Fractional derivative

• Fractional derivative
• Riemann Liouville sense derivative
• Caputo sense derivative
• Grünwald Letnikov sense derivative
• Riesz sense derivative
• Hadamard sense derivative
• Highlight on some algorithms

To get start with fractional derivative, you need to know that unlike Newtonian derivatives, fractional derivative is defined via integral.

Riemann Liouville sense derivative

Riemann Liouville sense derivative is built upon the Riemann Liouville sense integral.

\[_aD^\alpha_tf(t)=\frac{d^n}{dt^n}\ _aD^{-(n-\alpha)}_tf(t)=\frac{d^n}{dt^n}\ _aI^{n-\alpha}_tf(t)\]

\[_tD^\alpha_bf(t)=\frac{d^n}{dt^n}\ _tD^{-(n-\alpha)}_bf(t)=\frac{d^n}{dt^n}\ _tI^{n-\alpha}_bf(t)\]

We can use FractionalCalculus.jl to compute Riemann Liouville sense $0.5$ order fractional derivative of $f(x)=x$ at $x=1$ with step size $0.0001$:

julia> fracdiff(x->x, 0.5, 1, 0.0001, RLDiff_Approx())
1.1283791670955168

Caputo sense derivative

There many types of definitions of fractional derivative, Caputo is one of these useful definitions. The Caputo fractional derivative is first be proposed in Michele Caputo's Paper,

\[^CD_t^\alpha f(t) = \frac{1}{\Gamma(n-\alpha)}\int_0^t\frac{f^{(n)}(\tau)d\tau}{(t-\tau)^{\alpha+1-n}}, n=\lceil{\alpha}\rceil\]

In FractionalCalculus.jl, let's see, if you want to calculate the $0.5$ order fractional derivative of $f(x)=x$ at a $x=1$ with step size $0.0001$, simply typing these:

julia>fracdiff(x->x, 0.5, 1, 0.0001, Caputo_Piecewise())
1.128379167055761

We can see the result is closely resembling with the result in Riemann Liouville sense.

Grünwald Letnikov sense derivative

\[D^\alpha f(t)=\displaystyle \lim_{h\rightarrow0}\frac{1}{h^\alpha}\sum_{0\leq m\lt\infty}(-1)^m {{\alpha}\choose{m}}f(t+(\alpha-m)h)\]

To compute the Grünwald Letnikov sense derivative, you can use FractionalCalculus.jl by:

fracdiff(x->x, 0.5, collect(0:0.01:1), 2, GL_High_Precision())
101-element Vector{Float64}:
 0.0
 0.11283791670955126
 0.15957691216057307
 0.19544100476116796
 0.22567583341910252
 0.252313252202016
 0.27639531957706837
 0.29854106607209235
 0.31915382432114614
 ⋮
 1.082303275961202
 1.0881694613449238
 1.094004191971426
 1.0998079684646789
 1.1055812783082735
 1.111324596323283
 1.1170383851240115
 1.1227230955528664
 1.1283791670955126

Here, we use the high precision algorithm, the fourth parameter means we set the precision order as p=2. The returned result means the derivative on the interval $[0, 1]$.

Riesz sense derivative

The Riesz sense symmetric fractional derivative is defined by Caputo derivative:

\[ \frac{d^\beta \phi(x)}{d|x|^\mu}=D_{R}^{\beta}\phi(x)=\frac{1}{2}\Bigl(_{a}D_{x}^{\beta}\phi(x)+{_{x}D_{b}^{\beta}}\phi(x) \Bigr) \]

In FractionalCalculus.jl, we can use the Riesz_Symmetric algorithm to compute the fractional derivative:

julia> fracdiff(x->x, 0.5, 1, 0.01, Riesz_Symmetric())

Hadamard sense derivative

The Hadamard sense derivative is defined using Hadamard sense integral:

\[ {_HD_{a+}^{-\alpha}}f(x)=\frac{1}{\Gamma(\alpha)}\int_a^x(\log\frac{x}{t})^{-(1-\alpha)}f(t)\frac{dt}{t},\ x > a \]

So we can know the Hadamard sense fractional derivative:

\[ {_HD_{a+}^\alpha}f(x)=\delta^n[{_HD_{a+}^{-(n-\alpha)}f(x)}],\ x>a \]

\[ \delta=x\frac{d}{dx},\ n-1\leq\alpha<n\in\mathrm{Z^+} \]

To compute Hadamard fractional derivative, we can use the Hadamard relating algorithms in FractionalCalculus.jl:

julia> fracdiff(x->x, 0.5, 0, 1, 0.01, Hadamard_LRect())
0.9165222777761635

There are different approximating methods being used in the computing, choose the one you need😉

Highlight on some algorithms

FractionalCalculus.jl support many algorithms to calculate fractional derivative, here, I want to highlight the Triangular Strip Matrix method proposed by Prof Igor to discrete fractional derivative.

julia> fracdiff(f, α, end_point, h, RLDiff_Matrix())

With this advancing algorithms, we can not only compute the fractional derivative, but also the integer derivative! Or more precisely, arbitrary order!!!!🙌 Higher order derivative is also a piece of cake!!!!!!🎉

Try to set $\alpha$ as an integer, arbitrary integer of course! I promise you would enjoy it😏