Adaptive collocation methods of volterra functional equations

物理和生物科学中的许多涉及记忆效应的数学建模问题导致普通或部分沃尔泰拉积分微分方程（系统），其解在方程必须求解的区间中的某些点处具有低正则性。 如果这些函数方程中的沃尔泰拉积分算子具有弱奇异（无界）核或/和延迟变元（Vito沃尔泰拉的延迟积分微分模型（1927年）是后一种情况的著名例子），则会发生这种正则性损失。 由于基于先验选择的（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-一般来说效率低下，因此基于“动态”选择的网格的所谓自适应方法已成为选择的方法，该网格是由后验误差估计技术产生的。本研究建议关注的数学和计算方面的后验误差估计和自适应时间步进计划的功能方程，涉及沃尔泰拉积分算子的上述类型。 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;学生和博士后研究员在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.