fractional integration


The basic idea of ”Riemann-Liouville” type fractional integration comes from the following observation:

Given any integrable function f: in one variable, we have the following Cauchy Integration Formula:

D-n(f)(x)=tn=0x𝑑tnt1=0t2f(t1)𝑑t1=1(n-1)!t=0xf(t)(x-t)n-1𝑑t

when switching the index from integer n to non-integer α gives the ideas of the following definitions:

Definition 1: Left-Hand Riemann-Liouville Integration

ILα(f)(s,t)=1Γ(α)u=stf(u)(t-u)α-1𝑑u=u=stf(u)𝑑gtα(u)

where

gtα(u)=tα-(t-u)αΓ(α+1)

Definition 2: Right-Hand Riemann-Liouville Integration

IRα(f)(s,t)=1Γ(α)u=stf(u)(u-s)α-1𝑑u=u=stf(u)𝑑htα(u)

where

htα(u)=sα+(u-s)αΓ(α+1)

Definition 3: Riesz Potential

ICα(f)(s,t;p)=1Γ(α)u=stf(u)|u-p|α-1𝑑u=u=stf(u)𝑑rpα(u)

where

rpα(u)=pα+sign(u-p)|u-p|αΓ(α+1)

,

sign(x)=1 for x>0, sign(x)=0 for x=0, sign(x)=-1 for x<0

and Γ(x) is the gamma functionDlmfDlmfMathworldPlanetmath of x

Title fractional integration
Canonical name FractionalIntegration
Date of creation 2013-03-22 16:17:47
Last modified on 2013-03-22 16:17:47
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 17
Author rspuzio (6075)
Entry type Definition
Classification msc 26A33
Synonym fractional integralDlmfDlmfMathworld