fractional differentiation

fractional differentiation

The idea of

Grunwald-Letnikov differentiation

comes from the following formulas of backward (

http://planetmath.org/BackwardDifference

) and forward

difference

. Within this entry,

[

⋅

]

will be used to denote the

greatest integer function

and

Γ

will be used to denote the

gamma function

.

Backward difference

D

-

⁢

(

f

)

⁢

(

x

)

=

lim

h

→

0

⁡

f

⁢

(

x

)

-

f

⁢

(

x

-

h

)

h

(1)

D

-

n

⁢

(

f

)

⁢

(

x

)

=

lim

h

→

0

⁡

1

h

n

⁢

∑

k

=

0

n

(

-

1

)

k

⁢

n

!

k

!

⁢

(

n

-

k

)

!

⁢

f

⁢

(

x

-

k

⁢

h

)

(2)

For

derivatives

of

integer

orders, we only requires to specifies one point

x

∈

ℝ

.

Fractional derivatives

, like fractional

definite integrals

, require an

interval

[

a

,

b

]

to be specified for the

function

f

:

ℝ

→

ℝ

we are talking about.

Definition

1: Left-hand Grunwald-Letnikov derivative

D

-

p

⁢

(

f

)

⁢

(

x

)

=

lim

h

→

0

⁡

1

h

p

⁢

∑

k

=

0

[

b

-

a

h

]

(

-

1

)

k

⁢

Γ

⁢

(

p

+

1

)

k

!

⁢

Γ

⁢

(

p

-

k

+

1

)

⁢

f

⁢

(

x

-

k

⁢

h

)

(3)

Forward difference

D

+

⁢

(

f

)

⁢

(

x

)

=

lim

h

→

0

⁡

f

⁢

(

x

+

h

)

-

f

⁢

(

x

)

h

(4)

D

+

n

⁢

(

f

)

⁢

(

x

)

=

lim

h

→

0

⁡

1

h

n

⁢

∑

k

=

0

n

(

-

1

)

k

⁢

n

!

k

!

⁢

(

n

-

k

)

!

⁢

f

⁢

(

x

+

(

n

-

k

-

1

)

⁢

h

)

(5)

Definition 2: Right-hand Grunwald-Letnikov derivative

D

+

p

⁢

(

f

)

⁢

(

x

)

=

lim

h

→

0

⁡

1

h

p

⁢

∑

k

=

0

[

b

-

a

h

]

(

-

1

)

k

⁢

Γ

⁢

(

p

+

1

)

k

!

⁢

Γ

⁢

(

p

-

k

+

1

)

⁢

f

⁢

(

x

+

(

m

-

k

-

1

)

⁢

h

)

(6)

Theorem

1:

Properties

of fractional derivatives

•

Linearity:

D

±

p

⁢

(

a

⁢

f

+

b

⁢

g

)

⁢

(

x

)

=

a

⁢

D

±

p

⁢

(

f

)

⁢

(

x

)

+

b

⁢

D

±

p

⁢

(

g

)

⁢

(

x

)

where

a

,

b

∈

ℝ

are any real constants

•

Iteration

:

D

±

p

⁢

D

±

q

⁢

(

f

)

⁢

(

x

)

=

D

±

p

+

q

⁢

(

f

)

⁢

(

x

)

•

Chain rule

:

d

β

⁢

f

⁢

(

g

⁢

(

x

)

)

d

⁢

x

β

=

∑

k

=

0

∞

Γ

⁢

(

1

+

β

)

Γ

⁢

(

1

+

k

)

⁢

Γ

⁢

(

1

-

k

+

β

)

⁢

d

β

-

k

⁢

1

d

⁢

x

β

-

k

⁢

d

k

⁢

f

⁢

(

g

⁢

(

x

)

)

d

⁢

x

k

•

Leibniz Rule

:

d

β

⁢

(

f

⁢

(

x

)

⁢

g

⁢

(

x

)

)

d

⁢

x

β

=

∑

k

=

0

∞

Γ

⁢

(

1

+

β

)

Γ

⁢

(

1

+

k

)

⁢

Γ

⁢

(

1

-

k

+

β

)

⁢

d

k

⁢

f

⁢

(

x

)

d

⁢

x

k

⁢

d

β

-

k

⁢

g

⁢

(

x

)

d

⁢

x

β

-

k

Theorem 2: Table of fractional derivatives

•

D

±

α

⁢

(

x

p

)

=

Γ

⁢

(

p

+

1

)

⁢

x

p

-

α

Γ

⁢

(

p

-

α

+

1

)

where

α

,

p

∈

ℝ

and

Γ

⁢

(

x

)

•

D

±

α

⁢

(

e

λ

⁢

x

)

=

λ

α

⁢

e

λ

⁢

x

for all

λ

∈

ℝ

•

D

±

α

⁢

(

sin

⁡

x

)

=

sin

⁡

(

x

+

α

⁢

π

2

)

•

D

±

α

⁢

(

cos

⁡

x

)

=

cos

⁡

(

x

+

α

⁢

π

2

)

•

D

±

α

⁢

(

e

i

⁢

x

)

=

cos

⁡

(

x

+

π

⁢

α

2

)

+

i

⁢

sin

⁡

(

x

+

π

⁢

α

2

)

Title

fractional differentiation

Canonical

name

FractionalDifferentiation

Date of creation

2013-03-22 16:18:46

Last modified on

2013-03-22 16:18:46

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

21

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 26A06

Synonym

Grunwald-Letnikov differentiation

Related topic

HigherOrderDerivativesOfSineAndCosine

Defines

fractional derivative

Defines

left-hand Grunwald-Letnikov derivative

Defines

left hand Grundwald Letnikov derivative

Defines

right-hand Grundwald-Letnikov derivative

Defines

right hand Grundwald-Letnikov derivative

Generated on Fri Feb 9 19:43:32 2018 by

LaTeXML