I try to solve an integro-differential equation system numerically (with LSODA.) The system is following: ( $'$ is the derivation w.r.t. $x$)
$y'(x,s_{1}) = f(x,y(x,s_{1}),I[y(x,s)])\\ y'(x,s_{2}) = f(x,y(x,s_{2}),I[y(x,s)])\\ \vdots \\ y'(x,s_{n}) = f(x,y(x,s_{n}),I[y(x,s)])\\$
LHS, inside of $f$, includes also integration of $y(x,s)$ over $s$ , $I[y(x,s)]$ (Thank you @Lutz for notation correction ). For the optimized numerical calculation, I discrete $y(x,s)$ $n$ times for both integral and number of system i.e. system has $n$ differential equation and integral at LHS is discreted $n$ times.
Do you suggest any generic condition for $n$ determination? If I take big $n$ for better integration, the system gets huge and numerical solving is getting slow. Small $n$ makes error of integral bigger which I do not want.
Edit 1:
Example: Equations
$y'(x,s_{1}) = e^{x}\int_{0}^{\pi}y(x,s)ds\\ y'(x,s_{2}) = e^{x}\int_{0}^{\pi}y(x,s)ds\\ \vdots \\ y'(x,s_{n}) = e^{x}\int_{0}^{\pi}y(x,s)ds\\$
Here, LHS function is $f(x,y(x,s_{n}))=e^{x}\int_{0}^{\pi}y(x,s)ds$.
Suppose that we will use Euler Method with initial conditions $y(0,s)=\cos{s}$ where $s\in[0,\pi]$ and $\Delta x = 1$:
$y(1,s_{1}) = y(0,s_{1}) + (1-0)\left[e^{0}\int_{0}^{\pi}y(0,s)ds \right]\\ y(1,s_{2}) = y(0,s_{2}) + (1-0)\left[e^{0}\int_{0}^{\pi}y(0,s)ds \right]\\ \vdots \\ y'(1,s_{n}) = y(0,s_{n}) + (1-0)\left[e^{0}\int_{0}^{\pi}y(0,s)ds \right]$