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

许多物理过程可以用微分方程组来模拟，而微分方程组的精确解是无法构造出来的。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，系统中都有一个自然的延迟，它可以是一个固定的时间延迟，或者在更复杂的模型中，延迟本身的长度取决于系统的当前或过去的状态。在研究空间离散介质（如晶格）中的行波解时，就会出现这样的问题。在那里，以固定形式和速度在介质中传播的行波由微分延迟方程定义，其解取决于过去和未来的状态。从神经纤维传播到固体/液体界面传播，从交通流量到相对论电动力学，这些直观上荒谬的方程出现在各种各样的问题中。

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.