Stability Of Multistep Runge-Kutta Methods For A Class Of Nonliner Functional-integro-differential Equations

Functional-integro-differential equations (FIDEs) are widely used in the fields of medical science, ecology, chemistry, electrical power system, etc. Since their analyt?ical solutions can be obtained only in very restricted cases. So it is very important to study the numerical solution of functional-integro-differential equations.In recent years, many scholars have discussed the stability, convergence and dissipativity of the general linear methods, Runge-Kutta methods, ? methods, and linear multistep methods to solve the differential-integro-differential-equations , and many results are obtained.In this thesis, we focus on the stability and dissipativity of multistep Runge-Kutta methods for a class of initial value problems of functional-integro-differential equations where T > 0 are given constants delay ,? : [to-T, t0]? Cd, f : [to, +?) ×Cd×Cd ??Cd, g : D× Cd?Cd are continuous function , set D := {(t,s) : t ?[t0, +?), s?[t-T, t]}.First, we consider the stability of multistep Runge-Kutta methods under the following conditions Re<u-u1 - (w-w1),f(t,u,v) -f(t, u1,v1)>? ??u - u1?2 + ?1?v - v1?2 + ?2?w - w1?2, t>t0, u,u1,v,v1,w,w1 ?Cd,?g(t,?,u) - g(t,?,u1)? ? ??u-u1?, (t,? ?D,u,u1 ? Cd where -?,?1,?2,? are real positive constants. The sufficient conditions that ensure an algebraically stable multistep Runge-Kutta method is stable and asymptotic stable are obtained, respectively.Second,we consider the dissipativity of multistep Runge-Kutta methods under the following conditions Re<u-w,f(t,u,v))? ? + ??u?2 ?+ ?v?2+?2?w?2, t?t0, u,v,w?Cd,?g(t,?,u))? ???u?, (t,?) ?D, u?Cd We obtain dissipativity results of an algebraically stable, irreducible multistep Runge-Kutta methods for this class FIDEs.Some numerical experiments are given and the numerical results further proved the correctness of the theoretical results .