Post on 22-Jun-2021
Concept of Fractional derivatives
1) Friedrich, C., H. Schiessel, et al. (1999). "Constitutive behavior modeling and fractional derivatives." Rheology Series: 429-466.
2) Sokolov, I. M., J. Klafter, et al. (2002). "Fractional Kinetics." Physics Today 55(11): 48-54.
Siddarth SrinivasanJune 10th 2010
1
Where do fractional derivatives occur?
- Subdiffusive systems1 2
0( , ) ( , )tP x t D P x tt
αακ
−∂= ∇
∂
Subdiffusive system described by red curve. Singular cusp, slower relaxation
- charge transport in anomalous semiconductors- spread of contaminants in geological formations- diffusing particle trapped in optical tweezers -displacement of a monomer of the Rouse model in solvent
Image from sokolov et al. (2002) 2
Introduction – Power law behaviour
Fig 1(a). Storage modulus and loss modulus for polyisobutylene
( )G ω′( )G ω′′
Fig. 1(b). Relaxation function and creep function for polyisobutylene
( )G t( )J t
Fig 2. of substituted polybutadiene.
PB300 – unsubstituted
PB302, PB304 – active groupsattached to backbone
and GG′ ′′
0 ,,
1G Gα β
α βω ω′ ∝ ′′ ∝
< <
2 ,GG ω ω′∝ ′′ ∝
Powerlaw
Graphs from Friedrich et al. (1999) 3
Role of fractional derivatives in constitutive equations
Assume :
(1 )( ) E tG t
β
β λ
−= Γ −
are constants,E,λ,β
1
0( )
( ) ( 1)!
x tx t e dt
n n
∞ − −Γ =
Γ = −∫Gamma function
Interpolates factorial
( )( ) ( )t d tt G t t dt
dtγτ
−∞
′= − ′ ′
′∫
Linear viscoelasticity
( )( ) ( )(1 )
tE d tt dt t tdt
ββλ γτ
β−
−∞
′= ′ − ′Γ − ′∫
( )( ) d tt Edt
ββ
β
γτ λ=
0β<1≤
Can be written as a fractional derivative
( ) ( )( )(1 )
1 td t d tdt t tdt dt
ββ
β
γ γβ
−
−∞
′= ′ − ′Γ − ′∫
Rheological constitutive equation with fractional derivatives
0, ( ) ( ), Hooke's law( )1, ( ) , Newton's law
t E td tt
dt
β τ γγβ τ η
= =
= = 4
Formal definitions – Part I
!( )!
nm m n
n
d mx xdx m n
−=−
( 1)( 1)
nn
n
d x xdx n
µ µµµ
−Γ +=Γ − +
( 1)( 1)
d x xdx
αµ µ α
α
µµ α
−Γ +=Γ − +
Repeated integer differentiation of an integral power
Repeated integer differentiation of a fractional power
Fractional derivative of an arbitrary power
Can handle any function which can be expanded in a Taylor series
More general technique is using fractional integrals Riemann-Liouville approach 5
Defining fractional derivatives – Part II
( )1
1 1
22 2 2 2 1
( )
( )( ) ( )
( )
( )
x
a
x x
a a
y
a
f y dy
If y dy f y dy
I
d
x
y
f
I f x
=
= =
∫
∫ ∫ ∫
First, define repeated integrals and fractional integrals:
Similarly, integrating n times,11
1( ) ( d) ynyyx
a a a
nn nI f x df y y
−
= ∫ ∫ ∫
By Cauchy’s Theorem for repeated integration,
11( ) ( ) ( )( 1)!
xn n
af x x y dy
nI f y−= −
− ∫ (Proof ?)
11( ) ( ) ( )( )
x
aI f x x y f y dyα α
α−= −
Γ ∫ by generalizing to arbitrary α6
Defining fractional derivatives – Part III
Notation: (potentially confusing!)
11( ) ( ) ( )( )
x
a x aD f x x y f y dyα α
α− −≡ −
Γ ∫ ‘fractional integral of arbitrary order’
n
na x a xn
dD Ddx
α α−≡ ‘α-th fractional derivative’, 0[ ] 1n α α+ >=
0α >
The fractional derivative!
87.4 0.6
0 08 x xdD Ddx
−=
Example:
7.48
.
0
0 6n
a
n
α
α
==− = −
= ie, to differentiate 7.4 times:
1) fractionally integrate first by 0.62) differentiate 8 times
Why integrate first?7
Alternate Notation
ddt
α
α Used by Friedrich et al.
1
1 ( )( ) ( )
td f t dtdt t t
α
α αα +−∞
′= ′Γ − − ′∫If 0,α <
If 0,α > ( )n n
n n
fd d ddt dt dt
tα α
α α
−
−
=
] 1[n α= +
Used in Rheological Constitutive Equations (RCE) because the lower limit of integration is where
t′ = −∞( ) 0tτ ′ =
8
Fractional derivatives of simple functions
Example 1: 1/20 1D
0.51
5)
0
0.( 1
a
nn
f x
α
α
==− = −
=
= 0.5
0
1/20
1 ( )(1/ 2)
xx xdD
dy dy
x−
= − Γ ∫
0 0.5
0.5
0
1 ( )(1/ 2)
1 ( )
1
2
x
x
ddx
dd
z dz
z dx
xdx
x
z
d
π
π
π
−
−
−= Γ
=
=
=
∫
∫
x y z− =substitute
Half-derivative of a constant is not 0 !9
Fractional derivatives of simple functions
Example 2: 1/20 D x
0.51
5(
0
0.)
a
nn
f x x
α
α
==− = −
=
=1/2
00.5
0
1 ( )(1/ 2)
xx dD x y ydy
dx−
= − Γ ∫
( )
0.5
0
0.5 0.5 0
0 0.5
0.5
0
0.5
0
.5
0
1
1 ( ) ( )(1/ 2)
1 ( ) ( )
2
1
.
x
x
x x
x
ddx
ddxd z dzd
x
x
z dz x
z x z dz
z x z dz
x z dz
x x
π
π
π
π
−
−
−
− −
−= − Γ
= − = −
= + −
=
∫
∫
∫
∫
∫ Liebniz Rule
x y z− =substitute
10
More examples of fractional calculus
From Sokolov et al. (2002)
{ ( )} ( ) ( )tD f t i fα αω ω−∞ =
Fourier transform
Laplace transform
0{ ( )} { ( )}tD f t u fα α ω− −=
Eqn 10 in Freidrich et al is probably incorrect
1/ , 1/ 2
d d d fdt dt df x
ft
µ ν µ ν
µ ν µ ν
µ ν
+
+≠
= = =
11
Back to RCEs
(1 )( ) E tG t
β
β λ
−= Γ −
For
( )( ) ( )(1 )
tE d tt dt t tdt
ββλ γτ
β−
−∞
′= ′ − ′Γ − ′∫
1 0 1aα β β
∞= − ≤−
<=
1
1
( )( ) d d tt Edt dt
ββ
β
γτ λ−
−=
1
1 ( )( )( ) ( )
tE d tt dtt t dt
β
α
λ γτα +−∞
′= ′Γ − − ′ ′∫
CE for assumed form of G Definition of fractional derivative
( )( ) d tt Edt
ββ
β
γτ λ=
*( ) ( ) ( )Gτ ω ω γ ω=
( )( ) ( )t d tt G t t
dtγτ
−∞
′= − ′
′∫
For SAOS 0( ) i tt e ωγ γ=
( ) ( ) ( )E i βτ ω ωλ γ ω=
*( ) ( )G E i βω ωλ=
12