The purpose of the research is to study the properties of numerical methods for ordinary and functional differential equations and Volterra integral and integrodifferential equations. An efficient and reliable general purpose algorithm for the numerical solution of neutral delay-differential equations and Volterra integro-differential equations based on variable stepsize variable order Adams formulas will be developed. The asymptotic behavior of recurrence relations resulting from application of various numerical methods to Volterra integral and integro-differential equations with nonconvolution kernels will be studied. Moreover, convergence, order and stability properties of time-point relaxations and waveform relaxation Runge-Kutta methods for large systems of ordinary differential equations will be examined. These methods are appropriate in a parallel computing environment. Finally, variable stepsize two-step Runge-Kutta methods for differential equations will be investigated. One of the main applications of these methods is to provide efficient error estimates for continuous Runge-Kutta methods. This research is in the area of numerical analysis, which has applications to a wide variety of physical and engineering applications.
