Dissipativity Of Numerical Methods For Volterra Functional Differential Equations

Volterra functional differential equations (VFDEs) arise widely in the fields of biology, economics, physics, chemistry, engineering and control theory and so on. It is meaningful to investigate the theory and application of numerical methods for VFDEs. Many dynamical systems are characterized by the property of possessing a bounded absorbing set which all trajectories enter in a finite time and thereafter remain inside. When considering the applicability of numerical methods for these systems, it is important to analye whether or not numerical methods inherit the dissipativity of the underlying system. This paper deals with the analytic and numerical dissipativity of following system:where the functional f satisfies thatThe main results in this paper are as follows:1. We obtain the analytic dissipativity result of (1): Whenα0 +β0<0,then for anyε>0, there exists a positive numbewhere2. We investigate the dissipativity ofθ-methods for (1), and obtain the dissipativity results of one-lag and linearθ-methods.3. We investigate the dissipativity of BDF methods for (1), and obtain the dissipativity results of BDF methods.4. Several numerical examples in which theθ-methods and BDF methods are applied respectively are given to confirm theoretical results presented in this paper.