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

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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.