Mathworld FractionalDerivative

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Fractional Derivative

The fractional derivative of of order (if it exists) can be defined in terms of the fractional integral

as

(1)

where is an integer , where is the ceiling function. The semiderivativecorresponds to .

The fractional derivative of the function is given by

(2)

(3)

(4)

(5)

(6)

for . The fractional derivative of the constant function is then given by

(7)

8


of Sine

Jorge Gamaliel Frade

Chvez


of a Power


Chvez

Numerical Solution of

Some FractionalDiffusion Equations

Santos Bravo Yuste

An OrdinaryFractional Differential

Equation


Chvez

Search MathWorld

THINGS TO TRY:

135/216 - 12/25

four thousand three hundred

twelve

int e^t sin(5t) dt
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13,540 entries

Last updated: Mon Dec 8 2014

Created, developed, a nd

nurtured by Eric Weisstein

at Wolfram Research

The fractional derivate of the Et-functionis gi ven by

(9)

for .

It is always true that, for ,

(10)

but notalways true that

(11)

A fractional integralcan als o be s imilarly defined. The study of fractional derivatives an d integrals is called fractionalcalculus.

SEE ALSO:

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

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 Y ork: Academic

Press, 1974.

Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives.Yverdon, Sw itzerland: Gordon and Breach,

1993.

Referenced on Wolfram|Alpha: Fractional De rivative

CITE THIS AS:

Weisstein, Eric W."Fractional Derivative." From MathWorld--A Wolfram Web Resource.http://mathw orld.w olfram.com/FractionalDerivative.html

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