| A delay differential equation (DDE) can provide us with a realistic model of many phenomena arising in applied mathematics. For example, a DDE can be used for the modeling of population dynamics, the spread of infectious diseases, and two-body problems of electrodynamics. In this thesis, we present a numerical method for solving DDEs and analysis for the numerical solution.;We have developed the method by adapting recently developed techniques for initial value ordinary differential equations (continuous Runge-Kutta formulas, defect error control, and an automatic handling technique for derivative discontinuities) and developing a new approach (that is based on an iterative scheme determined by extrapolation or a special interpolant) to handle vanishing delays. This results in a robust variable stepsize method which can be applied to problems with state dependent delays. This approach can also be applied to any continuous Runge-Kutta formula.;We determine convergence properties for the numerical solution associated with our method. We first analyze such properties for retarded type DDEs and then the analysis is extended to the case of neutral type DDEs. The main theoretical result we establish is that the global error of the numerical solution is bounded by a multiple of the prescribed tolerance. Our analysis can also be applied to other numerical methods based on Runge-Kutta formulas for DDEs.;We have developed an experimental Fortran code as a modified version of DVERK based on a 6th order continuous Runge-Kutta formula. An implementation of our method is described and numerical results for various kinds of DDEs are presented. Numerical comparisons with two other existing codes, ARCHI and DRKLAG, are considered and some advantages of our approach are identified. |
