Fractional Calculus

Denote the $n$th Derivative $D^n$ and the $n$-fold Integral $D^{-n}$. Then

$$D^{-1} f(t)=\int_0^t f(\xi)\,d\xi.$$ (1)

Now, if

$$D^{-n} f(t)={1\over(n-1)!} \int_0^t (t-\xi)^{n-1}f(\xi)\,d\xi$$ (2)

is true for $n$, then

$$D^{-(n+1)}f(t) = D^{-1}\left[{{1\over(n-1)!} \int_0^t (t-\xi)^{n-1}f(\xi)\,d\xi}\right]$$

$$= \int_0^t \left[{{1\over(n-1)!}\int_0^x (x-\xi)^{n-1}f(\xi)\,d\xi}\right]\,dx.$$ (3)

Interchanging the order of integration gives

$$D^{-(n+1)}f(t)={1\over n!}\int_0^t (t-\xi)^nf(\xi)\,d\xi.$$ (4)

But (2) is true for $n=1$, so it is also true for all $n$ by Induction. The fractional integral of $f(t)$ can then be defined by

$$D^{-\nu}f(t)={1\over\Gamma(\nu)}\int_0^t (t-\xi)^{\nu-1}f(\xi)\,d\xi,$$ (5)

where $\Gamma(\nu)$ is the Gamma Function.

The fractional integral can only be given in terms of elementary functions for a small number of functions. For example,

$$D^{-\nu} t^{-\lambda} = {\Gamma(\lambda+1)\over\Gamma(\lambda+\nu+1)} t^{\lambda+\nu} \quad{\rm for\ } \lambda>-1, \nu>0$$ (6)

$$D^{-\nu} e^{at} = {1\over\Gamma(\nu)} e^{at} \int_0^t x^{\nu-1}e^{-ax}\,dx\equiv E_t(\nu,a),$$ (7)

where $E_t(\nu,a)$ is the Et-Function. The fractional derivative of $f$ (if it exists) can be defined by

$$D^\mu f(t)=D^m[D^{-(m-\mu)}f(t)].$$ (8)

An example is

$$D^\mu t^\lambda = {\Gamma(\lambda+1)\over\Gamma(\lambda+m-\mu+1)}$$

$$= {\Gamma(\lambda+1)\over\Gamma(\lambda-\mu+1)} t^{\lambda-\mu}\quad {\rm for\ } \lambda>-1, \mu>0$$ (9)

$$D^\rho E_t(\nu,a) = E_t(\nu-\rho,a) \qquad{\rm for\ } \nu>0, \rho\not=0.$$ (10)

It is always true that, for $\mu,\nu>0$,

$$D^{-\mu} D^{-\nu} f(t) = D^{-(\mu+\nu)},$$ (11)

but not always true that

$$D^\mu D^\nu=D^{\mu+\nu}.$$ (12)

See also Derivative, Integral

References

Love, E. R. ``Fractional Derivatives of Imaginary Order.'' J. London Math. Soc. 3, 241-259, 1971.

McBride, A. C. Fractional Calculus. New York: Halsted Press, 1986.

Miller, K. S. ``Derivatives of Noninteger Order.'' Math. Mag. 68, 183-192, 1995.

Nishimoto, K. Fractional Calculus. New Haven, CT: University of New Haven Press, 1989.

Spanier, J. and Oldham, K. B. The Fractional Calculus. New York: Academic Press, 1974.