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From: "John D'Errico" <woodchips@rochester.rr.com>
Newsgroups: comp.soft-sys.matlab
Subject: Re: the mathematic relationship between two series of data
Date: Sun, 13 Jul 2008 22:52:01 +0000 (UTC)
Organization: John D'Errico (1-3LEW5R)
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"Per Sundqvist" <sunkan@fy.chalmers.se> wrote in message 
<g5dg4f$f6n$1@fred.mathworks.com>...

> It looks like you should have y(1)=30 and y(0)=100 and
> y'(0)=0. I think you could use the mechanical fourth order
> differential equation in 1D to model this, using appropriate
> BC and stiffnes parameters. Then fit your data to this
> analytic formula, its related to linear combinations of
> sinh, cosh, sin and cos in some way.

An interesting point is that this is just a spline.

The 4'th order differential equation described
is the same one that generates a cubic spline.

An axial tension term merely turns this into a
tension spline, the solutions to which can be
written in terms of tanh.

A smoothness term allows you to turn it into a
smoothing spline, minimizing a combination
of the residual errors plus the potential energy
due to bending stored in the spline. And of
course, the end conditions described are also
modeled easily as a spline.

So the idea of posing a model in terms of a
differential equation, then fitting the result,
is achieved more simply by just using a least
squares spline in some form. The boundary
conditions described are all achievable using
splines.

John