Fundamental time-domain approaches for variable time delay

The partial differential equation that characterizes variable time delay is analyzed. Analytical solution methods are applicable when endpoint conditions and variable delay time are known in advance; two such methods are given. When variable time delay is but one part of a larger system, however, the simulation of the larger system on a computer can be facilitated by the use of finite difference approximations of the variable delay element. A class of variable delay approximations is developed from which several commonly used approximations follow as special cases. A particular approximation having truncation error of the order of (Δx)⁴ is found to be near optimum. Simulation results are presented.