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.
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
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