Mathematical Sciences: Studies in Numerical Solution of Functional Differential Equations

本课题的目的是研究泛函微分方程和Volterra和Abel积分方程的数值解。首席研究员计划研究基于局部缺陷而非局部离散化误差估计的阶跃和阶跃变化策略的泛函微分方程的全隐式一步方法和预测校正方法的实现。最近，恩莱特在常微分方程中提出了这种方法。由于局部缺陷取决于网格点的近似解以及插值方案，因此可以预期，与ODE的情况一样，该技术将比仅基于局部离散化误差估计的技术更可靠。主要研究者还计划研究Volterra和Abel积分和积分微分方程的数值方法相对于各种测试方程的稳定性。这些包括卷积和非卷积测试方程，退化核方程和完全正核方程。在许多情况下，数值方法对这些方程的应用导致了难以研究的变系数递归方程。本文还将探讨研究这些差分方程解的行为的一些新方法。

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.