Runge-Kutta Discretization For Stiff Delay Integro-Differential-Equations

Extended Runge-Kutta methods are important ways for solving delay integro-differential equations that exist widely in physics, biology, control theory and so on, and this paper presents the following researches about the discretizations of these methods.In the first chapter, presents the background of some applications of delay differential equations (DDEs) and introduces the development of analytic and numerical solutions. Reviewing the important stabilities of numerical methods in linear and nonlinear cases. Especially various studies of nonlinear stabilities for delay differential equations with constant delay, neutral type and with infinite delay. At the same time, the researches of delay integro-differential equations are presented.In the second chapter, generalizing the contractilities and asymptotical stabilities for multi-delay integro-differential equations(MDIDEs). Under proper stepsize, we obtain the discretization schemes of Runge-Kutta methods with the compound quadrature formula and the Pouzet quadrature formula, and besides, derive the global and asymptotical stabilities. Moreover, the numerical experiments show that the presented methods are highly effective.In the third chapter, the contractilities and asymptotical stabilities of neutral delay integro-differential equations are concerned. The extended Runge-Kutta methods, with the compound quadrature formula and the Pouzet quadrature formula, are numerically stable under suitable conditions. The numerical experiments prove this result.In the forth chapter, by a change of variable, the infinite-delay integro-differential equations(IDIDEs) can be transformed into delay integro-differential equations with constant delays, which are special form of the equations discussed in the second chapter. Moreover, concerning the numerical stabilities of the extended Runge-Kutta methods with nonconstant stepsizes where the stepsizes are geometrically increasing.The end part concludes studies above, and show the directions of researching in future.