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'Fractional Integration'
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| Title of object: |
Fractional Integration |
| Canonical Name: |
FractionalIntegration |
| Type: |
Definition |
| Created on: |
2006-10-03 15:26:36 |
| Modified on: |
2006-10-08 10:09:50 |
| Classification: |
msc:26A33 |
| Keywords: |
Fractional Integration |
| Defines: |
Fractional Integration |
| Synonyms: |
Fractional Integration=Fractional Integration |
Preamble:
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Content:
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:
\begin{displaymath}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
\end{displaymath}
when switching the index from integer $n$ to non-integer $\alpha$ gives the ideas of the following definitions:
{\bf Definition 1:} {\rm Left-Hand Riemann-Liouville Integration}
\begin{displaymath}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) \end{displaymath}
where \begin{displaymath}g^{\alpha}_t(u)=\frac{t^{\alpha}-(t-u)^{\alpha}}
{\Gamma(\alpha+1)}\end{displaymath}
{\bf Definition 2:} {\rm Right-Hand Riemann-Liouville Integration}
\begin{displaymath}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) \end{displaymath}
where \begin{displaymath}h^{\alpha}_t(u)=\frac{s^{\alpha}+(u-s)^{\alpha}}
{\Gamma(\alpha+1)}\end{displaymath}
{\bf Definition 3:} {\rm Riesz Potential}
\begin{displaymath}I^{\alpha}_C (f)(s,t;p)=
\frac{1}{\Gamma(\alpha)}\int_{u=s}^tf(u)|u-p|^{\alpha-1}\,du
=\int_{u=s}^t f(u)\,dr^{\alpha}_p(u) \end{displaymath}
where \begin{displaymath}r^{\alpha}_p(u)=\frac{p^{\alpha}+{\rm sign}(u-p)
|u-p|^{\alpha}}{\Gamma(\alpha+1)}\end{displaymath}
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