Applications of delay differential equations

Question

Being interested in the mathematical theory, I was wondering if there are up-to-date, nontrivial models/theories where delay differential equations play a role (PDE-s, or more general functional differential equations).

It is clear that

• in biological (population) models usually pregnancy introduces a delay term, or
• in disease transition in networks the latent period introduces a delay, or
• in engineering in feedback problems signal processing introduces the time delay.

I would like to see a list of answers where each answer contains one reference/example.

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Comments

Is it possible to make this Community Wiki?

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"Community wiki" option is the check-box on the lower-right corner of the question. ... Just checkmark it and press 'Save Edit'

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+1 [My apologies: if you're asking specifically for "up-to-date, nontrivial models / theories" I retract my previous comments.]

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@abkati: Small editorial issue - use stars (i.e. *) to make an list, not hyphens (i.e. -).

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Thanks for the edit. It might be my inability, but I do not find that checkmark (I looked for it...). Can someone edit the question for me?

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questions can only be turned CW by mods now, i.e. use the flagging mechanism

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I'm curious, is there rigouros definition for a delay PDE? Or are all PDE's concidered delay once they exhibit this trait?

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Answer 1

In my corner of things what comes to mind is a recent paper by Atiyah and Moore A Shifted View of Fundamental Physics.

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Answer 2

I'm going to give a Material Modelling Example.

Rubber has the property that it takes time to adjust to the conditions it is applied to (Visco-Elasticity). The behaviour of these effects for static loads are known (Relaxation).

However with the high use of Rubbers for dynamic engineering applications (Example: Car tyres) it has been the new focus to study this property more thoroughly using dynamical loads and deriving a suitable Material Model (Lion & Höfer, Lion & Rendek).

The PDE's that arise from this are partly pure theoretical derived from a generalized Maxwell Modell combined with phenomenological effects.

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Answer 3

I've seen delayed differential equations used in modeling lasers, particularly quantum dot lasers. Here is a nice comparative view of the use of delayed differential equations verses a finite difference model in quantum dot lasers.

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Answer 4

In fluid dynamics it is often possible for a given geometry to isolate different degrees of freedom and model long range effects by a delayed influence. An example of this is the "delayed action oscillator" for the El Niño/Southern Oscillation phenomenon in oceanography.

For more details, see the page on the Azimuth wiki here.

Often fast degress of freedom are modeled by noise, i.e. the model consists of a delayed stochastic differential equation. An example for such a model can be seen here on the arXiv.

Besides the applications in climate science and abstract statistical physics, this kind of modelling is also important in engineering applications where the simplification is needed in order to get numerical results in time, for a successful control of a given system. Details can be found here:

• Harold J. Kushner: "Numerical methods for controlled stochastic delay systems." (Systems & Control: Foundations & Applications. Boston, MA: Birkhäuser)

Applications for biological systems can be found in this monograph:

• Hal: Smith: "An introduction to delay differential equations with applications to the life sciences." (Texts in Applied Mathematics 57. New York, NY: Springer.)

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Answer 5

I have found the following paper by Frederik Beaujean and Nicolas Moeller quite interesting.

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