Retarded differential equation

Retarded differential equations are a special type of differential equation , often abbreviated as DDE (Delayed Differential Equation) or referred to as a differential equation with a lagging argument . With them, the derivation of an unknown function at the point in time depends not only on the function value at this point in time, but also on function values ​​at earlier points in time or on integrals over the function over past time intervals. DDEs play a role in models in which the effect only follows the cause with a delay (retarded). Well-known examples can be found in epidemiology ( infection , incubation period ), population development in biology ( reproduction , sexual maturity) and control engineering (delay time). ${\displaystyle t}$ ${\displaystyle t-\tau _{i}}$

notation

A DDE with an unknown function and a point-by-point delay can be called ${\displaystyle x(t)}$

${\displaystyle {\dot {x}}=f(t,x(t),x_{\tau _{1}},\ldots ,x_{\tau _{n}})}$ be noted with

${\displaystyle {\dot {x}}={\frac {\rm {d}}{{\rm {d}}t}}x(t)}$and . ${\displaystyle x_{\tau _{n}}=x(t-\tau _{n})}$

A continuous delay DDE can be used as a

${\displaystyle {\dot {x}}=f\left(t,x(t),\int _{-\infty }^{0}x(t+\tau ){\rm {d}}\mu (\tau )\right)}$

to be written.

Examples

• Population development

Let be the population density of sexually mature individuals , the time it takes to reach sexual maturity, the per capita reproductive rate , the death rate and the likelihood of reaching sexual maturity. Then the population density develops accordingly ${\displaystyle x}$ ${\displaystyle \tau }$ ${\displaystyle \alpha }$ ${\displaystyle \mu }$ ${\displaystyle p}$

${\displaystyle {\dot {x}}=-\mu x(t)+\alpha px(t-\tau )}$

particularities

Population development of a species

Compared to the initial values ​​for non-delayed differential equations, the function of DDEs must be given over a time interval that is at least as long as the maximum delay. Since there are no starting values ​​as with non-delayed starting value problems, but starting functions with in principle an infinite number of parameters, one also speaks of infinite-dimensional systems. Another special feature is that discontinuities in the initial conditions are gradually shifted to higher derivatives. Is z. B. above DDE with the parameters with at and initialized, the result is the population development shown. At the point in time , the jump from to is carried over to the first derivative at , the discontinuity is carried over from the first derivative to the second and so on, see also the example of step-by-step integration . Initial discontinuities in DDEs subside over time. ${\displaystyle x(t)}$ ${\displaystyle n}$ ${\displaystyle \mu =-0{,}1,\alpha p=0{,}5,\tau =3}$ ${\displaystyle x(t)=0}$ ${\displaystyle -3\leq t<0}$ ${\displaystyle x(0)=10}$ ${\displaystyle t=3}$ ${\displaystyle t=0}$ ${\displaystyle x=0}$ ${\displaystyle x=10}$ ${\displaystyle {\dot {x}}}$ ${\displaystyle t=6}$

Solution methods

Most DDEs do not have an analytical solution, so that one has to rely on numerical methods .

Integrate step by step

If a separation of the variables is possible, a closed solution can be obtained through gradual integration. To illustrate this, consider a DDE with a delay time : ${\displaystyle \tau }$

${\displaystyle {\frac {\rm {d}}{\rm {{d}t}}}x(t)=f(x(t),x(t-\tau ))}$

and the initial condition . ${\displaystyle \phi (t)\colon [-\tau ,0]\to \mathbb {R} ^{n}}$

The solution on the interval is then by solving the inhomogeneous initial value problem ${\displaystyle x_{1}(t)}$ ${\displaystyle [0,\tau ]}$

${\displaystyle {\frac {\rm {d}}{\rm {{d}t}}}x_{1}(t)=f(x_{1}(t),\phi (t-\tau ))}$

given with . Now the solution can be used as the initial condition for the solution on the interval . By repeating these steps N times, a closed solution can be found on the interval . ${\displaystyle x_{1}(0)=\phi (0)}$ ${\displaystyle x_{1}(t)}$ ${\displaystyle \psi (t):=x_{1}(t)}$ ${\displaystyle x_{2}(t)}$ ${\displaystyle [\tau ,2\tau ]}$ ${\displaystyle [0,N\tau ]}$

example

The DDE with the initial condition for leads to the inhomogeneous differential equation ${\displaystyle {\frac {\rm {d}}{{\rm {d}}t}}x(t)=x(t-1)}$ ${\displaystyle x(t)=\phi (t)=1}$ ${\displaystyle t\leq 0}$

${\displaystyle {\frac {\rm {d}}{{\rm {d}}t}}x_{1}(t)=1}$for . ${\displaystyle t\in [0,1]}$

By separating the variables you win

${\displaystyle \int _{x_{1}(0)}^{x_{1}(t)}{\rm {d}}x'=\int _{0}^{t}1{\rm {d}}t'}$

${\displaystyle x_{1}(t)-1=t}$

${\displaystyle x_{1}(t)=t+1}$,

with which the solution for the interval is known. For the interval one finds ${\displaystyle 0\leq t\leq 1}$ ${\displaystyle 1\leq t\leq 2}$

${\displaystyle \int _{x_{2}(1)}^{x_{2}(t)}{\rm {d}}x'=\int _{1}^{t}t'{\rm {d}}t'}$

${\displaystyle x_{2}(t)-2={\frac {t^{2}}{2}}-{\frac {1}{2}}}$

${\displaystyle x_{2}(t)={\frac {t^{2}}{2}}+{\frac {3}{2}}}$,

and so on. The overall solution is then given as a composite function of these partial solutions:

${\displaystyle x(t)={\begin{cases}x_{1}(t),&0\leq t\leq 1\\x_{2}(t),&1\leq t\leq 2\\\dots ,&\dots \\x_{N}(t),&N-1\leq t\leq N\end{cases}}}$.

Rewrite as a non-delayed DGL system

Sometimes continuous DDE can be written as a system of ordinary differential equations.

example

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

Through the substitution one obtains through partial integration ${\displaystyle y(t)=\int _{-\infty }^{0}x(t+\tau )e^{\lambda \tau }{\rm {d}}\tau }$

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

Web links

• DDEs on Scholarpedia.org
• Partial DDEs on Scholarpedia.org
• Homepage of XPPAUT , a free program with which, among other things, DDEs can be numerically integrated.

swell

1. ↑ O. Arino, ML Hbid, E. Ait Dads (ed.): Delay Differential Equations and Applications. In: NATO Science Series II: Mathematics, Physics and Chemistry . Springer-Verlag, Netherlands 2006.
2. ^ MR Roussel: Delay-differential equations. (PDF; 110 kB) 2005.