The Grünwald–Letnikov definition of the fractional derivative of order
of
is given by the expression
,
,
where
represents the gamma function and
the increment. We verify that
,
.
The expression
can be viewed as the expected value of the discrete random variable that for
takes the value
with probability
.
The Grünwald–Letnikov definition gets the slope of a triangle with upper corners
and
. The factor
in the denominator expression means that, for large values of
, we have a slow variation, while for small values of
we have a fast variation.
The implementation of the Grünwald–Letnikov definition of the fractional derivative corresponds to an
-term truncated series given by
.
[1] K. B. Oldham and J. Spanier,
The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order, New York: Academic Press, 1974.
[2] I. Podlubny,
Fractional Differential Equations, San Diego: Academic Press, 1999.
[3] J. A. Tenreiro Machado, "Discrete-Time Fractional-Order Controllers,"
Journal of Fractional Calculus & Applied Analysis,
4(1), 2001 pp. 47–66.
[4] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
Theory and Applications of Fractional Differential Equations, Vol. 204, Amsterdam: Elsevier, 2006.
[5] J. A. Tenreiro Machado, "Fractional Derivatives: Probability Interpretation and Frequency Response of Rational Approximations,"
Communications in Nonlinear Science and Numerical Simulations,
14(9–10), 2009 pp. 3492–3497.
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