In mathematics, delay differential equations (DDEs) are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times.
DDEs are also called time-delay systems, systems with aftereffect or dead-time, hereditary systems, equations with deviating argument, or differential-difference equations.
They belong to the class of systems with the functional state, i.e. partial differential equations (PDEs) which are infinite dimensional, as opposed to ordinary differential equations (ODEs) having a finite dimensional state vector.
Four points may give a possible explanation of the popularity of DDEs:[1] A general form of the time-delay differential equation for
represents the trajectory of the solution in the past.
DDEs are mostly solved in a stepwise fashion with a principle called the method of steps.
For instance, consider the DDE with a single delay
Then the solution on the interval
which is the solution to the inhomogeneous initial value problem
This can be continued for the successive intervals by using the solution to the previous interval as inhomogeneous term.
In practice, the initial value problem is often solved numerically.
Then the initial value problem can be solved with integration,
ϕ ( s − τ )
Similarly, for the interval
we integrate and fit the initial condition,
{\displaystyle {\begin{aligned}x(t)=x(\tau )+\int _{s=\tau }^{t}{\frac {d}{dt}}x(s)\,ds&=(a\tau +1)+a\int _{s=\tau }^{t}\left(a(s-\tau )+1\right)ds\\&=(a\tau +1)+a\int _{s=0}^{t-\tau }\left(as+1\right)ds,\end{aligned}}}
In some cases, differential equations can be represented in a format that looks like delay differential equations.
Similar to ODEs, many properties of linear DDEs can be characterized and analyzed using the characteristic equation.
[5] The characteristic equation associated with the linear DDE with discrete delays
is the exponential polynomial given by
det ( − λ
The roots λ of the characteristic equation are called characteristic roots or eigenvalues and the solution set is often referred to as the spectrum.
Because of the exponential in the characteristic equation, the DDE has, unlike the ODE case, an infinite number of eigenvalues, making a spectral analysis more involved.
The spectrum does however have some properties which can be exploited in the analysis.
For instance, even though there are an infinite number of eigenvalues, there are only a finite number of eigenvalues in any vertical strip of the complex plane.
[6] This characteristic equation is a nonlinear eigenproblem and there are many methods to compute the spectrum numerically.
[7][8] In some special situations it is possible to solve the characteristic equation explicitly.
The characteristic equation is
There are an infinite number of solutions to this equation for complex λ.
where Wk is the kth branch of the Lambert W function, so: