Fractional Calculus: A Tutorial - Northwestern...

Synthesis and Processing of Materials

U.S. Army Research, Development and

Engineering Command

Fractional Calculus: A Tutorial Presented at

Network Frontier Workshop

Northwestern University December 4, 2013

Bruce J. West ST- Chief Scientist Mathematics

Army Research Office

[email protected]

919-549-4257

Collaborators: P. Grigolini

M. Bologna

M. Turalska

M.T. Beig

P. Pramukkul

A. Svenkeson

mailto:[email protected]


2

Fractional Calculus: A Tutorial

• Why a fractional calculus? new ways of thinking

dynamics and fractals

• Fractional dynamics fractional difference equations

simple fractional operators

fractional rate equation

• Fractional diffusion and probability turbulent diffusion

fractional Bloch equation

Lévy foraging

phase space fractional equations

• Conclusions


3


• Fractional thinking is in-between thinking:

− between integers there are non-integers

− between integer-order moments there are fractional moments

− between integer dimensions there are fractal dimensions

− between integer Fourier series are fractional Fourier transforms

− between integer-valued operators are fractional-order operators

• This tutorial is on how the fractional calculus provides

insight into complex dynamic networks.

• Complexity is emphasized, which highlights the inability of traditional analytic

functions to satisfactorily characterized the rich structure of complex dynamic

phenomena (networks) in both space and time.

2012


4


….A NEW WAY OF THINKING….

old new

Why is the fractional calculus entailed by complexity?


5


• Karl Weierstrass (1872): generalized by Mandelbrot (1977)

• Interesting properties

− continuous everywhere

− nowhere differentiable

− self-similar

• What are the dynamic equations for fractal functions?

1 ; cos11

)( 0

abtba

tW n

nn

tba

b

dt

tdW n

n

n

0cos)(

b

attWtaWbtW

log

log ; )()()(

….no equations of motion….


6


• Richardson at the London Expo, released 10,000 balloons

with a return address. From the data on where/when the

balloons landed he constructed Richardson Dispersion Law

• The solution yields the lateral growth of smoke plumes

• Molecular diffusion has a mean-square displacement

• Anomalous diffusion was therefore first observed in the study of turbulent fluid flow.

• Perhaps it could be described by a Weierstrass function?

3/22

2

)()(

tRdt

tRd

32)( ttR

ttR 2)(

….Turbulent diffusion….


7


Complex Webs: Anticipating the Improbable, B.J. West and P. Grigolini, Cambridge (2011).

Empirical Power Laws


8


• Physics: constitutive relationships

− Hooks law in ideal solids

− Ideal Newtonian fluid

− Newton’s law of motion

− One model for soft matter

kxF

y

uvF

2

2

dt

xdmF

20 ;

dt

xdF

Fractional-order calculus

Integer-order calculus


9



10


tt

dt

d

tdt

d

tdt

d

ttdt

d

2/1

2/1

2/1

2/1

2/1

2/1

2/1

11

0

1

1

• Example of Riemann-Liouville fractional derivative; using properties of

Gamma functions.

Curious results not consistent with ordinary calculus

Result obtain by Leibniz in response to question by L’Hopital

In 1695.

2003


11


• One way to capture complex dynamics

• Rate equation:

• Fractional rate equation (FRE):

• Caputo fractional derivative: defined in terms of Laplace transform

teQQ(t)tQdt

tdQ )0( )()(

integerfor ? )()(

Q(t)tQdt

tQd

)0()(ˆ;)( 1QuuQuu

dt

tQdLT

2011


12


• Laplace solution to fractional rate equation:

• Inverse Laplace transform:

• Solution first obtain by Mittag-Leffler in 1903:

)0()(ˆ )(ˆ)0()(ˆ1

1 Qu

uuQuQQuuQu

)()0()();(ˆ1 tEQtQtuQLT

0 1)(

k

k

k

ttE

exp[ ] as 0

1 as

t t

tt


13


• A second reason to learn the fractional calculus

• Consider the Caputo fractional derivative of the Generalized Weierstrass

Function whose Laplace transform is

• No analytic inverse but the inverse Laplace transform does scale

Fractional derivative α of fractal function of dimension µ is another fractal

function with fractal dimension µ−α; it does not diverge. Fractional calculus

yields the appropriate dynamics for fractal processes.

20

2

12

0)(;)(

nn

n

n

bu

u

a

buWuu

dt

tWdLT

tuWuLTtW );()( 1

DtWb

abtW )()(


14


• We have not changed very much.

• Hooke’s Law – anagram 1676 challenge to scientific community: ceiiinosssttuns

– Hooke was concerned that Newton would get the credit.

– solution anounced 1678: ‘ut tensio sic vis’

o ‘as stretch, so force’

• Fractional memory by phenomenological argument – Scott Blair et al., PRS A 187 (1947); fractional equation:

strain stress : t t t R t

d t

t Rdt


15


• Viscoelastic material experiments: generalized stress-strain relations

• Relaxation function G(t):

• Stress relaxation: fractional MLF smoothly joins two empirical laws Glöckle & Nonnenmacher (J. Stat. Phys. 71,1993; Biophys. J. 68, 1995)

( )( )

d G tG t

dt

( )G tte

t

Mittag-Leffler function (MLF)

Kohlrausch-Williams-Watts

Nutting


16


• Fractional Probability Density

• α-stable Lévy distribution

• Fractional Turbulence

• Lévy Foraging

),(ˆ),(ˆ ),(),( t,1tkPkKtkPtxPKtxP

FTxt

;exp );,(ˆ),( 11 xtkKFTxtkPFTtxP

Boettcher et al., Boundry-Layer Metero 108 (2003)

Gaussian

Lévy

Humphries et al., Nature 465 (2012).

Win

d s

peed c

hange


17


Human network Network model

• Two-state master equation decsion making model (DMM)

• DMM is member of Ising universality class

– phase transitions to consensus

– scaling behavior

– temporal complexity

• How does the network dynamics influence individual

dynamics?

• Another approach to the fractional calculus

2013


18


0)1( )1(1 sgnsnsgns n

• Subordination models numerical integration of individual opinion s(n) in

discrete operational time n:

• This is the time experienced by the individual and for is a

Poisson proces

• The influence of network dynamics on individual in chronological time t

is

'''0 0

dtnsttttsn

t

n

Pramukkul, Svenkeson, Grigolini, Bologna & West,

Advances in Mathematical Physics 2013, Article ID 498789

(2013).

1g

Probability density of

last of n events occurs

in time (0,t’)

Probability no event

occurs in (t-t’)


19


1 ; )()(

tstsdt

d

tT

Ttψ

tT

Tt

11

1

• The waiting-time distribution and survival probability are taken from numerics.

• Solve the subordination equation using Laplace transforms to obtain

fractional differential equation for average individual opinion:

• This is the predicted average dynamics of the single element within the

social network.

Turalska & West, Chaos, Solitions

& Fractals 55, 109 (2013)


20


• Solution to FDE is the Mittag-Leffler function

)()( tstsdt

d

Fractional Differential Equation (FDE)

tjptjptjs ,,, • Average opinion

cKK

1 1n

n

n

ttEts

K ≤ KC K = KC K ≥ KC

91.0 81.0 53.0

2 0.99r


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Conclusions

• The fractional calculus provides a new perspective

on complexity.

• It has been used to describe the dynamics of

turbulent and anomalous diffusion, optimal foraging,

viscoelastic relaxation, and on and on

• The fractional calculus provides a framework for

the dynamics of scale-free complex networks.

• The influence of a network on an individual is

described by a stochastic fractional differential

equation.

• Network dynamics transforms a Poisson-type

individual into a Mittag-Leffler-type person.

2011 2012 2013


22


How pervasive are non-integer phenomena?

…from integer to non-integer…


23


• Second example using Cauchy’s formula:

• Generalize Cauchy formula to Riemann-Liouville fractional integral

and to the Riemann-Liouville fractional derivative

but this is only one of many definitions of fractional operators

tfDdfdftn

n

tj

t t t n

jn

nt n

0 0 0 1

1

0

1 1

)()()!1(

1

dfttfD

t

t

1

0

1

ntfDDtfD n

t

n

tt 1 ;

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