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?
Best Answer
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.