Spectral Deferred Correction Methods For Discrete And Distributed Delay Systems

In this paper, a novel algorithm is developed to solve discrete and distributed delaysystems. This method is a improvement of spectral deferred correction method.There are so many methods to compute delay differential systems efficiently, such asone-leg method, general linear method, Runge-Kutta method. Besides, spectral deferredcorrection method is also a class of effective methods, which was proposed by Dutt, Green-gard, Rokhlin in 2000.The algorithm is in general designed based on the Picard integralequations and Gauss quadrature formula. In this paper,we considered the techniques usingLegendre-Gauss point to solve the delay problems.The rest of the paper is organized as follows. Chapter 1 introduces the backgroundand significance of delay differential equations, and also describes the spectral deferredcorrection method's research status. Chapter 2 consists of preparative knowledge, such asLegendre polynomial, Gauss points. In Chapter 3 and 4, we separately describe in detailthe implementation of the numerical method for discrete and distributed delay systems.We improve spectral deferred correction method then obtain two new algorithms. Then weinvestigate theoretical and numerical solution by the new algorithms for delay differentialequations with a variable delay and delay-integro differential equations. Numerical exper-iments confirm the methods'effectiveness and the theoretical results. Finally, conclusionsare made in chapter 5.