fractional differentiation

The idea of Grunwald-Letnikov differentiation comes from the following formulas of backward ( and forward difference . Within this entry, [] will be used to denote the greatest integer function and Γ will be used to denote the gamma functionDlmfDlmfMathworldPlanetmath.

Backward difference

D-(f)(x)=limh0f(x)-f(x-h)h (1)
D-n(f)(x)=limh01hnk=0n(-1)kn!k!(n-k)!f(x-kh) (2)

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

Definition 1: Left-hand Grunwald-Letnikov derivative

D-p(f)(x)=limh01hpk=0[b-ah](-1)kΓ(p+1)k!Γ(p-k+1)f(x-kh) (3)

Forward difference

D+(f)(x)=limh0f(x+h)-f(x)h (4)
D+n(f)(x)=limh01hnk=0n(-1)kn!k!(n-k)!f(x+(n-k-1)h) (5)

Definition 2: Right-hand Grunwald-Letnikov derivative

D+p(f)(x)=limh01hpk=0[b-ah](-1)kΓ(p+1)k!Γ(p-k+1)f(x+(m-k-1)h) (6)

Theorem 1: Properties of fractional derivatives

  • Linearity: D±p(af+bg)(x)=aD±p(f)(x)+bD±p(g)(x) where a,b are any real constants

  • Iteration: D±pD±q(f)(x)=D±p+q(f)(x)

  • Chain ruleMathworldPlanetmath: dβf(g(x))dxβ=k=0Γ(1+β)Γ(1+k)Γ(1-k+β)dβ-k1dxβ-kdkf(g(x))dxk

  • Leibniz RulePlanetmathPlanetmath: dβ(f(x)g(x))dxβ=k=0Γ(1+β)Γ(1+k)Γ(1-k+β)dkf(x)dxkdβ-kg(x)dxβ-k

Theorem 2: Table of fractional derivatives

  • D±α(xp)=Γ(p+1)xp-αΓ(p-α+1) where α,p and Γ(x)

  • D±α(eλx)=λαeλx for all λ

  • D±α(sinx)=sin(x+απ2)

  • D±α(cosx)=cos(x+απ2)

  • D±α(eix)=cos(x+πα2)+isin(x+πα2)

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