50 - Delay Equations

Abstract

This chapter focuses on ordinary differential delay equations. There are several standard techniques for delay equations. The standard methods for solving delay equations are by the use of: (1) Laplace transforms, (2) Fourier transforms, (3) generating functions, (4) general expansion theorems and (5) the method of steps. For the first two methods, the technique is the same as it is for ordinary differential equations. That is, the transform is taken of the delay equation, by algebraic manipulations the transform is explicitly determined; and then an inverse transformation is taken. For a delay equation with a single delay, the method of steps consists of solving the delay equation in successive intervals, whose length is the time delay. In each interval, only an ordinary differential equation needs to be solved. The method of generating functions is frequently used when only integral values of the variables are of interest. The technique is similar to the technique for integral transforms. For generating functions the integration is replaced by a summation, and the “inverse transformation” is generally a differentiation.