 Delay differential equation

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.
A general form of the timedelay differential equation for is
where represents the trajectory of the solution in the past. In this equation, f is a functional operator from to
Contents
Examples
 Continuous delay
 Discrete delay

 for .
 Linear with discrete delay

 where .
 Pantograph equation

 where a, b and λ are constants and 0 < λ < 1. This equation and some more general forms are named after the pantographs on trains.
Solving DDEs
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
with given initial condition . Then the solution on the interval [0,τ] is given by ψ(t) which is the solution to the inhomogeneous initial value problem

 ,
with ψ(0) = ϕ(0). 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.
Example
Suppose f(x(t),x(t − τ)) = ax(t − τ) and ϕ(t) = 1. Then the initial value problem can be solved with integration,

 ,
i.e., x(t) = at + 1, where we picked C = 1 to fit the initial condition x(0) = ϕ(0). Similarly, for the interval we integrate and fit the initial condition to find that x(t) = at^{2} / 2 + t + D where D = (a − 1)τ + 1 − aτ^{2} / 2.
Reduction to ODE
In some cases, delay differential equations are equivalent to a system of ordinary differential equations.
 Example 1 Consider an equation
 Introduce to get a system of ODEs
 Example 2 An equation
 is equivalent to
 where
The characteristic equation
Similar to ODEs, many properties of linear DDEs can be characterized and analyzed using the characteristic equation^{[1]}. The characteristic equation associated with the linear DDE with discrete delays
is

 .
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 a 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 to the right of any vertical line in the complex plane.
This characteristic equation is a nonlinear eigenproblem and there are many methods to compute the spectrum numerically^{[2]}. In some special situations it is possible to solve the characteristic equation explicitly. Consider, for example, the following DDE:
The characteristic equation is
There are an infinite number of solutions to this equation for complex λ. They are given by
 λ = W_{k}( − 1),
where W_{k} is the kth branch of the Lambert W function.
Notes
 ^ Michiels, Niculescu, 2007 Chapter 1
 ^ Michiels, Niculescu, 2007Chapter 2
References
 Bellman, Richard; Cooke, Kenneth L. (1963). Differentialdifference equations. New YorkLondon: Academic Press. ISBN 9780120848508.
 Driver, Rodney D. (1977). Ordinary and Delay Differential Equations. New York: Springer Verlag. ISBN 0387902317.
 Michiels, Wim and Niculescu, SilviuIulian (2007). Stability and stabilization of timedelay systems. An eigenvalue based approach. ISBN 9780898716320.
External links
 DelayDifferential Equations at Scholarpedia, curated by Skip Thompson.
Categories: Differential equations
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