An Overview Of The Numerical Methods For Delay Differential Equations

The delay differential equations are widely applied in many fields, such as life sciences, medicine, traffic adjustment, engineering control and so on, which play a very important role in describing various phenomenons in natural and social science. From 1950's, more and more scholars carried the theoretical researches about the delay differential equations, and lots of results were obtained. There will be more difficulties for theoretical researches on the delay differential equations because of the existence of delay. However, the researches on numerical methods may be to perfect the theoretical researches, so it is necessary to carry out a research on the numerical methods for delay differential equations.The dissertation mainly sum up the researches on the numerical methods for delay differential equations which carried by the domestic researches. Firstly, to discuss the significance of the numerical methods, meanwhile, to sum up several kinds of classical schemes for solving delay differential equations. Secondly, to sum up the researches on the numerical methods for delay differential equations and to discuss the parallel algorithms as well as Runge-Kutta-Nystr?m method briefly for the initial value problem, besides, the convergence of some numerical methods for delay differential equations were discussed also. Thirdly, to sum up the researches on the stability of the numerical methods, for instance, the Euler method,θ-method, Runge-Kutta method. Finally, to sum up the works for the preserve property of Hopf bifurcations under the numerical discretization.The dissertation gives out the research report to establish the theoretical foundation for the deeply research in this field though collecting and summing up the works carried by the domestic scholar mainly.