Dissipativity Of Numerical Methods For Neutral Delay Integro-differential Equations

Many dynamical systems in science and engineering are dissipative. These systems are characterized by possessing a bounded absorbing set which all trajectoriesenter in a finite time and thereafter remain inside. For example, the twodimentionalNavier-Stokes equation and other important systems such as Lorenz equation are dissipative.The research of the dissipativity has always been important topic in the dynamicalsystems research. When considering the applicability of numerical methodsfor these systems, it is hope that the numerical methods inherit the dissipativityof the underlying systems.Neutral delay integro-differential equations arise widely in the fields of biology,medical, economics, physics, chemistry and control theory and so on. It is meaningful to investigate the theory and application of numerical methods for neutral delay integro-differential equations.This paper investigate the dissipativity of the numerical methods for neutral delay integro-differential equationswhere <·,·>denotes inner product and‖·‖denotes the corresponding norm, N∈C^d×dis constant matrix and‖N‖<1,τis a positive constant,(?):[-τ,0]→C^dis a continuous function, f:C^d×C^d×C^d→C^d is a locally Lipschitz continuousfunction, g : [0,+∞)×[-τ,+∞)×C^d→C^d is a continuous function, f and g satisfyandwhereγ,α,β,ω, c are real constants andβ≥0,γ≥0,ω≥0.In this paper we obtain numerical dissipativity of the algebraically stable Runge-Kutta methods and the algebraically stable and step-irreducible Multistep Runge-Kutta methods when they apply these problems. The results can be directly applied neutral delay differential equations, delay integro differential equations and delay differential equations and so on. We also get the same results that have appeared in references currently. The numerical experiments are given for checking the results.