One-step Discretization Methods For Several Classes Of Stochastic And Delay Dynamical Systems

Differential equations 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 real world, dynamical systems are usually influenced by delay phe-nomenon or random perturbation. Therefore, differential equations containing these factors can better simulate the real dynamical systems. However, the increasing complexity of equations makes it almost impossible to get their explicit solutions, and hard to study their dynamical properties. Thus, it is of great significance to develop numerical methods for solving these differential equations, and analyze the properties of numerical solutions, such as convergence and stability. In this thesis, we focus on several types of differential equa-tions including stochastic 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 stochasticθ-methods, we propose a class of split-steθ-methods for solving stochastic delay differential equations. It is proved that the methods are convergent with strong order 0.5 in the mean-square sense.Chapter 3 is devoted to study the index-1 stochastic delay differential-algebraic equa-tions (SDDAEs) of retarded type. The existence and uniqueness of strong solutions are derived under uniform Lipschitz condition and some general assumptions.In chapter 4, we develop a general framework for a class of one-step methods for solving index-1 SDDAEs of retarded type, and establish the strong convergence criterion. Based on the criterion, we construct some specific schemes for solving index-1 SDDAEs of retarded type and semi-explicit stochastic differential-algebraic equations of index-1.In chapter 5, by adapting the existed numerical methods of ordinary differential equa-tions, a class of extended Rosenbrock numerical simulation methods for solving discrete-distributed delay systems of neutral type are constructed, and some criteria, for judging that the numerical methods are asymptotically stable, are obtained. It is shown that meth-ods extended from A-stable classical Rosenbrock methods can preserve the asymptotically stability of the underlying linear system.In chapter 6, for a class of nonlinear functional-integro-differential equations, a type of mixed Runge-Kutta methods are presented by combining the underlying Runge-Kutta methods and the compound quadrature rules. Based on the non-classical Lipschitz condi-tion, some global and asymptotical stability criteria are derived for the methods. Several specific mixed Runge-Kutta methods of high precision and good stability are derived.With numerical experiments, efficiency of the proposed methods and applicability of the theoretical results are further illustrated.