- #1

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- TL;DR Summary
- What is a good approach for approximating a non-polynomial function appearing in an ODE in order to find a closed-form approximate solution?

In case of an integral ##\rightarrow## differential equation of the type:

$$ f(t) = \int_0^t g(f(\tau)) d\tau $$

$$ \rightarrow \frac{df(t)}{dt} = g(f(t)) $$

which turns out not to be solvable in exact form because ##g(f(t))## is a non-polynomial function (but it would if ##g(f(t))## was a polynomial), how would you approximate ##g(f(t))##?

The purpose is to get a closed-form approximate solution with no iterative processes.

Given the "dynamic evolution / evolving nature" of an ODE, very loosely speaking, I would assume that it is better to consider a Taylor polynomial centered in ##t=0## (lower limit of the integral, to be considered as the "starting instant" of a physical evolving system described by the above equations), whereas — for instance — a multilinear ("polynomial") regression would provide a "wider" overall accuracy for the "static" ##g(f(t))## (so to speak) but its worse approximation at ##t=0## would add greater and greater error as time passes.

$$ f(t) = \int_0^t g(f(\tau)) d\tau $$

$$ \rightarrow \frac{df(t)}{dt} = g(f(t)) $$

which turns out not to be solvable in exact form because ##g(f(t))## is a non-polynomial function (but it would if ##g(f(t))## was a polynomial), how would you approximate ##g(f(t))##?

The purpose is to get a closed-form approximate solution with no iterative processes.

Given the "dynamic evolution / evolving nature" of an ODE, very loosely speaking, I would assume that it is better to consider a Taylor polynomial centered in ##t=0## (lower limit of the integral, to be considered as the "starting instant" of a physical evolving system described by the above equations), whereas — for instance — a multilinear ("polynomial") regression would provide a "wider" overall accuracy for the "static" ##g(f(t))## (so to speak) but its worse approximation at ##t=0## would add greater and greater error as time passes.