Fractional Derivative
المؤلف:
Kilbas, A. A.; Srivastava, H. M.; and Trujiilo, J. J.
المصدر:
Theory and Applications of Fractional Differential Equations. Amsterdam, Netherlands: Elsevier, 2006.
الجزء والصفحة:
...
12-8-2018
3726
Fractional Derivative
The fractional derivative of 
of order 
(if it exists) can be defined in terms of the fractional integral 
as
![D^muf(t)=D^m[D^(-(m-mu))f(t)],](http://mathworld.wolfram.com/images/equations/FractionalDerivative/NumberedEquation1.gif) |
(1)
|
where 
is an integer ![>=[mu]](http://mathworld.wolfram.com/images/equations/FractionalDerivative/Inline5.gif)
, where ![[x]](http://mathworld.wolfram.com/images/equations/FractionalDerivative/Inline6.gif)
is the ceiling function. The semiderivative corresponds to 
.
The fractional derivative of the function 
is given by
for 
. The fractional derivative of the constant function 
is then given by
The fractional derivate of the Et-function is given by
 |
(9)
|
for 
.
It is always true that, for 
,
 |
(10)
|
but not always true that
 |
(11)
|
A fractional integral can also be similarly defined. The study of fractional derivatives and integrals is called fractional calculus.
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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