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 $\mathrm{\Gamma }$ will be used to denote the gamma function.

Backward difference

$$D_{-}(f)(x)=\underset{h\to 0}{lim}\frac{f(x)-f(x-h)}{h}$$(1)

$$D_{-}^n(f)(x)=\underset{h\to 0}{lim}\frac{1}{h^n}\sum _{k=0}^n\frac{{(-1)}^{k}n!}{k!(n-k)!}f(x-kh)$$(2)

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

Definition 1: Left-hand Grunwald-Letnikov derivative

$$D_{-}^p(f)(x)=\underset{h\to 0}{lim}\frac{1}{h^p}\sum _{k=0}^{\left[\frac{b-a}{h}\right]}\frac{{(-1)}^k\mathrm{\Gamma }(p+1)}{k!\mathrm{\Gamma }(p-k+1)}f(x-kh)$$(3)

Forward difference

$$D_{+}(f)(x)=\underset{h\to 0}{lim}\frac{f(x+h)-f(x)}{h}$$(4)

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

Definition 2: Right-hand Grunwald-Letnikov derivative

$$D_{+}^p(f)(x)=\underset{h\to 0}{lim}\frac{1}{h^p}\sum _{k=0}^{\left[\frac{b-a}{h}\right]}\frac{{(-1)}^k\mathrm{\Gamma }(p+1)}{k!\mathrm{\Gamma }(p-k+1)}f(x+(m-k-1)h)$$(6)

Theorem 1: Properties of fractional derivatives

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

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

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

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

Theorem 2: Table of fractional derivatives

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

• $D_{\pm }^{\alpha }(e^{\lambda x})={\lambda }^{\alpha }{e}^{\lambda x}$ for all $\lambda \in ℝ$

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

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

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

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