Fractional dynamics. Applications to the study of some transport phenomena

Dana ConstantinescuDepartment of Applied Mathematics

University of Craiova, 13 A. I. Cuza Street, 200585, Craiova Romaniadconsta@yahoo.com

Outline

1. Basic elements of Fractional Calculus

2. Fractional Diffusion Equation

3. Applications in Physics

4. Applications in Economics

5. Numerical method to solve fractional diffusion equations

000 ':0

xxxyxyytx

),(,],[: 0 baxRbay

h

hxyxy

h

hxyxyxy

hh

left

0000

00

0' lim

h

xyhxy

h

xyhxyxy

hh

right0000

00

0' lim

0'

0'

0' xyxyxy rightleft

00'

0, :0

xxxyxyyt leftxleft

00'

0, :0

xxxyxyyt rightxright

The derivation operator is local

1.1 Basic definitions. Local and non-local operators

The models described by ordinary or partially differential equations involve only local properties of the system

h

htxytxytx

t

y

,,

,

22

2 ,2,,

k

tkxyxytkxytx

x

y

(Local spatial effect)

(Local temporal effect-loss of memory)

The models described by integral or integro-differential equations involve non-local properties of the system

The integration operator is non-local

h

xbNhkxyhhkxyhdttyxyI b

N

k

N

khh

b

x

b

bb

,lim)(

0

0

0

0

00

0

0

Volterra integral equation

How could be included non-local effects in mathematical models similar with ordinary or partially differential equations?

Using fractional derivatives:

2

2

1

1

x

u

t

u

2

2

x

u

t

u

Memory-effectsat any time

x

u

t

u

1

1

Spatial effectsfor any position

x

u

t

u

Global spatial and global memory effectsincluded in any position and in any moment

1.2 History (memory effect?)

• 1695-1697 correspondence Leibniz-Johann Bernoulli, Leibniz-J Wallis

………………………………………………….

• 1695 correspondence l’Hospital ( ) Leibniz?2/1 xd xdxxxd :2/1

(One day very useful consequences will be drawn from this paradox, since there are little paradoxes without usefulness)

………………………………………………….• 1783 L. Euler introduces the Gamma-function as generalization of factorials (makes some comments on something which is “more curious than useful”)

• 1812 P.S. Laplace – Theorie Analitique des Probabilites

• 1819 F. S. Lacrois – Traite du calcul Differentiel et du calcul integral (use the Gamma-function in the definition of derivatives of non-integer order)

• 1823 N.H. Abel uses the derivative of order ½ for solving the “Tautochrone problem”

•1832-1837 J. Liouville (8 papers, one of them dedicated to fractional derivatives)•1847 G. F. B. Riemann proposed a formula for the fractional integral

xdttytxxyI

x

a

a

1)(1

• 1867 A.K. Grunwald (Ueber “begrenzte”Derivationen und deren Anverdung)

• 1868 A.V. Letnikov (Theory of differentiation with an arbitrary index )(Russian)

• 1869 N. Y. Sonin used the Cauchy’s Integral formula as starting point to reach differentiation of arbitrary order

•1872 A. V. Letnikov extended the idea on Sonin

•1884 H. Laurent obtained the fractional Riemann-Liouville integral and derivatives

• 1927 A. Marchaud developed an integral version of Grundwald-Letnikov derivatives• 1938 M. Riesz considered the “Riesz potential” which can be expressed in terms of Riemann-Liouville derivatives……………………………………………………………

• 1967 M. Caputo proposed the formula (Caputo derivative)

),1[1 1 nndttytx

dx

d

nxyD

x

a

nn

a

),1[1 1 nndttytxn

xyD

x

a

nna

C

• 1974 The first conference dedicated to FRACTIONAL CALCULUS (New Haven)

•1974 The first book about FRACTIONAL CALCULUS (Oldham and Spanier)

• 1998 First issue of “Fractional Calculus and Applied Mathematics”

1.3 Fractional integrals

x

a

x

a

n

t

a

n

t

a

t

a

na dttytx

ndtdtdtdttfxyI

n

1121 !1

1.......)(

1 12 (Cauchy)

b

x

b

x

n

b

t

n

b

t

b

t

nb dttyxt

ndtdtdtdttfxyI

n

1121 !1

1.......)(

1 12

x

a

a dttytxxyI 1)(1

b

x

b dttyxtxyI 1)(1

Integrals of order “n”

Riemann-Liouville integrals of any non-integer (fractional) order (1847)

0

1 dtet t 1...1 nn Nnnn !1

1.4 Fractional derivatives

aD

bD

Riemann (1847)-Liouville (1832-1837) H Laurent (1884)

),1[1 1 nndttytx

dx

d

nxyD

x

a

nn

a

Left derivative

(left spatial effect)

b

x

nnn

b nndttyxtdx

d

nxyD ),1[)(

1 1

Right derivative(right spatial effect)

),1[)( 21 nnxyDcxyDcxyD baR Riesz (1938)

xyxyD nna

xyxyID aa

n

j

jjnn

aaa ax

j

aIxyxyDI

11

NaxxDa

)(1

11

R-L derivatives are not compatible with Laplace transform

Caputo (1967)

],1(1 1 nndttytxn

xyD

x

a

nna

C

b

x

nnn

bC nndttyxt

nxyD ],1()(

1 1

Left derivative(left spatial effect)

Right derivative(right spatial effect)

nnaxk

ayxyDxyD

n

k

kk

aC

a ,11

1

0

nnxbk

byxyDxyD

n

k

kk

bC

b ,11

1

0

Relationswith

Riemann_Liouville derivatives

00)(1 xDaC

nnysysyssLyssyDL nnC 1)0(...0'0 121

0

nnnb

Cnb

nna

Cna yyDyDyyDyD 1

Marchaud(1927) Grundwald (1867) Letnikov(1868)

)(

12

!

C

nn dt

zt

ty

i

nzy

nk

k

hh

n

h

khxyk

knnn

xy

0

00

!

1...11

lim

xy n

xy Riemann-LiouvilleCaputo

Grundwald-Letnikov

h

khxyk

h

khxyk

xyD

aa N

k

k

hh

notation

N

k

k

hh

aGL

0

00

0

00

1

lim

!1...1

1

lim

h

khxyk

h

khxyk

xyD

bb N

k

k

hh

notation

N

k

k

hhb

GL

0

00

0

00

1

lim

!

1...11

lim

h

axNa

h

xbNb

yDyDyDyD bGL

baGL

a

Example

xxyRy sin],0[:

yDaGL0

4 Linear fractional differential equations

5)0('

0)0(

82''' 0

y

y

xyxyDxyxy

0....,,,)(....)( 21210011 kk

Ck

C RAAAxfxyDAxyDA K

nnsysLyssyDL

n

j

jjC

10

1

0

10

The direct Laplace transform

The inverse Laplace transform

0

1,

11

11k

k

k

xxEx

s

sL

Mittag-LeflerFunctions

(1927)

2. Fractional Diffusion Equation

Continuous Time Random Walk (CTRW)

(Stochastic processes, disordered systems-the position of a particle is influenced by its (random) interaction with other particles and/or sources)

0x xx 0

t

t = probability density function of waiting times

x = probability density function of jumps

txP , = probability of finding the particle in the position “x” at the moment “t”

t

tdtttdxtxPxxdttxtxP

0

'''',''',

Montroll-Weiss equation

Particles that did not move from “x” in the time interval [0,t]

Particles that came in “x” in the time interval [0,t]

t

tdtttdxtxPxxdttxtxP

0

'''',''',

Laplace Transform in time

0

,,~

, dttxPesxPstxPL st

Fourier Transform in space

dxtxPetkPktxPF ikx ,,ˆ,

kss

sP

ˆ~1

1~1~ˆ

skPkskPs ,~ˆ

21,

~ˆ 2

tet

Poisson distribution for waiting time

2

2

2

2

1

x

ex

Gaussian distribution for the jumps

2

2

2 x

P

t

P

t

x

et

txP 22

1,

CTRW are equivalent to Brownian motion

on large spatio-temporal scales

11

tt

Levy flight distribution for waiting times

1

||

1

xx

Levy flight distribution for the jumps

ses s 1~ )(

||1ˆ |)(| kek k

skPksskPs ,~ˆ||,

~ˆ 1

PDPD xtC

||0

PDPD xtC

||0

dketkEdkeds

ks

setxP ikxikxst

||

||,

0

1

Analytical solution

0 1k

k

k

xxE

Mittag-Lefler Function (1927)

0

1dttex xt

Matrix approach to discrete fractional calculusI. Podlubni, A. Chechkin, T Skovranek, Y. Chen, Journal of Computational Physics 228 (2009) 3137-3153

Numerical solution txfPDPD xtC ,||0

jhikffnmMfF ijij ,,][

n=number of time steps (of length h)m=number of spatial discretization intervals (of length k)

jhikPPnmMPG ijijP ,,][

txfPDPD xtC ,||0

FGRIdIdB pmnmn )( FRIdIdBG mnmnP 1)(

3. Applications in Fusion Plasma Physics (Tokamaks)

The tokamaks are toroidal devices used for producing energy through controlled thermonuclear fusion

+Atomic

nucleus

Atomic

nucleus

Atomicnucleus + Energy

nHeTD 10

42

31

21

Section of the ITER Project Tokamak reactor

The fuel is heated to temperatures in excess of 150 millions °C

The hot plasma is confined in the core region using a magnetic field

toroidal field

(traveling around the torus)

superconducting coils surrounding the

vessel

poloidal field (traveling in circles orthogonal to the toroidal field )

electrical current driven through the plasma

Tokamaks were invented in 1950s in Soviet Union

tokamak =toroidal’naya kamera s magnitnymi katushkami

(toroidal chamber with magnetic coils)

215 tokamaks over the world

ITER (France, Cadarache, the world’s largest tokamak, in construction) joint program of China, European Union, India, Japan, Korea, Russia, USA

http://en.wikipedia.org/wiki/Tokamak www.tokamak.info www.iter.org

A main problem:

to study the heat propagation (transport)

-core region- diffusive transport

-edge region-nondiffusive transport

Fractional diffusion model of non-local transport

(D. Del Castillo Negrete, Fractional diffusion models of nonlocal transport, Physics of Plasmas 13 (2006), 082308)

txSTx

TDT

txSTx

TDrTDlT

xR

t

xrxlt

,)(

,)(

2

2

||

2

2

1000

Finite size model ),( txSqqqT drxt

Tq xdd TDlq xl1

00 TDrq xrr

110

Standard diffusion termNon-local fluxes

2,1

Fractional diffusion

1

Fokker-Planck

2

Classical diffusion

Balistic transport

22 tx

Difusive transport

12 tx /22 tx

Superdifusive scaling

T(x,t)=averaged temperature on the surface 1,0xr at the moment t

Collisional dominated transportFree streaming regime

Experiment Power modulation

Electron temperaturetime [s]

Te

[keV

]T

e [k

eV]

Fractional model

1.25

Source: D. Del Castillo Negrete, Physics of Plasmas 13 (2006), 082308

1.25

Experiment

Te 0.03keVModel

Consistent with the experiment, the fractional model gives a delay of the order of 4ms for cold pulses

Comparison of a radial fractional transport model with tokamak experiments(A. Kullberg, G. J. Morales, and J. E. Maggs PHYSICS OF PLASMAS 21, 032310 (2014))

Radial diffusion model was derived by azymuthally averaging an isotropic Laplacian operatorKullberg, D. del Castillo-Negrete, G. J. Morales, and J. E. Maggs,

Phys. Rev. E 87, 052115 (2013).

trStrTtrTt r ,,),(

2

3 2/

0

2/ ''','1

drrTrrKrr

rrrr

',max,)',min(,1,

2,

2

2

2/1

2/',

21rrrrrr

r

rF

rrrK

Temperature profile in Rijnhuizen Tokamak Project (RTP device) for the plateau in core temperature

Temperature profile in RTP device for different locations of the heating source

(Black dark symbols are experimental temperature measurements)

4. Applications in Economics

Fractional diffusion models of option prices in markets with jumps A. Cartea et al, Physica A 374 (2007), 749-763

An important problem in finance is pricing the financial instruments that derive their value from financially traded assets( stocks for example)

S(t)=stock price/unit B(t)= price/unit of the bond

)()( tyBtxSV Financial portofolio

rBdtdB SdzSdtdS

Main assumption: the stock price follows a Brownian process

average growth

of the stock the volatility of the returns from holding S(t)

increment of

Brownian motion r = risk-free rate

ydBxdSdV (direct computation)

dzS

VSdt

S

VS

S

VS

t

VdV

2

222

2

1(Ito Lemma)

2

222 ),(

2

1),(,),(

S

tSVS

S

tSVrS

t

tSVtSrV

Sx ln

2

222 ),(

2

1

2

1,),(

S

tSV

S

Vr

t

txVtxrV

Black-Scholes (BS) model

)()( tyBtxSV Financial portofolio

Assumption: the stock price follows a Levy process with the density

0

0,||

1

1

xxpD

xxqDxw

20,1]1,1[,,0 qpqpD

SdLSdtvrdS rBdtdB r = risk-free rate v = convexity adjustment

tgVDS

Vr

t

txVtxrV x ,2/sec

2

12/sec

2

1,),(

Fractional Black-Sholes model

Nobel Prize in Economic Sciences (1997)

tgVDS

Vr

t

txVtxrV x ,2/sec

2

12/sec

2

1,),(

Numerical integration using Grundwald-Letnikov approximation (matrix approach)

TtBxKe

TtBxfortxV

dx

d

0,max

00,

Option prices: European call

The owner has the right (but not obligation) to buy/sell a unit of stock at the future time T for a pre-specified price K

Restriction imposed in the “down and out” call

52/1,52/2,12/1,12/2,12/3

25.0,435.1

T

Bd

Differences between classical BS and fractional BS in down-out model

Interpretation of the numerical results

(K=50, S(0)=50 and barrier bellow the strike price K) :

- BS options are more expensive that LS options for S<K

- LS options are more expensive for

S>53

3D financial model

Ma JH, Chen YS, Study for the bifurcation topological structure and the global complicated character of nonlinear financial systems , Appl Math Mech 22 (2001, 1375-1382

czxz

xbyy

xayzx

'

1'

'2

x=interest rate

y=investment

z=inflation

a=saving amount

b = cost per investment

c = elasticity of demand of commercial markets

czxzD

xbyyD

xayzxD

qt

qt

qt

3

2

1

21

Wei-Ching Chen, Nonlinear dynamics and chaos in a fractional-order, Chaos, Solitons and Fractals 36 (2008) 1305–1314

fixed points,periodic orbits, and chaotic dynamics

czxzD

xbyyD

xayzxD

t

t

t

21

0.1)(99.0)(96.0)(

93.0)(85.0)(84.0)(

fed

cba

11.03 cba

2,3,2,, 000 zyx

2,3,2,, 000 zyx

Parameters

Initial point

Conclusion

Chaos exists in fractional systems with order

3321 qqq

55.23 In this case

130.0max The largest Lyapunov exponent

zxzD

xyyD

xyzxD

t

t

qt

1

21 1.01

31

zxzD

xyyD

xyzxD

t

qt

t

1

2

1

1.01

3

2

65.0)(70.0)(80.0)(90.0)( 1111 qdqcqbqa 89.0)(90.0)(95.0)(99.0)( 22 dcqbqa

5. Numerical method to solve fractional diffusion equations

1D symmetric diffusion equation

mmifd

t

xT

mtxTD

ma

mtCt

1,1

,1

00

n=number of time steps (of length h)m=number of spatial discretization intervals (of length k)

],0[0,

00,,0

,

0

||0

LxxTxT

ttLTtT

txSTDATD xtC

jhikTPnmMTG ijijP ,,][ jhikSSnmMSS ijij ,,][

SGRIdIdB pmnmn )(

SRIdIdBG mnmnP 1)(

],0[0,

00,,0

,

0

,,0 2211

LxxTxT

ttLTtT

txSTDBTDATD rlRxrl

Rx

Ct

1D non-symmetric diffusion equation

ppdrdltxTD pbpb x

tT

x

bp

x

tT

a

xp

xprlRx

11, 11

,,1,

txTT ,

2D-non-symmetric diffusion equation

.

],0[],0[,0,,,0,

],0[],0[,0,,,,0

],0[],0[,,00,,

,,2211 ,,0

txy

tyx

yx

rlRyrl

Rx

Ct

LLtxtLxTtxT

LLtytyLTtyT

LLyxyxTyxT

tyxSTDTDTD

tyxTT ,,

tyxijkijk hkhjhiNnkmjpiNG ,,},...,2,1{,}....,2,1{,},...2,1{,

3D grid

1,,2,,1,,,,)()(

1,,)(2,,

)(1,,

)(,, ...... jijinjinjinjijinjinji TTTTBTTTT

the Caputo derivative in time at these nodes can be approximated using discretized Grundwald-Letnikov

operators

.

00......0

00...0

..................

......0

......

1

1

21

121

121

)(

w

ww

www

wwww

hB

n

nn

tn

11

11

sss

sw

kikikmikmimkikikmikmi

kjkjkjpkjppkjkjkjpkjp

TTTTRLTTTT

TTTTRLTTTT

,1,,2,,1,,,)(,1,

)(,2,

)(,1,

)(,,

,,1,,2,,1,,)(,,1

)(,,2

)(,,1

)(,,

......

lyrespective

......

The spatial derivative at these nodes can be also approximated using discretized Grundwald-Letnikov operators

)(1

)(1

ppp FrBlRL )(

2)(

2mmm FrBlRL

)()(

1

21

121

121

)( and

00......0

00...0

..................

......0

......

1

pp

p

pp

xp BF

w

ww

www

wwww

hB

. and

00......0

00...0

..................

......0

......

1 )()(

1

21

121

121

)(

mm

m

mm

ym BF

w

ww

www

wwww

hB

tyxSTDTDTD rlRyrl

Rx

Ct ,,

2211 ,,0

pmnpmnmnpmpnpmn FURLIdIdRLIdIdB

},...,2,1{,}....,2,1{,},...2,1{,,, nkmjpikhjhihTU tyx pmnpmnMU pmn ,

},...,2,1{,}....,2,1{,},...2,1{,,, nkmjpikhjhihSF tyx pmnpmnMFpmn ,

pmnmmmpnpmn SRLIdIdRLIdIdBpmnU 1

},...,2,1{,}....,2,1{,},...2,1{,,, nkmjpikhjhihTU tyx

Constantinescu D., Negrea M., Petrisor I., Theoretical and numerical aspects of fractional 2D transport equation. Applications in fusion plasma theory (to be published in Romanian Journal of Physics)

]1,0[]1,0[,0,1,,0,

]1,0[]1,0[,0,,1,,0

]1,0[]1,0[,3sin10,,

0||||0

txtxTtxT

tytyTtyT

yxyxxyxT

TDyT

xT

tC

y =polo idal angle

x=

rad

ius

=1, = 2 , = 2 tim e t=0 .2

0 0 .2 0 .4 0 .6 0 .8 10

0.1

0 .2

0 .3

0 .4

0 .5

0 .6

0 .7

0 .8

0 .9

1

-0 .09

-0 .08

-0 .07

-0 .06

-0 .05

-0 .04

-0 .03

-0 .02

-0 .01

0

20.0,2,1 t

y =polo idal angle

x=

rad

ius

=1, = 1 .9 , = 1 .9 tim e t=0 .2

0 0 .2 0 .4 0 .6 0 .8 10

0.1

0.2

0.3

0 .4

0 .5

0 .6

0 .7

0 .8

0 .9

1

-0 .14

-0 .12

-0 .1

-0 .08

-0 .06

-0 .04

-0 .02

0

0.02

20.0,90.1,1 t0 0.2 0.4 0.6 0.8 1

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

y

T(0.

5,y,

0.20

)

==2

==1.90

==1.85

==1.80

20.0,,2/11 yT

TDTD xtC

||0 Radial heat transport(Average on poloidal direction)

Heat transport in radial and poloidal directions