fractional dynamics. applications to the study of some transport phenomena dana constantinescu...
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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
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