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Fractional Derivative

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The fractional derivative of f(t) of order mu>0 (if it exists) can be defined in terms of the fractional integral D^(-nu)f(t) as

D^muf(t)=D^m[D^(-(m-mu))f(t)],
(1)

where m is an integer >=[mu], where [x] is the ceiling function. The semiderivative corresponds to mu=1/2.

The fractional derivative of the function t^lambda is given by

D^mut^lambda=D^m[D^(-(m-mu))t^lambda]
(2)
=D^m[(Gamma(lambda+1))/(Gamma(lambda+m-mu+1))t^(lambda+m-mu)]
(3)
=(Gamma(lambda+1)(lambda-mu+m)(lambda-mu+m-1)...(lambda-mu+1))/(Gamma(1+m+lambda-mu))t^(lambda-mu)
(4)
=(Gamma(lambda+1)(1+lambda-mu)_m)/(Gamma(1+m+lambda-mu))t^(lambda-mu)
(5)
=(Gamma(lambda+1))/(Gamma(lambda-mu+1))t^(lambda-mu)
(6)

for lambda>-1,mu>0. The fractional derivative of the constant function f(t)=c is then given by

D^muc=clim_(lambda->0)(Gamma(lambda+1))/(Gamma(lambda-mu+1))t^(lambda-mu)
(7)
=(ct^(-mu))/(Gamma(1-mu)).
(8)

The fractional derivate of the Et-function is given by

D^rhoE_t(nu,a)=E_t(nu-rho,a)
(9)

for nu>0,rho!=0.

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

D^(-mu)D^(-nu)=D^(-(mu+nu)),
(10)

but not always true that

D^muD^nu=D^(mu+nu).
(11)

Fractional derivatives are implemented in the Wolfram Language as FractionalD.

A fractional integral can also be similarly defined. The study of fractional derivatives and integrals is called fractional calculus.

See also

Fractional Calculus, Fractional Differential Equation, 'Fractional Integral, Semiderivative

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References

Kilbas, A. A.; Srivastava, H. M.; and Trujiilo, J. J. Theory and Applications of Fractional Differential Equations. Amsterdam, Netherlands: Elsevier, 2006.Love, E. R. "Fractional Derivatives of Imaginary Order." J. London Math. Soc. 3, 241-259, 1971.Miller, K. S. "Derivatives of Noninteger Order." Math. Mag. 68, 183-192, 1995.Oldham, K. B. and Spanier, J. The Fractional Calculus: Integrations and Differentiations of Arbitrary Order. New York: Academic Press, 1974.Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, 1993.

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Fractional Derivative

Cite this as:

Weisstein, Eric W. "Fractional Derivative." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FractionalDerivative.html

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