Dissipativity Analysis Of One-leg Methods And Runge-Kutta Methods For A Class Of Neutral Delay Integro-differential Equations

Let X be a real or complex Hilbert space with the inner product<·,·>and the corre-sponding norm ?·?.Consider the nonlinear neutral delay integro-differential equations(NNDIDEs)(?)where ? is a positive constant,the functions f:[0,+?)×X×X×X×X?X,g:[0,+?)×[-?,+?)× X ? X,?:[-?,0]? X are assumed to be continuous and for any t ? 0,y,u,v,w ?X,f and g satisfies the conditions:Re(f(t,y,u,v,w),y)? ? ?y?2+??f(t,0,u,v,w)?2,?f(t,y,u,v,w)?2 ? ?1 + Ly ? y ?2 +?? f(t,0,u,v,w)?2,?f(t,0,u,v,w)?2 ? ?2 + Lu ? u ?2+ Lv ?v ?2 +Lw ? w ?2 and?g(t,?,u)????u?,t-????t,where-?,?,?1,?2,Lu,Lv,Lw,Ly,?,? are nonnegative real constants.In this paper,we study dissipativity of NNDIDEs itself and numerical methods for solving nonlinear neutral delay integro-differential equations.First,the sufficient condition which ensures the system to be dissipative is given.Second,the G(c,p,0)-algebraically stable one-leg methods for solving above prob-lems are dissipative when(?+?(Lu+LvLy+Lw?2?2)/1-Lv?)h<p/2,and(k,l)-algebraically stable Runge-Kutta methods are dissipative when(?+?(Lu+LvLy+Lw?2?2)/1-Lv?)h<l.Finally,the numerical experimentations are given by used G(c,p,0)-algebraically sta-ble one-leg method and(k,l)-algebraically stable Runge-Kutta method for the initial value problem equations,and the numerical results verify correctness of the theoretical results.