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 time-delay differential equation for is
where represents the trajectory of the solution in the past. In this equation, is a functional operator from to
- 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.
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 is given by which is the solution to the inhomogeneous initial value problem
with . 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.
Suppose and . Then the initial value problem can be solved with integration,
i.e., , where we picked to fit the initial condition . Similarly, for the interval we integrate and fit the initial condition to find that where .
Reduction to ODE
In some cases, delay differential equations are equivalent to a system of non-delay differential equations.
- Example 1 Consider an equation
- Introduce to get a system of ODEs
- Example 2 An equation
- is equivalent to
The characteristic equation
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. 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
where Wk is the kth branch of the Lambert W function.
- Bellman, Richard; Cooke, Kenneth L. (1963). Differential-difference equations. New York-London: Academic Press. ISBN 978-0-12-084850-8.
- Driver, Rodney D. (1977). Ordinary and Delay Differential Equations. New York: Springer Verlag. ISBN 0-387-90231-7.
- Michiels, Wim and Niculescu, Silviu-Iulian (2007). Stability and stabilization of time-delay systems. An eigenvalue based approach. ISBN 978-0-89871-632-0.