Block Boundary Value Methods For Discrete And Distributed Delay Systems

Time-delay systems form a wide class of evolution equations which belong to the class of functional differential equations (FDEs) that are infinite dimensional, as apposed to ordi-nary differential equations (ODEs). They share the property that the rate of the present state depends on a discrete or distributed set of values of the solution itself in the past. These types of equations arise widely in many science and engineering fields and efficient numeri-cal methods for them become very important. Traditional numerical methods such as linear multistep methods (LMMs) and Runge-Kutta methods (RKMs) have attracted much atten-tion during the past decades. However, high order LMMs often have bad stability property, RKMs share both high accuracy and excellent stability but cost much in the implementation.Boundary value methods (BVMs) and block boundary value methods (BBVMs) are relatively new numerical methods which have gained much attention in computational and engineering sciences during the last two decades. They have been shown to be very efficient for many kinds of problems such as initial value problems (IVPs) and boundary value prob-lems (BVPs) for ODEs, differential-algebraic equations (DAEs), Hamilton problems and partial differential equations (PDEs).The purpose of this dissertation is to develop new (block) boundary value methods for solving a variety of dynamical systems with discrete and distributed delays, such as delay differential equations (DDEs), delay differential-algebraic equations (DDAEs) and Volterra delay integro-differential equations (VDIDEs). The aim of this thesis is twofold. Firstly, we want to look for new numerical schemes for dynamical systems with delay. Secondly, we need to know whether the new algorithms can outperform the traditional ones. Through careful and rigorous convergence analysis, stability analysis and accurate numerical experiments, we confirm that our algorithms are very efficient, accurate and stable. Numerical results show that the algorithms based on BBVMs are more efficient than the algorithms based on Runge-Kutta methods (RKMs), although the latter algorithms are, in general, somewhat more accurate under the condition of the same order and stepsize.