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Probabilistic Interpretation of a Fractional Derivative

This Demonstration explores the Grünwald–Letnikov definition of the fractional derivative and its numerical approximation. A geometric and probabilistic interpretation is depicted. The displayed tabular data supports the numerical calculation.

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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
.
References:
[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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