Fully Discretised Collocation Methods For Volterra Functional Integral Equations With Non-vanishing Delays

Volterra integral equations are widely used in natural sciences and social sciences,such as demographics,mechanics,ecology,automatic control,economic management,biopharmaceuticals,and so on.Volterra integral equations with delays not only cover the classical Volterra integral equation,but also cover some differential equations with delays.Due to the existence of the delay terms,the regularity of the equations is usually lower,and its theoretical research and numerical methods are more complex and are more and more widely concerned.In this paper,collocation methods for Volterra functional integral equations with non-vanishing delays are studied,focusing on the convergence of the full discretised collocation scheme.In the first chapter,we first introduce the reserch background and significance of the Volterra integral equations.Then we briefly analyze and summarize the research results of the Volterra integral equations with delays,and propose the research content of this paper.In the second chapter,we perform secondary discretisation of the integral occurring in the exact collocation solutions for Volterra functional integral equations with non-vanishing delays,thereby obtaining the fully discretised collocation scheme and corresponding iterative collocation scheme,then the global convergence and local superconvergence are analyzed in deail.The results show that they have the same convergence properties as the exact collocation solutions.In particular,the fully discretised collocation method based on the m Gauss points only has superconvergence in the first macro-interval,but cannot reach the optimal global superconvergence order m + 1 and the local superconvergence of order2 m in other macro-intervals.However,if the fully discretised collocation method selects m Radau II points as the collocation parameters,the local superconvergence order at the mesh points can reach up to 2m-1.In Chapter 3,some typical numerical examples are given.We consider the collocation method based on the Radau II and Gauss points to verify the convergence order of fully discretised collocation method and fully discretised iterative collocation method.The results show that the convergence order of numerical experiments is consistent with the theoretical analysis.