Application Of Waveform Relaxation Methods To Delay Differential Equations Of Neutral Type

In this paper, we discuss the applications of waveform relaxation method, which originated from the simulation of large scale integral circuits, in solving of delay differential systems with neutral type. For this method essentially a parallel method, it is very flexible for solving differential systems with high dimension. And thus in the last two decades, both mathematicians and engineers have given enough attention and interest to this method. As a numerical method, there are many aspects that need to be investigated from the mathematic point of view, and two of them aspects are the convergence and the rate of convergence of this method. By investigating these two aspects, we can formate better method and improve the efficiency of original method by mathematic proving and computers.For ordinary differential equations(ODEs), waveform relaxation method was utilized in linear ODEs at first, and then it was introduced to nonlinear ODEs and delay ODEs of nonneutral type. In the last ten years, the applications of waveform relaxation method in delay ODEs of neutral type have obtained a broad attention by many authors. By careful analysis and large numbers of numerical experiments, it turns out that many of these existing conditions for convergence and superlinear convergence are too rigorous and can not be verified easily. Therefore, we will pay special attention in this paper to find more general and flexible conditions.In Chapter two, based on the hypothesis of time-dependent and delay-dependent Lipschitz conditions of the so called splitting function, we investigate the properties of the convergence and superlinear convergence of the continuous-time waveform relaxation method applied to neutral differential-functional systems. We will see that, our results are more general and can be verified easily.In Chapter three, we investigate the convergence properties of continuous-time and discrete-time WR method for the linear neutral systems. For continuous-time case, we get a more accurate evaluation of spectral radius of the operators. And for the discrete-time case, we discuss the influence of splitting and the character of delays to the convergence and the rate of convergence.In this paper, our main results obtained in Chapters two and three are validated by numerical results.