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fractional integration

(Definition)

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

Given any integrable function $f:{\mathbb{R}}\mapsto {\mathbb{R}}$ in one variable, we have the following Cauchy Integration Formula:

$\displaystyle D^{-n}(f)(x)=\int_{t_n=0}^x dt_n\ldots \int_{t_1=0}^{t_2} f(t_1)\,dt_1 =\frac{1}{(n-1)!} \int_{t=0}^x f(t)(x-t)^{n-1}\,dt$

when switching the index from integer to non-integer $\alpha$ gives the ideas of the following definitions:

Definition 1: Left-Hand Riemann-Liouville Integration

$\displaystyle I^{\alpha}_L (f)(s,t)= \frac{1}{\Gamma(\alpha)}\int_{u=s}^tf(u)(t-u)^{\alpha-1}\,du =\int_{u=s}^t f(u)\,dg^{\alpha}_t(u)$

where

$\displaystyle g^{\alpha}_t(u)=\frac{t^{\alpha}-(t-u)^{\alpha}} {\Gamma(\alpha+1)}$

Definition 2: Right-Hand Riemann-Liouville Integration

$\displaystyle I^{\alpha}_R (f)(s,t)= \frac{1}{\Gamma(\alpha)}\int_{u=s}^tf(u)(u-s)^{\alpha-1}\,du =\int_{u=s}^t f(u)\,dh^{\alpha}_t(u)$

where

$\displaystyle h^{\alpha}_t(u)=\frac{s^{\alpha}+(u-s)^{\alpha}} {\Gamma(\alpha+1)}$

Definition 3: Riesz Potential

$\displaystyle I^{\alpha}_C (f)(s,t;p)= \frac{1}{\Gamma(\alpha)}\int_{u=s}^tf(u)\vert u-p\vert^{\alpha-1}\,du =\int_{u=s}^t f(u)\,dr^{\alpha}_p(u)$

where

$\displaystyle r^{\alpha}_p(u)=\frac{p^{\alpha}+{\rm sign}(u-p) \vert u-p\vert^{\alpha}}{\Gamma(\alpha+1)}$

${\rm sign}(x)=1$ for , ${\rm sign}(x)=0$ for , ${\rm sign}(x)=-1$ for

and $\Gamma(x)$ is the gamma function of

"fractional integration" is owned by rspuzio. [ full author list (2) | owner history (1) ]

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Other names:

fractional integral

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Cross-references: gamma function, potential, definitions, integer, index, integration formula, variable, function, observation, type
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This is version 14 of fractional integration, born on 2006-10-03, modified 2007-06-23.
Object id is 8416, canonical name is FractionalIntegration.
Accessed 2166 times total.

Classification:

26A33 (Real functions :: Functions of one variable :: Fractional derivatives and integrals)

Pending Errata and Addenda

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