The Method Of Integral Differential Equations Fugitive,

Many dynamical systems in physics and engineering are dissipative. These systems are characterized by possessing a bounded absorbing set which all trajectories enter in a finite time and thereafter remain inside. For example, the two-dimentional Navier-Stokes equation and other important systems such as Lorenz are dissipative.The research of the disssipativity has always been important topic in the dynamical systems research (see Teman). When considering the applicability of numerical methods for these systems, it is important to analyze whether or not numerical methods inherit the dissipativity of the underlying systems.Integro-differential equations (IDEs) arise widely in the fields of Physics, Engineering, Biology, Medical Science, Economics and so on. The theory of computational methods has decisive importance in Numerical Integro-differential equations. Recently, many scholars have pay careful attention to it. For the Integro-differential equation S.Gan first studied the dissipativity of the system and theθ-methods applied to the above system.The thesis is concerned with the dissipativity of numerical methods for the underlying system on the basis of the study of S.Gan. Our main results in the thesis are as follows:(1) When the integration term is approximated by the CQ formula, it is proved that the (k,l)-algebraically stable Runge-Kutta methods are dissipative in finite-dimentional space for k≤1 and are dissipative in infinite- dimentional space for k＜1.(2) When the integration term is approximated by PQ formula, we proved that the (k,l)-algebraically stable methods are dissipative in finite-dimentional space for k≤1.(3) We considered the dissipativity of the one-leg methods. When the integration term is approximated by the CQ formula, we proved that the G(c,p,0)-algebraically stable one-leg methods are dissipative in finite-dimentional space for k≤1 and are dissipative in infinitedimentional space for k＜1.(4) We studied the dissipativity of the linear multistep methods. A sufficient condition for the dissipativity of the methods (the integration term is approximated by the CQ formula) is given.(5) We investigated the dissipativity for the multistep Runge-Kutta methods when the integration term is approximated by the CQ formula. The sufficient conditions for the dissipativity of the methods in finite-dimentional and infinite-dimentional space are given.(6) Numerical experiments are given for checking the dissipativity of Runge-Kutta methods, one-leg methods and linear multistep methods, which confirm the theoretical results obtained in this paper.