Adaptive collocation methods of volterra functional equations

volterra函数方程的自适应配置方法

• 批准号: 9406-2006
• 负责人: Brunner, Hermann
• 金额: $ 1.68万
• 依托单位: Memorial University of Newfoundland
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=363123
• 关键词: Adaptive; collocation; methods; volterra; functional

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.

物理和生物科学中的许多数学建模问题涉及记忆效应，导致常或偏沃尔泰拉积分微分方程（组），其解在方程求解区间的某些点上具有低正则性。如果这些函数方程中的沃尔泰拉积分算子具有弱奇异性，则会出现这种正则性的丧失（无界）核或/和延迟参数（Vito沃尔泰拉的两个竞争种群的延迟积分微分模型（1927）是后一种情况的著名例子）。（分级的或约束的）时间网格--其数学理论现在已经被很好地理解--通常是低效的，基于由后验误差估计技术产生的“动态”选择的网格的所谓自适应方法已经成为选择的方法。目前的研究计划是关注的数学和计算方面的后验误差估计和自适应时间步长计划的功能方程，涉及沃尔泰拉积分算子的上述类型。这项工作将进行合作，在中国，意大利，爱沙尼亚，苏格兰和加拿大的研究人员，它也将暴露博士学位。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。