Let's say I have a function $N$ whose future value at a time $t + t_{d}$ obeys the relation
$N(t + t_{d}) = A(t)N(t)$
where $A(t)$ is also a function of $t$ whose value can be calculated. One can discreetly calculate $N(t)$ for all integer multiples of $t_{d}$ but I would like to be able to approximate this by a differential closed form. If one considers the Taylor expansion then
$N(t + t_{d}) = N(t) + t_{d}N'(t) + \frac{t_{d}^2}{2}N''(t) +....$
If I drop all terms higher than $N'(t)$, I can naively combine these to get
$N'(t) \approx \frac{1}{t_{d}}(A(t) - 1)N(t)$
which I could readily solve by numerical methods. However, I'm not sure of the validity of this approach; $t_{d}$ is small but not infinitesimal but I'm not sure whether this matters. More importantly, if this is a valid approach I also don't know how to determine whether this simplification is justified. I could, for example, keep in $N''(t)$ to yield the second order identity
$N''(t) + \frac{2}{t_{d}}N'(t) - \frac{2}{t_{d}^2}(A(t)-1)N(t) \approx 0$
or I could go higher order. I imagine the errors decrease as one adds more terms, but I am unfamiliar with any method of quantifying these errors when one truncates the series. My specific query is then three-fold
(1) Is such an approach valid for the problem I've outline?
(2) If so, is there a way to quantify the expected error as one truncates?
(3) Following on from this, is there a rule of thumb for where one should choose to truncate ?
I'd be very grateful if you could point in the right direction here!
Try methods for first-order homogeneous difference equations? Define $n_k = N( t+k\,t_D )$, $a_k = A( t + k\,t_D )$.
$$n_{k+1} - a_k n_k = 0.$$
There is a closed-form solution, but I don't want to deprive you of all the fun! :-)