Fractional integral
Riemann Liouville fractional integral is defined as follow:
\[_aD_t^{-\alpha}f(t)=\frac{1}{\Gamma(\alpha)}\int_a^t(t-\tau)^{\alpha-1}f(\tau)d\tau\]
In FractionalCalculus, you can compute the integral of a function with order $\alpha$:
julia> fracint(x->x, 0.5, 0, 1, 0.0001, RL_Direct())A tuple contains result and estimating error will be returned.
julia> fracint(x->x, 0.5, 0, 1, 0.0001, RL_())
(0.7522527785271369, 8.022170098417246e-9)Linear interpolation to approximate function
Highlight on some algorithms
FractionalCalculus.jl support many algorithms to calculate fractional integral, here, I want to highlight the Triangular Strip Matrix method proposed by Prof Igor to discrete fractional derivative.
julia> fracdiff(f, Ξ±, end_point, h, RLInt_Matrix())With this advancing algorithms, we can not only compute the fractional integral, but also the integer integral! Or more precisely, arbitrary order!!!!π Higher order integral is also a piece of cake!!!!!!π
Try to set $\alpha$ as an integer, arbitrary integer of course! I promise you would enjoy itπ