Probabilistic Interpretation of a Fractional Derivative

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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.

Contributed by: L. A. Mendes Afonso and J. A. Tenreiro Machado (March 2011)
After work by: J. A. Tenreiro Machado
Open content licensed under CC BY-NC-SA



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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