## Integrating a differential-functional equation

I need to solve a differential-functional equation:

$\partial_t x_m(t,s) = \sum_n A_{mn} x_n(t,s) + \int_0^t \sum_n \sum_{n'} B_{mnn'}(t,s') x_n(t,s) x_{n'}(t,s') ds'$

with $t > s$ and initial condition $x_m(t,t) = C_m$.

What is the best numerical procedure?

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Huh, that's one strange equation. Where does it come from? Are you sure there's not a simple form? (in particular, one you can differentiate to give a simple differential equation) Anyway, look up litterature for integro-differential equation. Or just bruteforce it with an Euler method and piecewise constant integration, if you don't need high precision.

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 Pls note that the differentiation is in the first parameter, and integration over the second. – Katastrofa Dec 4 2011 at 11:23 Right, sorry. So your equation just doesn't make any sense, because there is no constraint on the s-evolution of x. Think first of the simplest case : in one variable, A = 1, B = 0, $\partial_t x(t,s) = x(t,s)$. Impose an equation for $\partial_s x$ or a constraint. – Antoine Levitt Dec 4 2011 at 12:00 There is an initial condition for x(t,t). – Katastrofa Dec 4 2011 at 17:13

I have solved it in a trivial way: using Euler discretization in the $t$ direction and trapezoid quadrature in $s$ direction, with equal steps. Works quite well for the data I use.

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