New fractional derivative algorithms

[ErikQQY]

This issue would be used to record algorithms need to be implemented.

• Fractional derivative based on Chebyshev Polynomial for 0<alpha<1 in paper: https://doi.org/10.1016/j.entcs.2008.12.077
• And 1<alpha<2 in paper: https://www.researchgate.net/publication/262049501_Numerical_algorithm_for_fractional_calculus_based_on_Chebyshev_polynomial_approximation
• Algorithms in Prof Li Changpin’s book: Numerical methods for fractional calculus
• Algorithms in Prof Xue Dingyu’s book: Fractional-order Control Systems - Fundamentals and Numerical Implementations #9

[LeeLizuoLiu]

It seems the paper for the Fractional Derivative based on Chebyshev Polynomial need further check, especially for how it obtain the Cheybshev coefficients in equation (15) in paper https://doi.org/10.1016/j.entcs.2008.12.077.

[LeeLizuoLiu]

The better Chebyshev Polynomial derivation may refer to the paper: https://d-nb.info/1171325665/34

[pedromxavier]

I would also suggest including fractional derivatives by spectral methods, i.e.
$$\partial_{t}^{\alpha} f(t) = \mathscr{F}^{-1}\left\lbrace{}(i \omega)^{\alpha}\mathscr{F}\left\lbrace{}f\right\rbrace{}\right\rbrace$$
This could leverage some neat julia FFT libraries. I've seen a MATLAB implementation for this kind of thing some years ago.

[ErikQQY]

@LeeLizuoLiu Thanks for your suggestions! I will check the better Chebshev polynomial derivations in that paper😄

[ErikQQY]

@pedromxavier Thanks for you suggestions on adding new algorithms to this package, and it would be nice if you can help implement Fourier spectral methods for evaluating fractional order derivatives.😆

[pedromxavier]

@ErikQQY This one seems neat to warm up on spectral fracdiff:
https://advancesindifferenceequations.springeropen.com/articles/10.1186/s13662-020-02590-4