Fitting via differential equations

[moustachio-belvedere]

Proof of concept already done many months ago. To implement:

• [ ] improve fractional derivative algorithm (use Podlubny book) and add tests
• [ ] Implement LHS and RHS operator function/structures for each RheoModelClass that can be applied to the data
• [ ] Write the modeldiffeqfit functions

[akabla]

Here is a package to explore for fractional derivatives: https://github.com/JuliaApproximation/ApproxFun.jl/blob/master/src/Extras/fractional.jl

See also: https://discourse.julialang.org/t/is-there-a-package-for-fractional-calculus/19702/6

[moustachio-belvedere]

I had another look at the link, I'm not sure the approach will work for us directly as it seems to require the function to be an actual Julia function (that is smooth) and then approximates that, presumably at Chebyshev nodes. Happy to be wrong on that though.

At the very least, we can write some test functions in Julia, compute them on a range of values, and compare our numerical approach with the above. So, still useful there. Maybe that's what you intended already?

Edit: the above might be useful for testing our numerical algorithms against functions we can't analytically find fractional derivative of.

[akabla]

Thanks Louis. You're right, probably not the way to go for sampled data. As you said, it may be useful if we need to differentiate some of the moduli functions.