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
919-549-4257
Collaborators: P. Grigolini M. Bologna
M. Turalska
M.T. Beig
P. Pramukkul
A. Svenkeson
mailto:[email protected]
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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
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Fractional Calculus: A Tutorial
• 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
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Fractional Calculus: A Tutorial
….A NEW WAY OF THINKING….
old new
Why is the fractional calculus entailed by complexity?
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Fractional Calculus: A Tutorial
• 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….
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Fractional Calculus: A Tutorial
• 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….
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Fractional Calculus: A Tutorial
Complex Webs: Anticipating the Improbable, B.J. West and P. Grigolini, Cambridge (2011).
Empirical Power Laws
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Fractional Calculus: A Tutorial
• 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
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Fractional Calculus: A Tutorial
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Fractional Calculus: A Tutorial
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
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Fractional Calculus: A Tutorial
• 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
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Fractional Calculus: A Tutorial
• 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
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Fractional Calculus: A Tutorial
• 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.
202
12
0)(;)(
nn
n
n
bu
u
a
buWuu
dt
tWdLT
tuWuLTtW );()( 1
DtWb
abtW )()(
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Fractional Calculus: A Tutorial
• 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
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Fractional Calculus: A Tutorial
• 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
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Fractional Calculus: A Tutorial
• Fractional Probability Density
• α-stable Lévy distribution
• Fractional Turbulence
• Lévy Foraging
),(ˆ),(ˆ ),(),( t,1 tkPkKtkPtxPKtxP 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
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Fractional Calculus: A Tutorial
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
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Fractional Calculus: A Tutorial
0)1( )1(1 sgnsnsgnsn
• 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’)
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Fractional Calculus: A Tutorial
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)
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Fractional Calculus: A Tutorial
• 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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Fractional Calculus: A Tutorial
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
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Fractional Calculus: A Tutorial
How pervasive are non-integer phenomena?
…from integer to non-integer…
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Fractional Calculus: A Tutorial
• 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 ntntt 1 ;