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Date: 17-6-2019
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Date: 23-4-2019
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Date: 30-3-2019
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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 semiderivative corresponds to .
The fractional derivative of the function is given by
(2) |
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(3) |
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(4) |
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(5) |
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(6) |
for . The fractional derivative of the constant function is then given by
(7) |
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(8) |
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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