Delay differential equation

Delay differential equation
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "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 $x(t)\in R^n$ is

$\frac{d}{dt}x(t)=f(t,x(t),x_t),$

where $x_t=\{x(\tau):\tau\leq t\}$ represents the trajectory of the solution in the past. In this equation, $f$ is a functional operator from $R\times R^n\times C^1$ to $R^n.\,$

The following form is called — after the pantographs on trains — the pantograph equation:

$\frac{d}{dt}x(t) = ax(t) + bx(\lambda t),$

where a, b and λ are constants and 0 < λ < 1. Often some more general forms of this equation are used, also being called pantograph equation.

content

1 Examples
2 Reduction to ODE
3 The characteristic equation
4 External links

Examples

Continuous delay

$\frac{d}{dt}x(t)=f\left(t,x(t),\int_{-\infty}^0x(t+\tau)d\mu(\tau)\right)$

Discrete delay

$\frac{d}{dt}x(t)=f(t,x(t),x(t-\tau_1),\ldots,x(t-\tau_n))$ for $\tau_1>\ldots>\tau_n\geq 0$ .

Reduction to ODE

In some cases, delay differential equations are equivalent to a system of ordinary differential equations.

Example 1 Consider an equation

$\frac{d}{dt}x(t)=f\left(t,x(t),\int_{-\infty}^0x(t+\tau)e^{\lambda\tau}d\tau\right).$

Introduce $y(t)=\int_{-\infty}^0x(t+\tau)e^{\lambda\tau}d\tau$ to get a system of ODEs

$\frac{d}{dt}x(t)=f(t,x,y),\quad \frac{d}{dt}y(t)=x-\lambda y.$

Example 2 An equation

$\frac{d}{dt}x(t)=f\left(t,x(t),\int_{-\infty}^0x(t+\tau)\cos(\alpha\tau+\beta)d\tau\right)$

is equivalent to

$\frac{d}{dt}x(t)=f(t,x,y),\quad \frac{d}{dt}y(t)=\cos(\beta)x+\alpha z,\quad \frac{d}{dt}z(t)=\sin(\beta) x-\alpha y,$

where

$y=\int_{-\infty}^0x(t+\tau)\cos(\alpha\tau+\beta)d\tau,\quad z=\int_{-\infty}^0x(t+\tau)\sin(\alpha\tau+\beta)d\tau.$

The characteristic equation

Solutions of linear DDEs can be studied by analyzing the characteristic equation, similarly to the ODEs. However, for DDEs, the characteristic equation can have continuum of solutions, making the spectral analysis hard. Consider, for example, the following equation:

$\frac{d}{dt}x(t)=-x(t-1).$

As for ODEs, we seek a solution of the form x(t) = e^λt. This results in the characteristic equation for λ:

$\lambda=-e^{-\lambda}.\,$

There are an infinite number of solutions to this equation for complex λ (they are given by the Lambert W function).

External links

Delay-Differential Equations at Scholarpedia, curated by Skip Thompson.

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Contents

Examples

Reduction to ODE

The characteristic equation

External links