Analysis and computation of state-dependant retarded & advanced-retarded differential boundry value problems

状态相关延迟的分析与计算

• 批准号: 261389-2008
• 负责人: Humphries, Antony
• 金额: $ 1.31万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2009
• 资助国家: 加拿大
• 起止时间: 2009-01-01 至 2010-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=426577
• 关键词: Analysis; computation; state; dependant; retarded

Many physical processes can be modelled by systems of differential equations for which exact solutions cannot be constructed. Nevertheless either, analytical techniques can be used to deduce characteristic features of the solution, or numerical techniques can be used to find highly accurate approximations to solutions. But, in many of these processes there is a natural delay built into the system, which can be either be a fixed time delay, or in more complex models the length of the delay itself can depend on the current or past state of the system. One situation where such problems arise is in the study of travelling wave solutions in spatially discrete media, such as crystal lattices. There, the travelling wave, which propagates through the medium with a fixed form and velocity, is defined by differential delay equations where the solution depends on both past and future states. Such intuitively absurd equations arise in a surprisingly wide variety of problems from nerve fibre propagation to solid/liquid interface propagation to traffic flow to relativistic electro-dynamics.

许多物理过程可以通过微分方程组来建模，但无法构造精确的解。然而，分析技术可以用来推断解的特征，或者数值技术可以用来找到解的高度精确的近似值。但是，在许多这些过程中，系统中内置有自然延迟，它可以是固定时间延迟，或者在更复杂的模型中，延迟本身的长度可以取决于系统的当前或过去状态。出现此类问题的一种情况是在空间离散介质（例如晶格）中行波解的研究。在那里，以固定形式和速度通过介质传播的行波由微分延迟方程定义，其中解取决于过去和未来的状态。这种直观上荒谬的方程出现在各种各样的问题中，从神经纤维传播到固/液界面传播，再到交通流，再到相对论电动力学。