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In mathematics , the Grünwald–Letnikov derivative is a basic extension of the derivative in fractional calculus that allows one to take the derivative a non-integer number of times. It was introduced by Anton Karl Grünwald (1838–1920) from Prague , in 1867, and by Aleksey Vasilievich Letnikov (1837–1888) in Moscow in 1868.
The formula
f
′
(
x
)
=
lim
h
→
0
f
(
x
+
h
)
−
f
(
x
)
h
{\displaystyle f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}}
for the derivative can be applied recursively to get higher-order derivatives. For example, the second-order derivative would be:
f
″
(
x
)
=
lim
h
→
0
f
′
(
x
+
h
)
−
f
′
(
x
)
h
=
lim
h
1
→
0
lim
h
2
→
0
f
(
x
+
h
1
+
h
2
)
−
f
(
x
+
h
1
)
h
2
−
lim
h
2
→
0
f
(
x
+
h
2
)
−
f
(
x
)
h
2
h
1
{\displaystyle {\begin{aligned}f''(x)&=\lim _{h\to 0}{\frac {f'(x+h)-f'(x)}{h}}\\&=\lim _{h_{1}\to 0}{\frac {\lim \limits _{h_{2}\to 0}{\dfrac {f(x+h_{1}+h_{2})-f(x+h_{1})}{h_{2}}}-\lim \limits _{h_{2}\to 0}{\dfrac {f(x+h_{2})-f(x)}{h_{2}}}}{h_{1}}}\end{aligned}}}
Assuming that the h 's converge synchronously, this simplifies to:
=
lim
h
→
0
f
(
x
+
2
h
)
−
2
f
(
x
+
h
)
+
f
(
x
)
h
2
{\displaystyle =\lim _{h\to 0}{\frac {f(x+2h)-2f(x+h)+f(x)}{h^{2}}}}
which can be justified rigorously by the mean value theorem . In general, we have (see binomial coefficient ):
f
(
n
)
(
x
)
=
lim
h
→
0
∑
0
≤
m
≤
n
(
−
1
)
m
(
n
m
)
f
(
x
+
(
n
−
m
)
h
)
h
n
{\displaystyle f^{(n)}(x)=\lim _{h\to 0}{\frac {\sum \limits _{0\leq m\leq n}(-1)^{m}{n \choose m}f(x+(n-m)h)}{h^{n}}}}
Removing the restriction that n be a positive integer, it is reasonable to define:
D
q
f
(
x
)
=
lim
h
→
0
1
h
q
∑
0
≤
m
<
∞
(
−
1
)
m
(
q
m
)
f
(
x
+
(
q
−
m
)
h
)
.
{\displaystyle \mathbb {D} ^{q}f(x)=\lim _{h\to 0}{\frac {1}{h^{q}}}\sum _{0\leq m<\infty }(-1)^{m}{q \choose m}f(x+(q-m)h).}
This defines the Grünwald–Letnikov derivative.
To simplify notation, we set:
Δ
h
q
f
(
x
)
=
∑
0
≤
m
<
∞
(
−
1
)
m
(
q
m
)
f
(
x
+
(
q
−
m
)
h
)
.
{\displaystyle \Delta _{h}^{q}f(x)=\sum _{0\leq m<\infty }(-1)^{m}{q \choose m}f(x+(q-m)h).}
So the Grünwald–Letnikov derivative may be succinctly written as:
D
q
f
(
x
)
=
lim
h
→
0
Δ
h
q
f
(
x
)
h
q
.
{\displaystyle \mathbb {D} ^{q}f(x)=\lim _{h\to 0}{\frac {\Delta _{h}^{q}f(x)}{h^{q}}}.}
An alternative definition
In the preceding section, the general first principles equation for integer order derivatives was derived. It can be shown that the equation may also be written as
f
(
n
)
(
x
)
=
lim
h
→
0
(
−
1
)
n
h
n
∑
0
≤
m
≤
n
(
−
1
)
m
(
n
m
)
f
(
x
+
m
h
)
.
{\displaystyle f^{(n)}(x)=\lim _{h\to 0}{\frac {(-1)^{n}}{h^{n}}}\sum _{0\leq m\leq n}(-1)^{m}{n \choose m}f(x+mh).}
or removing the restriction that n must be a positive integer:
D
q
f
(
x
)
=
lim
h
→
0
(
−
1
)
q
h
q
∑
0
≤
m
<
∞
(
−
1
)
m
(
q
m
)
f
(
x
+
m
h
)
.
{\displaystyle \mathbb {D} ^{q}f(x)=\lim _{h\to 0}{\frac {(-1)^{q}}{h^{q}}}\sum _{0\leq m<\infty }(-1)^{m}{q \choose m}f(x+mh).}
This equation is called the reverse Grünwald–Letnikov derivative. If the substitution h → −h is made, the resulting equation is called the direct Grünwald–Letnikov derivative:[ 1]
D
q
f
(
x
)
=
lim
h
→
0
1
h
q
∑
0
≤
m
<
∞
(
−
1
)
m
(
q
m
)
f
(
x
−
m
h
)
.
{\displaystyle \mathbb {D} ^{q}f(x)=\lim _{h\to 0}{\frac {1}{h^{q}}}\sum _{0\leq m<\infty }(-1)^{m}{q \choose m}f(x-mh).}
References
The Fractional Calculus , by Oldham, K.; and Spanier, J. Hardcover: 234 pages. Publisher: Academic Press, 1974. ISBN 0-12-525550-0 From Differences to Derivatives , by Ortigueira, M. D., and F. Coito. Fractional Calculus and Applied Analysis 7(4). (2004): 459-71. 
