The Existence And Uniqueness Of Numerical Solutions For A Class Of Implicit Neutral Integro-differential Equations

Implicit neutral integro-differential equations(INIDEs) which belong to FIDEs could model the problems in aeroelasticity, ecology, bioengineering and so on. However,INIDEs are so complex that their analytic solutions can't be obtained by theoretical analysis. Hence, it is necessary to discuss the numerical solutions of INIDEs. In this paper, the solvability of the extended general linear methods(GLMs) for a class of INIDEs is discussed.In chapter one, the background and the development process of FIDEs are introduced, so as well the research situation of the existence and uniqueness of the numerical solutions.In chapter two, extended GLMs for a class of INIDEs are derived by an adaption of basic GLMs combined with mechanical quadrature formulas which are general and include both compound quadrature formulas and pouzet quadrature formulas, even some unknown quadrature formulas.In chapter three, the solvability of the numerical solutions is discussed when the INIDEs satisfy the Lipschitz conditions. And the sufficient conditions for the existence and uniqueness of the solutions are given. Next, the existence of the numerical solutions solved by Newton interation is discussed and the sufficient conditions for the existence and uniqueness of the solutions are given.In chapter four, corollaries about the existence and uniqueness of the numerical solutions of some special cases of extended GLMs are given, such as, mixed Runger-Kutta methods, Pouzet-Runger-kutta methods, the extended linear multisteps methods and a class of diagonally implicit multistage integration methods. In the end,numerical experiments further illustrate the theories results and the methodical effectiveness.