Adaptive collocation methods of volterra functional equations

在物理和生物科学中，许多涉及记忆效应的数学建模问题导致（系统）普通或部分Volterra积分-微分方程，其解在必须求解方程的区间内的某些点上具有低程度的规律性。如果这些泛函方程中的Volterra积分算子具有弱奇异（无界）核或/和延迟参数（Vito Volterra的两个竞争物种的延迟积分-微分模型（1927）是后一种情况的著名例子），则会发生正则性损失。由于基于先验选择（分级或约束）时间网格的数值方案-其数学理论现在已经很好地理解-通常效率低下，所谓的自适应方法基于由后判误差估计技术产生的“动态”选择网格已成为首选方法。目前的研究计划涉及涉及上述类型Volterra积分算子的泛函方程的后验误差估计和自适应时间步进方案的数学和计算方面。这项工作将与中国、意大利、爱沙尼亚、苏格兰和加拿大的研究人员合作进行；它还将使博士生和博士后了解现代数值分析和科学计算的许多重要发展。

Many mathematical modelling problems in the physical and biological sciences that involve memory effects lead to (systems of) ordinary or partial Volterra integro-differential equations whose solutions possess a low degree of regularity at certain points in the interval on which the equations have to be solved.  This loss of regularity occurs if the Volterra integral operators in these functional equations possess weakly singular (unbounded) kernels or/and delay arguments (Vito Volterra's delay integro-differential model (1927) for two competing species is a celebrated example of the latter case).  Since numerical schemes that are based on a priori selected (graded or constrained) temporal meshes - for which the mathematical theory is now quite well understood - are in general inefficient, so-called adaptive methods based on `dynamically' selected meshes resulting from a posteriori error estimation techniques have become the methods of choice. The present research proposal is concerned with the mathematical and computational aspects of a posteriori error estimation and adaptive time-stepping schemes for functional equations involving Volterra integral operators of the types described above.  The work will be carried out in collaboration with researchers in China, Italy, Estonia, Scotland and Canada; it will also expose Ph.D. students and postdoctoral fellows to many important developments in modern numerical analysis and scientific computation.