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

calculation. Thanks in advance,

Herbert