Fractional derivative

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

Non-Local Operators

It is noteworthy that the fractional derivatives are not local operators, which means that we cannot calculate the fractional derivative solely on the basis of function values of $f(x)$ taken from neighborhood of the point $x$. Instead, we have to take the entire history of $f(x)$ (i.e., all function values $f(x)$ for $0<x<a$) into account.

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, RLDiffApprox())
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, CaputoPiecewise())
1.128379167055761

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

Info

The Riemann Liouville sense derivative and Caputo sense derivative have the following relationship:

\[_{RL}D^{\alpha}_{a, t}f(t)={_CD^\alpha_{a, t}f(t)}+\sum_{k=0}^{m-1}\frac{f^{(k)}(a)(t-a)^{k-a}}{\Gamma(k+1-\alpha)}\]

Here $m-1<\alpha<m$, $f^{(m)}$ is integrable on $[a, t]$.

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:

julia> fracdiff(x->x, 0.5, collect(0:0.01:1), 2, GLHighPrecision())
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 $[0, 1]$.

Info

If the function $f(t)$ is suitably smooth, then the Grünwald Letnikov sense derivative and the Riemann Liouville sense derivative is equivalent.

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 RieszSymmetric algorithm to compute the fractional derivative:

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

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, HadamardLRect())
0.9165222777761635
Non-singular kernel

When we are using the "fractional derivative" with non-singular kernel, like Caputo-Fabrizio and Atangana-Baleanu-Caputo sense, we should know these operators are in fact integer order operators, for more details we recommend you to read the paper Why Fractional Derivatives with Nonsingular Kernels Should Not Be Used

Caputo-Fabrizio sense derivative

Caputo-Fabrizio sense fractional order derivative is defined by:

\[{^{CF}_0D^\alpha_t u(t)}=\frac{M(\alpha)}{1-\alpha}\int^t_0\frac{d}{d\tau}u(\tau)\exp[-\frac{\alpha}{1-\alpha}(t-\tau)]d\tau\\ M(\alpha)=1-\alpha+\alpha/\Gamma(\alpha)\]

To compute the Caputo-Fabrizio sense derivative, we can use the CaputoFabrizioAS algorithm in FractionalCalculus.jl:

julia> fracdiff(x->x, 0.5, 1, 0.01, CaputoFabrizioAS())
0.9887564512257243

Atangana-Baleanu sense derivative

Atangana-Baleanu sense fractional order derivative is defined by:

\[{^{ABC}_0D^\alpha_t f(t)}=\frac{AB(\alpha)}{1-\alpha}\int^t_0\frac{d}{d\tau}f(\tau)E_\alpha[-\frac{\alpha}{1-\alpha}(t-\tau)^\alpha]d\tau\\ AB(\alpha)=1-\alpha+\alpha/\Gamma(\alpha)\]

To compute the Atangana-Baleanu sense derivative, we can use the AtanganaSeda algorithm in FractionalCalculus.jl:

julia> fracdiff(x->x, 0.5, 1, 0.01, AtanganaSeda())
-0.8696378200415389
Note

Here we need to specify the start point and end point

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