# Help for integro-differential Equations

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Posted by Herbert Nachbagauer on June 18, 1998 at 07:09:24:

Hello,
I need some help on the numerical solution of
a system of integro-differential equations since
I am not expert on numerical problems.

I came across that beasty thing

d/dt y (t) = f ( y (t) ) + int_0^t F( y(t) , y(t') ) dt' .

with some initial conditions y(0).

y(t) is a vector I'm trying to solve the equation for,
and f and F are some given non-linear vector-valued functions
of the unknown y. (There is no explicit time dependence
if that helps, and F and f are both smooth functions
in y(t),y(t')). Is there any particular numerical strategy
to solve such type of equations. I think, the nasty thing
about it is the dependence of F on both, the parameter
t' integrated over, and the value of y(t) at the upper
limit of integration. I've tried predictor-corrector
methods, but they turn out to be unstable even for
values of t being 'not too large'. By that I mean that
the solution oscillates a couple of times before the
integration procedure breaks down, probably due to
exponentiation of errors. Is there some idea of how
the integrator for the integral at the r.h.s
and the integrator for integrating the differential
equation should be -- or should not be related ?
It would help if something could be said about the
case of scalar y,f and F. I'm quite sure that the
solution remains bounded for large t, and just
oscillates a bit, but I do want to show that by