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 τ is a positive constant,the functions f:[0,+∞)×X×X×X×X→X,g:[0,+∞)×[-τ,+∞)× X → X,φ:[-τ,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)≤ α ‖y‖2+β‖f(t,0,u,v,w)‖2,‖f(t,y,u,v,w)‖2 ≤ γ1 + Ly ‖ y ‖2 +σ‖ f(t,0,u,v,w)‖2,‖f(t,0,u,v,w)‖2 ≤ γ2 + Lu ‖ u ‖2+ Lv ‖v ‖2 +Lw ‖ w ‖2 and‖g(t,ξ,u)‖≤λ‖u‖,t-τ≤ξ≤t,where-α,β,γ1,γ2,Lu,Lv,Lw,Ly,σ,λ 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(α+β(Lu+LvLy+Lwλ2τ2)/1-Lvσ)h<p/2,and(k,l)-algebraically stable Runge-Kutta methods are dissipative when(α+β(Lu+LvLy+Lwλ2τ2)/1-Lvσ)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.