Numerical Stability Analysis On Some Kinds Of Delay Differential Equations

This thesis comprises of four parts: Firstly, the delay-dependent stability ofdelay-integro-di?erential equation and it'sθ?method are considered; Secondly,the stability of analytic solution of the delay-integro-di?erential-algebraic equa-tions and the numerical stability of Runge-Kutta methods are studied; Thirdly,we investigate the numerical stability of explicit and semi-implicit Runge-Kuttamethods for the delay-di?erential equations with piecewise continuous argument(EPCA); Finally, the stability of a new classθ?method for the delay-di?erential-algebraic equation is discussed.The thesis is organized as follow.In chapter one, we present many applications of delay-di?erential equationsand the research of both analytic stability and numerical stability theory of delay-di?erential equations for the recent forty years. Moreover, we have given anintroduction to the background of the study problems.There are relatively a large number of papers devoted to the study of thedelay-independent stability. Nevertheless, the delay-dependent stability analy-sis is sharper since it is more suitable for giving a complete description of theasymptotic behavior of the numerical methods considered. Therefore, the delay-dependent stability of the delay-integro-di?erential equation and the numericalstability ofθ?method are deeply investigated respectively in chapter two andchapter three.In chapter four, we consider the stability of analytic solution of delay-integro-di?erential-algebraic equation and the numerical stability of Runge-Kutta meth-ods.About the stability of the numerical methods for EPCA, many conclusionshave been obtained, nevertheless, until now no result has been obtained for thestability of explicit and semi-implicit Runge-Kutta methods. Therefore, in chap-ter five, we discuss the Order star of the explicit and semi-implicit Runge-Kuttamethods, and study the numerical stability of these numerical methods.In chapter six, a new classθ?method has been constructed to solve the delay-differential-algebraic equation, and the numerical stability is also considered.