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 function.
Backward difference
D-(f)(x)=limh→0f(x)-f(x-h)h | (1) |
Dn-(f)(x)=limh→01hnn∑k=0(-1)kn!k!(n-k)!f(x-kh) | (2) |
For derivatives of integer orders, we only requires to specifies one point x∈ℝ. Fractional derivatives, like fractional definite integrals, require an interval [a,b] to be specified for the function
f:ℝ→ℝ we are talking about.
Definition 1: Left-hand Grunwald-Letnikov derivative
Dp-(f)(x)=limh→01hp[b-ah]∑k=0(-1)kΓ(p+1)k!Γ(p-k+1)f(x-kh) | (3) |
Forward difference
D+(f)(x)=limh→0f(x+h)-f(x)h | (4) |
Dn+(f)(x)=limh→01hnn∑k=0(-1)kn!k!(n-k)!f(x+(n-k-1)h) | (5) |
Definition 2: Right-hand Grunwald-Letnikov derivative
Dp+(f)(x)=limh→01hp[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∈ℝ are any real constants
-
•
Iteration: Dp±Dq±(f)(x)=Dp+q±(f)(x)
-
•
Chain rule
: dβf(g(x))dxβ=∞∑k=0Γ(1+β)Γ(1+k)Γ(1-k+β)dβ-k1dxβ-kdkf(g(x))dxk
-
•
Leibniz Rule
: 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∈ℝ and Γ(x)
-
•
Dα±(eλx)=λαeλx for all λ∈ℝ
-
•
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 |