The Analysis And Numerical Test Of Contractivity And Asymptotic Stability Properties Of Numerical Methods For Stiff Volterra Functional Differential Equations

Stiff Volterra functional differrntial equations (SVFDEs) were well investigatedin many fields, such as control theory, biology, medicine, economics and engineering. It is meaningful to study the theory and application of numerical methods for SVFDEs. In recent 30 years, the theory of numeical methods for SVFDEs, especially for the stiff delay differential equations (DDEs), has been widely discussed by many authors such as Barwell, Bellen, Torelli, Zennaro, Spijker,Watanabe, in't Hout, Koto, Shoufu Li, Jiaoxun Kuang, Mingzhu Liu, ChengmingHuang, Chengjian Zhang, Hongjiong Tian, Guangda Hu, Siqing Gan and so on. The main results can be found in the monograph of Bellen, Zennaro and Jiaoxun Kuang. Because it's hard to get the theory solution of Volterra functional differential equations, so we pay more attention to the numerical solution of actual problems.This papar is concerned with the contractivity and asymptotic stability propertiesof several numerical methods for stiff Volterra functional differential equationsin finite-dimensional Euclidean spaces. The main results obtained in this paper are as follows:(1) We study the contractivity and asymptotic stability properties of linearθ-methods for stiff Volterra functional differential equations in finite-dimensions Euclidean spaces, and obtained that linearθ- methods is asymptotic stability ifθ∈(1/2, 1] . Then we gave some numerical tests to confirm the theory.(2) We study the contractivity and asymptotic stability properties of one-legθ- methods for stiff Volterra functional differential equations in finite-dimensions Euclidean spaces, and obtained that one-legθ- methods is asymptotic stability ifθ∈(1/2, 1] . Then we gave some numerical tests to confirm the theory.(3) We study the contractivity and asymptotic stability properties of BDF methods of order two for stiff Volterra functional differential equations in finitedimensionsEuclidean spaces.(4) Some numerical tests were presented to confirm Professor Li Shoufu's newest theory: the contractivity and asymptotic stability properties of RungeKuttamethods for stiff Volterra functional differential equations in finite-dimensional Euclidean spaces.