# fractional differentiation

The idea of Grunwald-Letnikov differentiation comes from the following formulas of backward (http://planetmath.org/BackwardDifference) and forward difference . Within this entry, $[\cdot]$ will be used to denote the greatest integer function and $\Gamma$ will be used to denote the gamma function.

Backward difference

 $D_{-}(f)(x)=\lim_{h\to 0}\frac{f(x)-f(x-h)}{h}$ (1)
 $D^{n}_{-}(f)(x)=\lim_{h\to 0}\frac{1}{h^{n}}\sum_{k=0}^{n}\frac{(-1)^{k}n!}{k!% (n-k)!}f(x-kh)$ (2)

For derivatives of integer orders, we only requires to specifies one point $x\in{\mathbb{R}}$. Fractional derivatives, like fractional definite integrals, require an interval $[a,b]$ to be specified for the function $f:{\mathbb{R}}\to{\mathbb{R}}$ we are talking about.

Definition 1: Left-hand Grunwald-Letnikov derivative

 $D^{p}_{-}(f)(x)=\lim_{h\to 0}\frac{1}{h^{p}}\sum_{k=0}^{\left[\frac{b-a}{h}% \right]}\frac{(-1)^{k}\Gamma(p+1)}{k!\Gamma(p-k+1)}f(x-kh)$ (3)

Forward difference

 $D_{+}(f)(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$ (4)
 $D^{n}_{+}(f)(x)=\lim_{h\to 0}\frac{1}{h^{n}}\sum^{n}_{k=0}\frac{(-1)^{k}n!}{k!% (n-k)!}f(x+(n-k-1)h)$ (5)

Definition 2: Right-hand Grunwald-Letnikov derivative

 $D^{p}_{+}(f)(x)=\lim_{h\to 0}\frac{1}{h^{p}}\sum_{k=0}^{\left[\frac{b-a}{h}% \right]}\frac{(-1)^{k}\Gamma(p+1)}{k!\Gamma(p-k+1)}f(x+(m-k-1)h)$ (6)

Theorem 1: Properties of fractional derivatives

• Linearity: $D^{p}_{\pm}(af+bg)(x)=aD^{p}_{\pm}(f)(x)+bD^{p}_{\pm}(g)(x)$ where $a,b\in{\mathbb{R}}$ are any real constants

• Iteration: $D^{p}_{\pm}D^{q}_{\pm}(f)(x)=D^{p+q}_{\pm}(f)(x)$

• Chain rule: $\displaystyle{\frac{d^{\beta}f(g(x))}{dx^{\beta}}=\sum_{k=0}^{\infty}\frac{% \Gamma(1+\beta)}{\Gamma(1+k)\Gamma(1-k+\beta)}\frac{d^{\beta-k}1}{dx^{\beta-k}% }\frac{d^{k}f(g(x))}{dx^{k}}}$

• Leibniz Rule: $\displaystyle{\frac{d^{\beta}(f(x)g(x))}{dx^{\beta}}=\sum_{k=0}^{\infty}\frac{% \Gamma(1+\beta)}{\Gamma(1+k)\Gamma(1-k+\beta)}\frac{d^{k}f(x)}{dx^{k}}\frac{d^% {\beta-k}g(x)}{dx^{\beta-k}}}$

Theorem 2: Table of fractional derivatives

• $\displaystyle{D^{\alpha}_{\pm}(x^{p})=\frac{\Gamma(p+1)x^{p-\alpha}}{\Gamma(p-% \alpha+1)}}$ where $\alpha,p\in{\mathbb{R}}$ and $\Gamma(x)$

• $\displaystyle{D^{\alpha}_{\pm}(e^{\lambda x})=\lambda^{\alpha}e^{\lambda x}}$ for all $\lambda\in{\mathbb{R}}$

• $\displaystyle{D^{\alpha}_{\pm}(\sin x)=\sin\left(x+\frac{\alpha\pi}{2}\right)}$

• $\displaystyle{D^{\alpha}_{\pm}(\cos x)=\cos\left(x+\frac{\alpha\pi}{2}\right)}$

• $\displaystyle{D^{\alpha}_{\pm}(e^{ix})=\cos\left(x+\frac{\pi\alpha}{2}\right)+% i\sin\left(x+\frac{\pi\alpha}{2}\right)}$

 Title fractional differentiation Canonical name FractionalDifferentiation Date of creation 2013-03-22 16:18:46 Last modified on 2013-03-22 16:18:46 Owner Wkbj79 (1863) Last modified by Wkbj79 (1863) Numerical id 21 Author Wkbj79 (1863) Entry type Definition Classification msc 26A06 Synonym Grunwald-Letnikov differentiation Related topic HigherOrderDerivativesOfSineAndCosine Defines fractional derivative Defines left-hand Grunwald-Letnikov derivative Defines left hand Grundwald Letnikov derivative Defines right-hand Grundwald-Letnikov derivative Defines right hand Grundwald-Letnikov derivative