scifracx / fractionalcalculus.jl Goto Github PK

The newly added algorithm GL_High_Precision throw DimensionMissMatch error when p is bigger than 2

There are some algorithms we can refine using Richardson extrapolation, see: https://doi.org/10.1016/j.cma.2004.06.006

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See paper: http://dx.doi.org/10.1007/3-7643-7412-8_28

I want to make a request for symbolic computation of fractional derivatives with any alpha for n-1<alpha<n for a given number n using Caputo or Riemann-Liouville sense . E.g say semidiff(f(x), x, alpha, n, Caputo).

Examples on the FractionalCalculus.jl home page are returning functions not defined in Julia 1.9.2. For instance,

julia> fracdiff(x->x, 0.5, 1, 0.0001, RLDiffL1())
ERROR: UndefVarError: RLDiffL1 not defined
Stacktrace:
[1] top-level scope
@ REPL[2]:1

Another example:

julia> semidiff(log(x))
ERROR: UndefVarError: semidiff not defined
Stacktrace:
[1] top-level scope
@ REPL[5]:1

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
• #9

I'm doing the most simple Caputo derivative
$$^C_0 D_t^\alpha t = \frac{t^{1-\alpha}}{\Gamma(2-\alpha)}$$
and I get a relative error of $0.024$ for a time step of $10^{-9}$, is this supposed to be working properly?

using FractionalCalculus
using SpecialFunctions: gamma

T = 0.6; α = 0.9; Δt = 1e-9
numerical_caputo_der = fracdiff(t->t,α,0,T,Δt,CaputoDirect())[1]
theoretical_caputo_der = T^(1-α)/gamma(2-α)
abs(numerical_caputo_der-theoretical_caputo_der)/abs(theoretical_caputo_der) # 0.0239