# Solving integro-differential equation - numerically

I want to solve a integro-differential equation numerically.

The equation is given by :

$\dot{c}(t)=-\int_0^t \mathrm{d}t_1f(t-t_1)c(t_1)$

Hereby, $f(t-t_1)$ will be given a realisation of some random numbers, e.g. $f(t-t_1)$ originally was a rondom variable, and I want to solve the integer differential equation for one realisation. So the function will most certainly be a messy polynomial, where the solution cannot be obtained easily with the Laplace-transform used to solve the equation.

How is the best procedure to numerically solve this (or any) integer differential equation?

Thanks already Martin

• What data do you have? Values for $c(t)$ and $t_1$? – unseen_rider Sep 8 '17 at 15:26
• Also, what is the regularity of f? Is it, e.g. Continuous, bounded derivative? – Zach Boyd Sep 9 '17 at 18:13
• Thanks for the reply. I have none. I just have discrete values for $f(t-t_1)$ for each $t_1$ from $0$ up to $t$. In the end I want to make the steps of $t_1$ very small and 'smear out' my discrete values to get a continuous function for $f(t-t_1)$. Then I want to solve the integro-differential equation given. The functional dependence may be very messy, so solving it with the Laplace-transform is not my first choice (I would need the inverse Laplace-transform at some point, and that may be difficult because of many roots of the function). Thus I want to know if there are numberical approaches. – Martin Sep 9 '17 at 18:14