Mathematical Sciences: Studies in Numerical Solution of Functional Differential Equations

本项目的目的是研究泛函微分方程和Volterra和Abel积分方程的数值解。主要研究人员计划研究基于局部缺陷估计而不是局部离散化误差的全隐式一步法和预测-校正方法在具有步长和步长/阶变策略的泛函微分方程中的实现。这种方法最近由Enright在常微分方程式的背景下提出。由于局部缺陷依赖于网格点上的近似解和内插方案，因此，与常微分方程的情况一样，这种技术将比仅基于局部离散化误差估计的技术更可靠。主要研究人员还计划研究Volterra和Abel积分和积分-微分方程数值方法关于各种测试方程的稳定性。它们包括卷积和非卷积测试方程、具有退化核的方程和具有完全正核的方程。在许多情况下，将数值方法应用于这些方程导致了很难研究的变系数递推方程。还将探讨研究这些差分方程解的性态的一些新方法。

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.