Probabilistic Interpretation of a Fractional Derivative

position

x

1

derivative order

α

0.2

step increment

h

0.001

series truncation order

r

50

Grünwald–Letnikov

approximation formulas

α

D

f(x)

≈

f(x)

-

μ

α

h

=

CB

AC

μ ≈ -

r

∑

k=1

γ(α, k)f(x - kh)

γ(α, k) =

k

(-1)

Γ(α + 1)

k!Γ(α - k + 1)

h	0.001
α	0.200
r	50
x	1
f(x)	0.368
μ	0.227
α D f	0.560

k	x - kh	f(x - kh)	γ(α, k)	γ(α, k)f(x - kh)
1	0.99900	0.368620	-0.200000	-0.073723
2	0.99800	0.369350	-0.080000	-0.029548
…	…	…	…	…
r = 50	0.95000	0.405550	-0.001575	-0.000639
		r ∑ k=1	-0.607830	-0.227260
		∞ ∑ k=1	-1

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.