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I have a general question about delay differential equations. I know that even simple ones hardly have analytic solutions and mine clearly doesn't have any as it is a system of non-linear delay differential equations.

I am looking for ways of solving them numerically. I know a few algorithms to solve ODEs numerically.

Using a basic algorithm (say Euler's method), the equation is : $$f_{n+1} = f_n + hf'_n$$ where $h$ is the step size.

For DDE, is it possible to use $f_{n-a}$ instead of $f_n$ for some $a$ corresponding to the delay?

How does it work for more complicated methods such as the Runge-Kutta family or the PECE methods?

If it works, has there been any study on the stability of the solvers?

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There is an excellent monograph on the topic by Bellen and Zennaro.

You can find quite sophisticated FORTRAN code on the homepage of Nicola Guglielmi.

There is also a good analysis of numerical methods for stiff delay equations on the corresponding scholarpedia article.

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