Concept of Fractional derivativesweb.mit.edu/.../June_10_2010_FractionalDerivatives... ·...

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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 Srinivasan June 10 th 2010 1

Transcript of Concept of Fractional derivativesweb.mit.edu/.../June_10_2010_FractionalDerivatives... ·...

Page 1: Concept of Fractional derivativesweb.mit.edu/.../June_10_2010_FractionalDerivatives... · differentiation of an integral power Repeated integer differentiation of a fractional power.

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

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Page 2: Concept of Fractional derivativesweb.mit.edu/.../June_10_2010_FractionalDerivatives... · differentiation of an integral power Repeated integer differentiation of a fractional power.

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

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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

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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

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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

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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

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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

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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 ′ =

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Page 9: Concept of Fractional derivativesweb.mit.edu/.../June_10_2010_FractionalDerivatives... · differentiation of an integral power Repeated integer differentiation of a fractional power.

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

Page 10: Concept of Fractional derivativesweb.mit.edu/.../June_10_2010_FractionalDerivatives... · differentiation of an integral power Repeated integer differentiation of a fractional power.

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

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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

+

+≠

= = =

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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=

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