Mathematical Sciences: Studies in Numerical Solution of Functional Differential Equations

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The purpose of this project is to study the numerical solution of functional differential equations and Volterra and Abel integral equations. The principal investigator plans to examine the implementation of fully implicit one step methods and predictor-corrector methods for functional differential equations with step and step/order changing strategies based on the estimation of the local defect rather than the local discretization error. This technique has been recently suggested by Enright in the context of ordinary differential equations. Since the local defect depends on the approximate solution at the grid points as well as the interpolation scheme it is expected that, as in the case of ODE's, this technique will be more reliable than the technique based on the estimation of local discretization error alone. The principal investigator also plans to study stability properties of numerical methods for Volterra and Abel integral and integrodifferential equations with respect to various test equations. These include convolution and nonconvolution test equations, equations with degenerate kernels and equations with completely positive kernels. In many cases the application of numerical methods to these equations lead to recurrence equations with variable coefficients which are difficult to investigate. Some novel approaches to the study the behaviour of solutions to these difference equations will also be investigated.