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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, [] 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)
Dn-(f)(x)=limh01hnnk=0(-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

Dp-(f)(x)=limh01hp[b-ah]k=0(-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)
Dn+(f)(x)=limh01hnnk=0(-1)kn!k!(n-k)!f(x+(n-k-1)h) (5)

Definition 2: Right-hand Grunwald-Letnikov derivative

Dp+(f)(x)=limh01hp[b-ah]k=0(-1)kΓ(p+1)k!Γ(p-k+1)f(x+(m-k-1)h) (6)

Theorem 1: Properties of fractional derivatives

  • Linearity: Dp±(af+bg)(x)=aDp±(f)(x)+bDp±(g)(x) where a,b are any real constants

  • Iteration: Dp±Dq±(f)(x)=Dp+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

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

  • Dα±(eλx)=λαeλx for all λ

  • Dα±(sinx)=sin(x+απ2)

  • Dα±(cosx)=cos(x+απ2)

  • Dα±(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