Volterra 积分代数方程和积分微分代数方程的具体模型广泛存在于物理、化学和工程等众多科学技术领域。配置方法是一个高效的数值方法，广泛应用于各类积分和微分方程。本项目以具有弱奇异核、常延迟及比例延迟的积分代数方程和积分微分代数方程为研究对象。对这几类方程的指标、解耦及解的正则性等基本理论进行系统地分析和讨论；根据这几类积分代数方程的正则性设计合适的配置方法的步长，以获得较高的收敛阶；根据这几类积分微分代数方程的解耦信息选取合适的配置空间以获得较广泛的配置方法的收敛性和超收敛性。本项目涉及了一些新的研究课题，不仅可以使这些方程的理论分析得到完善，而且也对获得这些方程的其他高精度的数值方法具有指导和借鉴意义，为实际应用提供一定的理论依据。

Volterra integral algebraic equations (IAEs) and integro-differential algebraic equations (IDAEs) arise in many mathematical modeling processes, such as physics, chemistry and engineering, etc. Collocation methods, widely used for many kinds of integral and differential equations, are of high accuracy. In this project, we regard IAEs and IDAEs with weakly singular kernels, constant delays and pantograph delays as research subjects. Some fundamental theories, such as the definition of index, decoupling and regularity, will be thoroughly analyzed and discussed. For these kinds of IAEs, to obtain high convergence orders, some suitable stepsizes of collocation methods will be designed according to the regularity. For these kinds of IDAEs, to gain the general convergence and superconvergence of collocation methods, the natural collocation space will be chosen according to the decoupling. The project refers to some new research topics, and not only improves the connotation of the theoretical analysis, but also will be guidance and reference for other high accuracy numerical methods of these equations, and provides some theoretical evidences for practical applications.