From: oliveria@engin.umich.edu (Roque Donizete de Oliveira)
Subject: summary: system of first-order delay/neutral ODEs
Date: Tue, 15 Dec 92 19:18:17 EST
Organization: University of Michigan Engineering, Ann Arbor

   In a previous posting I had inquired about the numerical
solution of delay/neutral first-order ODEs. I received
some references and an state-of-art fortran code. 

References:
 1) George Marsaglia, Arif Zaman and John C.W. Marsaglia,
    Math. Comp. 53, 191-201 (1989).
 2) G.S. Virk,
    "Runge-Kutta method for delay differential systems",
    IEE Proc. D 132, 119-123 (1985).
 3) Irving R. Epstein and Yin Luo, J. Chem. Phys. 95, 244-254 (1991).
 4) Neves, K. W., Automatic Integration of Functional Differential
    Equations -- An Approach,
    ACM Trans. Math. Softw., Vol. 1, No. 4, 1975, pp. 357-368.
 5) K. W. Neves and  S. Thompson
    Solution of Systems of Functional Differential Equations
    With State Dependent Delays.

mroussel@alchemy.chem.utoronto.ca (Marc Roussel) described the first 3 references as :
> The first presents a series expansion method which may not be appropriate
> for large systems such as the poster suggested he wanted to examine.
> The second extends Runge-Kutta methods to delay-differential equations
> and may be the best approach available at the moment.  The third
> presents some ideas on integrating delay-differential equations in an
> appendix; it hardly contains a careful study, but may be useful anyhow.

Reference 4 is the famous DMRODE fortran code (get it from netlib, in
the TOMS directory, algorithm 497)

Reference 5 is a new software package called DRKLAG.
DRKLAG consists of a suite of FORTRAN subroutines for the numerical 
solution of systems of functional differential equations with state
dependent delays.
The package implements continuously imbedded Runge-Kutta methods which
are based on C^1 polynomial interpolants.
These interpolants are exploited by the software to
handle the necessary interpolations for delayed solution
values. The interpolants are used also to handle the root finding 
associated with locating the points of derivative discontinuity inherent 
in the solution of delay equations.
In addition, they are used to handle other root finding tasks in a
manner similar to several well-known ordinary differential equation solvers.

If you are interested in DRKLAG, send e-mail to
thompson@reed.science.runet.edu and he
will gladly send you a TeX paper (still unpublished, I believe,
it lists 46 references in this field) of about 40 pages describing 
the method, the fortran source and examples. It runs on unix workstations 
and IBM PCs (the author told me it has some nice graphics capabilities on 
the IBM PC).

I'm using DRKLAG for it quickly (less than 10 seconds) 
solved my system of 4 first-order delay (and neutral) ODEs. 

In addition to these, there are many books on
the subject ("Delay and Differential Equations", 
"Delay Differential Equations and Dynamical Systems", 
"Delay and Functional Differential Equations, by Klaus Schmitt, Academic Press, N.Y. 1972", 
"Ordinary and Delay Differential Equations", etc). Just
go to your library and search for the titles above (sorry
but I don't have the full information about some of these books
with me now).

I'm still interested in Maple or Mathematica macros that 
would help me to solve delay/neutral first-order ODEs.

   Roque
   oliveria@engin.umich.edu
   Nuclear Eng Dept, The U. of Michigan