipativity of theoretical solution and numericalsolution for a class of implicit neutral functional integro-differential equations (NFIDEs).Firstly, the chapter presents the background of functional integral and differentialequations, introduces the development of dissipativity, and gives the main framework.Secondly, a sufficient condition for the dissipativity of theoretical solution of thementioned problem above is given. Thirdly, the numerical dissipativity results areobtained for extended linear θ–methods and extended Pouzet-Runge-Kutta methodswhen they are applied to the kind of NFIDEs, by approximating the delay and integralterm with linear interpolation, compound trapezoidal formula, Pouzet formula. On theone hand, in theory,(1) assume that the method (3.1-3.3) satisfies12≤θ≤1, then the method is dissipative;(2) assume that Runge-Kutta for ordinary differential equationsis is algebraically stable,b_j>0, j=1,2,..., s,Then the extended Pouzet-Runge-Kutta (4.2-4.5) is dissipative.They proved that the numerical solution under proper conditions is dissipative. On theother hand, a set of case studies are provided to show the effectiveness of the proposedapproach.Finally, the ending concludes studies above, and shows the directions of researchingin future.