Dissipativity Analysis Of The Numerical Methods For A Class Of Nonlinear Functional-integro-differential Equations

Let Cd be a d dimensional complex Euclidian space with the inner product <·,·> and the cor-responding norm ‖·‖. Consider the nonlinear functional integro-differential equations (FIDEs)(?)(?)where τ > 0 is a given constant delay, the functions f : [t0, +∞) × Cd × Cd → Cd, g : D × Cd→Cd, and φ :[t0 -τ, t0] → Cd are assumed to be continuous and satisfies the conditions Re<f(t,u,v),u-w) ≤ r + α‖u‖2 +β‖v‖2 +η‖w‖2,t ≥t0,u,v,w ∈ Cd,‖g(t,ζ,u) ‖≤λ‖u‖,(t,ζ)∈D,u∈Cd,where γ, α, β,η,γ are real constant and γ, -α, η are nonnegative constant, λ > 0 with 2λ2τ2 <1,D={(t,ζ :t ∈ [t0,+∞), ζ∈ [t-τ,t]}.In this paper, we denote this problem class which satisfies the conditions as R(γ,α, β, η, λ)and study dissipativity of itself and numerical methods for solving nonlinear functional 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 method is dissipative when h(α + β+ +ηv2λ2) < p-(1 + v2λ2) and Runge-Kutta method is is dissipative when α+ β+ηv2λ2 < 0.Finally, as an example, the numerical tests are given by used G(c,p, 0)-algebraically stable one-leg method and Runge-Kutta method for the initial value problem equations and the numer-ical results verify correctness of the theoretical results.