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is anyone aware of a publication or one or several method(s) where one can construct ODE models based on a series or a function, defined over an interval in space [a,b]?

The function would be nonlinear and one-dimensional, and the respective plot would be of a wave-train with time-dimension on the x-axis and the amplitude on the y-axis.

So the interesting part is that these functions, which are trigometric complex, are particularly important on the interval $[a,b]$, where $a,b\in \mathbb{R}$. These functions are square integrable, and I have already used the metric $d(x,y)=\sqrt{x^2+y^2}$ to find the distance between them.

Now, since these are wave-functions for fluids, I take there can be higher order derivatives in the ODE. However, by EdiPiaf's suggestion we start by making the assumption that the ODE looks like:

$$k^2u+u′′=f$$

So say that I take function x, and subject it to the first derivative, and then add that to it multiplied by some constant, then I get the force function, as EdiPiaf says. Would this be a ODE that solves these functions, which are NON-linear? What other things should I take into account?

I want to add the plot of these functions, which is referred to here:

https://mathematica.stackexchange.com/questions/277880/how-can-i-find-an-intermediate-function-between-a-set-of-functions

Thanks

Luthier415Hz
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    The question lacks clarity. For PDEs, one should consider knowledge of a function of several variables, say $u(x,t)$. Note that such inverse problems are often ill-posed (non-uniqueness, instability, etc.), see this post. Could you tell us more about the underlying physics (system, boundary conditions, etc.)? – EditPiAf Jan 17 '23 at 10:34
  • You are right, it may be a far-fetched ambition without realism. I am interested in extracting, and have extracted, analytic functions from plots of waves. These functions are very long and complex, but even so, it would be a paramount result to relate these functions to a ODE of higher order. PDE is not a necessity, since these functions are actually one.dimensional. I should change the classification. – Luthier415Hz Jan 17 '23 at 10:47
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    That sounds already a bit more realistic. If we restrict ourselves to time-periodic solutions of linear wave equations over a bounded periodic domain, say $$ k^2 u + u'' = f$$ for $x$ in $[0,L]$ with $k=\omega/c$ denoting the wavenumber, then we are interested in solving the inverse problem for the forcing $f$, the boundary conditions, and possibly the wavenumber (still very ill-posed I suppose). It would be nice if you could update your post with assumptions of that kind! – EditPiAf Jan 18 '23 at 09:36
  • @EditPiAf this is very useful what you write. I added an update. I will use Mathematica, and will start by trying your method. Since these are nonlinear problems, would we need some extra function to be multiplied to u? Thanks – Luthier415Hz Jan 18 '23 at 09:51
  • Thanks Mariano, I will think of your comment. However, is this ONLY restricted to Penduli and second order diff. equation? The functions are actually plotted here https://mathematica.stackexchange.com/questions/277880/how-can-i-find-an-intermediate-function-between-a-set-of-functions and they may (and should) satisfy higher order ODEs too. – Luthier415Hz Jan 18 '23 at 10:21
  • Interesting, I wasn't aware of this capacity of mathematics. But say my functions are "ugly" and long and complex. I plan to convert them to series on a restricted interval, and by experience, the series looks nice. Would this method work with series too? I use Mathematica. Also, if it does work, do you know a source of various models to try for wave equations of fluid types, or should I just go ahead and use to most common, ie. KdW, NLSE ? – Luthier415Hz Jan 18 '23 at 15:49

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