Legendre-Gauss-Lobatto Collocation Method For Nonlinear Delay Differential Equations

Delay differential equations arise in many areas of mathematical modelings, for example, con-trol systems, cell biology, lasers and population growth. Usually, the solution of a delay differentialequation has no global smoothness, and hence has a much more complicated dynamics than that ofordinary differential equations. During the past few decades, many numerical schemes have beenintroduced and frequently used for delay differential equations. We particularly point out that theimplicit Runge-Kutta method is now accepted as one of the most effective approaches. However,to our knowledge, there are only very few numerical methods with the spectral accuracy for delaydifferential equations.Recently, Guo et al. designed several Legendre/Laguerre spectral collocation methods forordinary differential equations, which are numerically stable and possess the spectral accuracy.The purpose of this dissertation is to develop a Legendre-Gauss-Lobatto spectral collocationmethod for solving a class of nonlinear delay differential equations, motivated by the preciousLegendre spectral collocation method. We introduce a single-step Legendre-Gauss-Lobatto spec-tral collocation scheme and design an efficient algorithm. We also apply it to the multi-domaincase. As a theoretical result we obtain a general convergence theorem for the single-step case.Numerical results show that the suggested algorithm enjoys high order accuracy both in time andin the delayed argument, and can be implemented in a robust and efficient manner.The dissertation consists of four parts:In Chapter Ⅰ, we briefly introduce some existing numerical methods for delay differentialequations.In Chapter Ⅱ, we propose the single-step Legendre-Gauss-Lobatto spectral collocation schemeand carry out the error analysis. Some numerical experiments are given to illustrate the theoreticalresults.In Chapter Ⅲ, we describe the multi-domain scheme and present some numerical results todemonstrate its high accuracy.In Chapter Ⅳ, we give some concluding remarks.