1、DifferentiAlequations are powerful tools to model the time evolution of dynamical
　　systems, which have arisen widely in mechanics, physics, biology, ecology and the other
　　scientific fields. In the reAlworld, d

2、ynamicAlsystems are usually in?uenced by delay phe-
　　nomenon or random perturbation. Therefore, differentiAlequations containing these factors
　　can better simulate the reAldynamicAlsystems. However, the increas

3、ing complexity of
　　equations makes it almost impossible to get their explicit solutions, and hard to study their
　　dynamicAlproperties. Thus, it is of great significance to develop numericAlmethods for
　　sol

4、ving these differentiAlequations, and analyze the properties of numericAlsolutions, such
　　as convergence and stability. In this thesis, we focus on severAltypes of differentiAlequa-
　　tions including stochastic

5、and delay arguments, develop one-step methods for solving these
　　equations, and study the convergence and stability of these methods.
　　 In chapter 2, by combining the idea of split-step methods with stochastic

6、θ-methods, we
　　propose a class of split-step θ-methods for solving stochastic delay differentiAlequations. It
　　is proved that the methods are convergent with strong order 0.5 in the mean-square sense.
　　 Ch

7、apter 3 is devoted to study the index-1stochastic delay differential-algebraic equa-
　　tions (SDDAEs) of retarded type. The existence and uniqueness of strong solutions are
　　derived under uniform Lipschitz condi

8、tion and some generAlassumptions.
　　 In chapter 4, we develop a generAlframework for a class of one-step methods for
　　solving index-1SDDAEs of retarded type, and establish the strong convergence criterion.
　　

9、Based on the criterion, we construct some specific schemes for solving index-1SDDAEs of
　　retarded type and semi-explicit stochastic differential-algebraic equations of index-1.
　　 In chapter 5, by adapting the

10、existed numericAlmethods of ordinary differentiAlequa-
　　tions, a class of extended Rosenbrock numericAlsimulation methods for solving discrete-
　　distributed delay systems of neutrAltype are constructed, and som

11、e criteria, for judging
　　that the numericAlmethods are asymptotically stable, are obtained. It is shown that meth-
　　ods extended from A-stable classicAlRosenbrock methods can preserve the asymptotically
　　s

12、tability of the underlying linear system.
　　 In chapter 6, for a class of nonlinear functional-integro-differentiAlequations, a type
　　of mixed Runge-Kutta methods are presented by combining the underlying Runge-

13、Kutta
　　methods and the compound quadrature rules. Based on the non-classicAlLipschitz condi-
　　tion, some globAland asymptoticAlstability criteria are derived for the methods. Several
　　specific mixed Runge-