Stability Of Numerical Methods For Several Classes Of Nonlinear Functional Differential Equations

Functional differential equation are widely used in the field of science and engineering technology.In recent years,the theoretical research and numerical analysis of such equations have been highly valued by scholars and have achieved rich research results.However,due to its wide variety and structure complexity,there are still a large number of new algorithms and theories to be explored and discovered.The study of the analytical and numerical stability(including dissipativity)is an important topic in the numerical analysis for functional differential equations.This thesis focuses on the stability(including dissipativity)and convergence of numerical methods for several classes of functional differential equations,and main results are as follows:A new generalized continuous Halanay-type inequality is proposed,and by applying this inequality,the dissipativity results are derived for a class of nonlinear delay integrodifferential equations and a class of nonlinear Volterra integro-differential equations,respectively.By applying the above generalized Halanay inequality,the dissipativity results of the theoretical solution of a class of Hale-type neutral functional differential equations in Banach space are obtained.In order to study the dissipativity of the numerical methods to solve such problems,a new generalized discrete Halanay inequality is proposed,and the sufficient conditions for dissipativity of the implicit Euler method to solve this kind of problem are obtained.For above neutral functional differential equations,its exponential stability of theoretical solution is obtained by the use of general continuous Halanay-type inequality.Meanwhile,sufficient conditions of the exponential stability for the linear -method to solve this kind of problem are gained with the help of the above generalized discrete Halanay inequality.For a class of composite stiff Volterra functional differential equations in Hilbert space,a splitting one-leg -method is constructed,and its corresponding stability,consistency and convergence results are discussed.By comparison with the traditional Im Ex one-leg -method,some numerical experiments are shown that our method is more efficient.For a class of stiff Volterra functional differential equations in Hilbert space,the contractivity of general linear method is studied,and the corresponding sufficient conditions are obtained.As a special case,the contractivity result of the multi-step Runge-Kutta method is also derived,and a cluster of contractive step 2 stage 2 Runge-Kutta method is constructed.