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Mathematics
Differential Equations
ODE with non-exact solution: closed-form, non-iterative approximations
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[QUOTE="FranzS, post: 6870220, member: 684446"] [B]TL;DR Summary:[/B] 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. [/QUOTE]
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ODE with non-exact solution: closed-form, non-iterative approximations
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