Homogeneous linear discrete delay differential equation with constant coefficients

Definition

A linear discrete delay differential equation with constant coefficients is a particular kind of delay differential equation described as follows:

$\sum _{k=0}^{r}\left(\sum _{j=1}^{m_{k}}A_{k,j}{\frac {d^{k}(x(t-\tau _{k,j}))}{dt^{k}}}\right)=0$

where $m_{k}$ are nonnegative integers for $k\in \{0,1,\dots ,r\}$ and $A_{k,j}$ are real numbers while $\tau _{k,j}$ are nonnegative real numbers.

Procedure for finding smooth solutions

We construct a characteristic equation as follows:

$\sum _{k=0}^{r}\left(\sum _{j=1}^{m_{k}}A_{k,j}z^{k}e^{-z\tau _{k,j}}\right)=0$

We treat this as an equation in $z$. For every real solution $\alpha$, the function $x(t)=e^{\alpha t}$ satisfies the original differential equation. Since the equation is linear, linear combinations of such solutions are also solutions.

Further, if $a\pm ib$ is a pair of complex conjugate solutions to the characteristic equation, then $x(t)=e^{at}\cos(bt)$ and $x(t)=e^{at}\sin(bt)$ solve the original delay differential equation.

In general, the characteristic equation is not a polynomial equation, hence we cannot say anything offhand about the nature or number of its roots. It is often the case that there are infinitely many roots of the characteristic equation. Two cases where it reduces to a polynomial situation are given below.

Case of no delay: linear differential equation with constant coefficients

The homogeneous linear differential equation with constant coefficients can be thought of as a special case of this where there are no delay terms, i.e., all the $\tau _{k,j}$ are equal to zero. In this case, the characteristic equation becomes a polynomial equation in $z$, and this case is extensively studied.

Case of integer delays and zeroth order

This case is often studied as a linear difference equation, though that is more properly viewed as its discrete analogue. Explicitly, this is a situation where $r=0$ (so no derivatives appear) and all the $\tau _{0,j}$ are integers. In this case, the characteristic equation becomes polynomial in $e^{-z}$.