Medical use of 2nd order ODE

[TheodoreM.Bones]

Much effort to use well known math formulas for the half-life of a marker chemical in renal excretion studies that were written in MathLab into MathSoft has been studied for over a month and essential programming was done. Conventional modeling of the raw data file into 2-term exponential form was used, generating a second order ODE and the solution of the ODE was accomplished. This also necessitated conference with the University of Alabama at Birmingham and research for important publications, including a 1955 paper at the Ohio State University. Two examples are given in the worksheet, posted in MC11.

[ptc-1368288]

For the data set you tried, there are two immediate superb fits, at right yellow. I read quickly the remaining of the work sheet but don't understand immediately.
For what it's worth.

Jean

[TheodoreM.Bones]

The idea was to develop a 2nd order ODE from the raw data, not to just fit it, and then solve the ODE. This process is not well understood in these recent times. I recall learning it in 1946 while in a Servomechanisms course in General Electric Co., but I had a book by Ince published in 1900 in Egypt that showed how an ODE could be regarded as a polynominal and solved that way. That is, a 3rd order ODE would be solved for its roots as if it was a cubic polynominal. That is too weighty, but I use deProny to get the coeffs. and roots fast and easily. I will post this shortly today, going as far as 8th order ODEs and the coeffs. and roots might be complex values, but still works,

[ptc-1368288]

Getting an nth order polynomial as the solution of a DE is what they call "collocation". I have an example in Mathcad. An nth order DE is decomposed in lower order DE's, that's rK method, but no DE has solution via polynomial unless considering Frobenius, that we have in Mathcad.

Jen

[TheodoreM.Bones]

In many medical and chemical reactnion processes, the
rate or convergence is well expressed by a 2-term sum of
exponentials, leading to the formation of a 2nd order ODE.

To investigate if any higher order ODE would benefit an understanding of the data, the deProny algorithm is used to easily obtain very high order ODEs and compare with the 2nd order ODE.
The deProny method is a semi-circular Hankel matrix with upward sloping diagonals. As a matrix solution it is free from any Minerr
least squares errors. It is necessary to rework the data prepared from raw lab reports taken at irregular times into a file with evenly spaced values of the x-axis.

In general not much benefit in accuracy of the final results may appear on most real life physical or medical work. The worksheet here allows ODEs from 2nd to 9th order or more.

[TheodoreM.Bones]

What I did was just like the Frobenius Method but much easier with the deProny method. Not soi much typing needed. Thanks, Jean, you sdre do know meth.