essence of fractional calculus in applied sciences
TRANSCRIPT
Essence of FRACTIONAL CALCULUS in applied sciences
Part-I
WORK SHOP ONFRACTIONAL ORDER SYSTEM
28-29 March, 2008
IEEE KOLKATA CHAPTERDRDL HYDERABAD
BRNS(DAE) MUMBAI
Shantanu DasReactor Control Division
BARC
Salute to Indian Mathematicians of Fractional Calculus
Anil GangalKiran KolwankarH.M.SrivastavaO.P.AgarwalS.C Dutta RayL.DebnathR.K.SaxenaRasajit Kumar Bera…………
………………and to all exponents around the globe to have given this wonderful subject to us applied scientists and engineers, a language what nature understands the best, to communicate with nature in better and efficient way.
Essence of fractional calculus is…….
………….in understanding nature better.
………….in making effort to have this subject as Popular Science.
………….in simple teaching and evolving the future methods in mathematics and making working systems
……………in realizing that our physical understanding is limited and mathematical tools go far beyond our understanding
……………in appreciating the wonderful world of mathematics that lays between integer order differentiation and integration.
Fractional Calculus does not mean the calculus of fractions, nor doest it mean a fraction of any calculus, differentiation, integration or calculus of variations.
The FRACTIONAL CALCULUS is a name of theory of integration and derivatives of arbitrary order, which unify and generalize the notion of integer order n-fold repeated differentiation and n-fold repeated integration.
FRACTIONAL CALCULUS isGENERALIZED differentiation and integration.
GENERALIZED DIFFERINTEGRATIONS
What is not FRACTIONAL CALCULUS
THE GENERALIZED CALCULUS
( )
,
i
i
d f zd z
α β
α β
α β
+
+
∈
( )d f tdt
α
α
α ∈
( )n
n
d f xd x
n ∈
Integer OrderNewtonianPoint property
Fractional order
Non-localDistributedHistory/heredityNon-Markovian
Complex-Order
Generalization of theory of numbers and calculations
32 2 2 2 8= × × = Can be visualized0.52 exp{(0.5) ln 2} 1.414= = Number exists but hard to visualize how.
5! 1 2 3 4 5 120= × × × × = Is a visualized quantity, but what about (5.5)!
Generalized factorial as GAMMA FUNCTION (5.5)! (1 5.5) (6.5) 287.88=Γ + =Γ =ln
1
0
,! ( 1 ) ( )
( )
( ! )( ) l i m( 1 ) ( 2 ) . . . ( )
r r x
t x
x
n
x e rx x x x
x e t d t
n nxx x x x n
∞− −
→ ∞
= ∈= Γ + = Γ
Γ =
Γ =+ + +
∫
f dfdx
2
2
d fdx
3
3
d fdx
fdx∫fdx∫∫
Wonderful universe of mathematics lays in betweenOne full integration and one full differentiation
Fractional calculus gives continuum between full differ-integration
2
/
/ /
( )( ) 2( ) 2
f x xf x xf x
=
=
=
( )
0 1
d f xdx
α
α
α≤ ≤
0 2( ) ( )f x D f x x= =/ 1( ) ( ) 2f x D f x x= =
/ 1( ) ( ) 2f x D f x x= =
// 2( ) ( ) 2f x D f x= =
1.5 ( )D f x
Curve fitting will be effective by use of fractional differential equation, as compared with polynomial regression and integer order differential equation. The reason is extra freedom to closely track the the curvature in continuum.Could be a magnifier tool to observe the formation of discontinuity.
Application-IGeneralization of Newtonian mechanics and differential equations
// /0( ) ( ) ( ) ( )mx t b x t kx t f t+ + =
( )x t
/
( 0 ) 0(0 ) 0
xx
=
=
Mass concentrated at pointMass less springFrictionless springInfinite wall
( )f tm
20( ) ( ) ( )ms b s k X s F s+ + = Spring with friction ( ) ( )
0 1
qq spk s X s F s
q
=
≤ ≤
1 1
1 1 0
20
20
2
0
2
0
( ) ( ) ( )
( ... ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
n n
n n
n
q q qq q q q
Nq
nn
q
ms b s k s k X s F s
ms b s k s k s k s k X s F s
k s X s F s
k q s X s F s
−
−
=
=
+ + + =
+ + + + + + =
=
⎛ ⎞=⎜ ⎟
⎝ ⎠
∑
∫
Distributed massSpring with massSpring with frictionDamping with spring actionNon conservation systemLeaky wall/termination
Application-IISystem Identification & order distribution
// /0
20
00
( ) ( ) ( ) ( )
( ) ( ) ( )
{ [ ( 2) ( 1) ( )] } ( ) ( )q
mx t b x t kx t f t
ms b s k X s F s
m q b q k q s dq X s F sδ δ δ∞
+ + =
+ + =
− + − + =∫ 0 1 2q
( )k q0b
km
3 12 2 21 0 1 0
1 0 1 0
3 12 2 2
1 0 1 03 1222
( ) ( ) ( )
{[ ( 2) ( 1.5) ( 1) ( 0.5) ( )] } ( ) ( )
( ) ( ) ( ) ( ) ( )
q
ms b s b s k s k X s F s
m q b q b q k q k q s dq X s F s
d x t d x t dx t d x tm b b k k f tdt dt dtdt
δ δ δ δ δ
+ + + + =
− + − + − + − + =
+ + + + =
Integer Order:
Fractional Order
Continuous Order
0
1
0
( ) ( ) ( )
{ ( ) }* ( ) ( )
q
q
k q s dq X s F s
k q s dq x t f t
∞
∞−
⎛ ⎞=⎜ ⎟
⎝ ⎠⎛ ⎞
ℑ =⎜ ⎟⎝ ⎠
∫
∫ q
( )k q
Application-III
Order distribution based feed back control system
Reaction of a system depends on order value.Reaction of a system depends on amplitude of orderA first (integer) order system cannot go into oscillations.Presence of fractional order and its strength can give oscillations.Why not control system order and its strength?
A futuristic automatic controller
( )H s ( )G s
2
20
1( ) , ( ) ( ) qG s H s k q s dqs a
∞=
= =+ ∫
2
02
2
0
( ) ( )( )1 ( ) ( )
( )( )
1 ( ( ) )
q
q
H s G sT sH s G s
k q s d qT s
s k q s d q
=+
=+ +
∫
∫ Demanded order distribution-
( )k qq
Application-IVCircuit theory
Fractional order sourceFractional order loadFractional order connectivity
sLLRBBV
1V2V 3V
Ci
WWi
BRRBi
( ) ( )L BBdi tL R i t V
dt+ =
( )i t( ) ( )Ci t i t=
1 2 1 20
1( ) ( ) ( ) (0)t
Cv t v t i t dt vC −− = +∫
C
( ) ( ) ( )C W R Bi t i t i t= +
12
0 2 3( ) [ ( ) ( )]W ti t D v t v t= −
2 3( ) ( )( )RBB
v t v ti tR−
=
Inside battery
Application-V
Heat flux and temperature for semi infinite heat conductor.
12
0( ) [ ( ) ]t surfkQ t D T t T
kc
α
αρ
= −
=
0x =x = −∞
( )surfT t
2
2
2
2
0
0
0
( , ) ( , )(0, ) , (0, ) 0( ,0) ( ), ( ,0) ( )
( ,0)( )
surf surf
T Tc kt xu uc kt x
u t x T t x TT x T u xT t T t u t T t T
T tQ tx
ρ
ρ
∂ ∂=
∂ ∂∂ ∂
=∂ ∂
= −
= =
= = −
∂=
∂
Application-VI
Impedance RC distributed semi infinite transmission line
12
12
1( ) . ( )
1
1( )
a ti t D v tR
RCRZ sC s
α
α
=
=
⎛ ⎞= ⎜ ⎟
⎝ ⎠
Basic building block for fractional order immittance realization of arbitrary order to make fractional order analog function generator and fractional order analog PID controller.
2
2
1
( , ) ( , )
( , ) ( , )
(0, ) ( ), ( , ) 0
v x t i x t Rx
i x t v x tCx t
v i vR RCx x t
v t v t v t
∂=
∂∂ ∂
=∂ ∂
∂ ∂ ∂= =
∂ ∂ ∂= ∞ =
Application-VII
Fuel efficient control system
( )( 1)
KG sJs sτ
=+
21
1( ) K sH s Ksα+
=
Set speed Output speed
The constant close loop phase gives a feature of ISO-DAMPING where the peak overshoot is invariant on parametric spreads, giving fuel efficiency, avoidance of plant spurious excursions and trips, enhances safety and increases plant operational longevity.
Application-VIII
Fractional Divergence
To define non-local flux of material flowing through an isotropic media, loss volume and heterogeneous ambient.
Non Fickian diffusion phenomena
Anomalous diffusion
Anomalous random walk with unrestricted jump length per time.
( )
2
1lim . ( )
( ) ( ) 0
11 2
V REVS
div J J J ndS xV
d x B xdx
α α
β
β
β αβ
→
Δ →= ∇ ≡ = Φ
Φ+ Φ =
= +≤ ≤
∫
Application-IX
Electrode Electrolyte interface, derivation of Warburg lawApplication in Electrochemistry.Non-Fickian reaction kinetics.Power law in anomalous diffusionTime constant aberrationMagnetic flux diffusion studies in geophysics
2
0 2( , ) ( , )
1( , )
t NFD C x t D C x tx
C x tt
α
α
∂=
∂
≈
( , )C x t
tReaction to impulse excitationNon exponential reaction
0( , ) (0,0)exp( / )1
1( , )
F
C x t C t
D
C x ttα
τ
τ
≠ −
≈
≈
Application-X
Fractional CurlIn between dual solution in electrodynamics
( , , , , , ) ( , , , , , )( , ) ( , )
1 ( )( )
1 ( )( )
e m
fd
fd
E H D B H E B DE H H E
E Eik
H Hik
αα
αα
ρ ρμ ε ε μ
η η
η
↔↔ − −
↔ −
= ∇×
= ∇×
Future R&D in in-between mapping of Right Handed Maxwell systems and Left Handed Maxwell Systems (RHM)-(LHM)
Application-XI
ElectrodynamicsWave propagation in media with losses.
2 2
0 0 0 0 02 2 0
1 2
E E Et t x
α
αμ ε μ ε χ
α
∂ ∂ ∂+ + =
∂ ∂ ∂≤ ≤
Power factor modeling in AC machines, a new field of R&D.
sin sin2
D E t E tα α παω ω ω⎛ ⎞= +⎜ ⎟⎝ ⎠ 2
πα
Application-XII
ElectrodynamicsMultipole expansion
X
Y
Z
θa rR P
2 2
22
2 3
0
( , )2 cos
(cos ) (3 cos 1) ...2
(cos )k
kk
q qrR r a ar
q qa qar r r
q a Pr r
θθ
θ θ
θ∞
=
Φ ∝ =+ −
→ + + − +
⎛ ⎞= ⎜ ⎟⎝ ⎠
∑
( )kP x
1x
21 (3 1)2
x −
31 (5 3 )2
x x−
4 21 (35 30 3)8
x x− +
Fractional mutipoleFractal charge distribution
0
12
0
1( )4
(1 ) (cos )4 ( )
qa Dr
qa Pr
αα
α
αα
πε
α θπε
−∞
+
⎛ ⎞ ⎛ ⎞Φ = ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠
Γ +Φ =
aa
aFractional Legendre polynomial, FractionalPoles, dipole, monopoleSelf similarity-fractal distribution
0
1
2
0 21 22 2
α
α
α
= →
= →
= →
MonoDipoleQuadra
Application-XIIIFractal Geometry & Fractional Calculus 0
loglim1log
FND
ε
ε→
=⎛ ⎞⎜ ⎟⎝ ⎠
1FD = 2FD =
2N =
3N =
2, 1/2r ε= =
3, 1/3r ε= =4
2, 1/2Nr ε== =
93, 1/3
Nr ε== =
32, 1/ 2
log3 log3/ log2 1.5851log 12
F
Nr
D
ε== =
= = =⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
Application to graph theory and reliability analysis of software, data structure, cancer cellgrowth as future R&D topic on use of Local Fractional Calculus.
Application-XIVRelation of fractal dimensions and order of fractional calculus
Time constant aberration and transfer function of flow through a Fractal structure and relation to its fractal dimension.
1
2
1( )1
1( )1 ( )
G ss
G ss
D
λ
τ
τλ
=+
=+
↔ Relation of order to the fractal dimension
Application-XVFractional calculus and multifractal functions
Fractals and multifractal functions and corresponding curves or surfaces are found in numerous non-linear, non-equilibrium phases like low viscous turbulent fluid motion, self similar and scale independent processes, continuous but nowhere differentiable curves.
0( ) cos( )
30 1, 0, 12
loglog 2
n n
nf x a b n
a b ab
aDb
π
π
∞
=
=
< < > > +
=+
∑
Fractality implies D>1 and it is scale independent, has no smaller scale
Weistrauss
Application XVViscoelasticity
0
00
10
( ) ( )0 1
( ) ( ) ( )
( ) ( ) ( )
t
t
t
t K D t
t Y t Y D tdt t D td t
ασ εα
σ ε ε
σ η ε η ε
=< <
= =
= =
Pure solid Hook’s law
Newtonian fluid
Ideally no matter is pure solid nor is pure fluid
Y
Y η Y η
Application-XVIBiology
Muscles and joint tissues in musco-skeletal system seem to behave as visco-elastic material, with fractional integrator, then this could be compensated by fractional order differentiator dynamics of neurons.
0
/0
1 2
1
( )
( )( 1)( )
( ) 1
X X
G s X ssR s s
V s s
α
α
β α
ω ω
τ ττ
−
−
−
=
=+
=+
Membrane reaction relation as power law to frequency of current
Motor discharge rate to rate of change of position
And several more…….
Observations
Distributed systems behave as fractional orderRepresentation of distributed system is better with fractional calculus.Distribution can be in space or in time.Almost all semi-infinite system gets representations in half derivative.Good field of study as to why?Can ambient changes manifest the order of calculus from say half to other value?What is the physics behind that change?This order value changes can be instrumented to study or make the instruments or instrumentation systems for measurement and control.
Generalized repeated differ-integration of monomial
1
22
2
1
2
( )
( )
( ) ( 1)
11
1( 1)( 2)
m
m m
m m
m m
m m
f x xd x mxdxd x m m xdx
x dx xm
x dxdx xm m
−
−
+
+
=
=
= −
=+
=+ +
∫
∫∫
( ) ( 1)( 2)...( 1)
( 1) ( 1)( 2)....( 1) ( 1)
nm m n
n
d x m m m m n xdx
m m m m m n m n
−= − − − +
Γ + = − − − + Γ − +
( 1) ( 1)...( 1)( 1)
m m m m nm nΓ +
= − − +Γ − +
For any arbitrary index ,mn∈
Euler formulation (1730)
( 1)( )( 1)
nm m n
n
d mx xdx m n
−Γ +=Γ − +
Differ-integration is:
Examples of Euler formula:0.5 0.5
1 0.50.5
(1 1) (2) 1 2( )(1 0.5 1) (1 0.5) 0.5 (0.5)
d x xx x xdx π
−Γ + Γ= = = =Γ − + Γ + Γ
0.5 0.5 ( 0.5)
0.5
(0.5 1) 0.5 (0.5)(0.5 { 0.5} 1) (2) 2
d xx x xdx
π− − −
−
Γ + Γ= = =Γ − − + Γ
1 1 1 1 ( 1) 2
1
(1 1) (2) (1 1)1,(1 1 1) (1) (1 { 1} 1) 2
d x d x xx xdx dx
− − − −
−
Γ + Γ Γ += = = = =Γ − + Γ Γ − − +
Using monomial integration in solving differential equationExample classical oscillator
//
/
( ) ( ) ( )( ) ( )(0) 0, (0) 0( ) sin
x t x t f tf t tx xx t t
δ+ ==
= ==
( )
//
//
0 0 0 0
/
0 0 0
1 2 2 2 2 2 2@ 0
( ) ( ) ( )
( ) ( ) ( )
( ) (0) (0) ( ) ( )
( ) (0) [ ( )] ( ) ( ( )) ( ( ( ))) ..
t t
t
t t t t t t t t
x t f t x t
x t f t dt x t dt
x t x tx f t dt x t dt
x t x t d x t d f t d d f t d d d f t
→ → + +→
→ + +→
− − − − − −=
= −
= −
− − = −
= + + − + −
∫∫ ∫∫ ∫∫
∫∫ ∫∫
2td −
2td −
2td −
2td −
1
1−
1−
1−
∑
( )f t
(0)x/(0)tx
/( ) ( ), (0) 0, (0) 0f t t x xδ= = =
3 5 7
( ) .. sin3! 5! 7!t t tx t t t= − + − + ≈
1
2
23
34
( ) 1( )
( )2
( )3 2
d td t t
td t
td t
δ
δ
δ
δ
−
−
−
−
=
=
=
=×
0x
1x
2x
3x
Using monomial differ-integration to solve fractionalDifferential equation:Example oscillator with fractional loss component
1// 2
3/ 2 2 2
/
( ) ( ) ( ) ( )
( ) (0) ( ) ( ) ( ) ( )(0) 0, (0) 0, ( ) ( )
x t d x t x t f t
x t x tx t d f t d x t d x tx x f t tδ
−− −
+ + =
= + + − −
= = =
2d −
3 22( )d d− −+
3 22( )d d− −+
1
1−
1−
0( )x t
0( )x t
0( )x t
/(0)x
( )f t
∑(0)x
( )x t
( 1)( 1)
m nn m m xd x
m n
−Γ +=Γ − +
Euler’s generalization
20
5/2 33 221
5 93 4 52 23 222
( ) ( )
( ) ( )( )(7/2) (4)
( ) ( ) 2(7/2) (4) (5) (11/2) (6)
x t d t t
t tx t d d t
t t t t tx t d d
δ−
− −
− −
= =
⎛ ⎞=− + =− +⎜ ⎟Γ Γ⎝ ⎠
⎛ ⎞⎜ ⎟= + + = + +⎜ ⎟Γ Γ Γ Γ Γ⎝ ⎠
2.5 3 4 4.5 5
( ) 2 ..(3.5) (4) (5) (4.5) (6)t t t t tx t t= − − + + + +
Γ Γ Γ Γ Γ
Fractional oscillator an example:
LV
0r→
0
2
2
1 ( )( ) lim ( )
1 ( )( ) ( )
r
d i ti t d t ri t L VC dt
d i t dVi t L V tC dt dt
δ
→+ + =
+ = =
∫
0r→
C
CVL
Long CRO cable as Semi infinite TL half derivative12 ( )TLv kd i t−=
12
1 22
1 22
1 ( )( ) ( )
1 ( ) ( )( ) ( )
di ti t dt kd i t L VC dt
d i t d i t dVi t k L V tC dt dtdt
δ
−+ + =
+ + = =
∫
C R O
C R O
Short CRO cable circuit as oscillator
First order system and monomial integration/
1 1 1 1
1 1 1 1 1 1
( ) ( ) ( )( ) ( ) ( )
( ) (0) ( ) ( ) ( ) ..(0) 0, ( ) ( )
x t x t f td d x t d x t d f tx t x d f t d d f t d d d f tx f t tδ
− − −
− − − − − −
+ =
+ =
− = − + −= =
( )f t 1+
1−
1−
1d −
1d −
1d −
(0)x
∑
0x
1x
2x
2 3
( ) 1 ... exp( )2! 3!t tx t t t= − − + − ≈ −
First order system with fractional loss term monomial solution1/ 2
11 1 2
1
( ) ( ) ( ) ( )( ) ( ), (0) 0
( ) (0) ( ) ( 1) ( ) ( )n n
n
x t d x t x t f tf t t x
x t x d f t d d f t
δ∞
−− −
=
+ + == =
= + + − +∑1
0121 11 12 2
1 0
1 12 21 1 11 12 2 2
2 1
( ) 1
( ) (1) (1)(1.5)
( )(1.5) (1.5)
x d t
tx d d x d d t
t tx d d x d t d d t d
δ−
− −− −
− − −− −
= =
= + = + = +Γ
= + = + + +Γ Γ
( 1)( 1)
nm m n
n
d mx xdx m n
−Γ +=Γ − +
Euler relation
3 322 2
(2.5) (2) 2 (2.5)t t t t
= + + +Γ Γ Γ
31 22 22( ) 1 ....(1.5) (2) (2.5) 2t t t tx t t= − − + + + −
Γ Γ Γ
C
R
12
12
1 ( ) ( ) ( )
( ) 1( ) ( ) ( )
i t dt Ri t kd i t VC
di t dVR kd i t i t V tdt C dt
δ
−+ + =
+ + = =
∫
V
Distributed effect of long TL comes as fractional derivative/integral term.behaves as half order element , will it give II order response for I order system?
Poles in first order system with fractional lossConcept of w-plane conformal mapping
12
12
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
dx t ad x t x t f t
sX s as X s X s F s
+ + =
+ + =
Characteristic equation is: in s-plane1s a s+ +let
12s w= then 2 1w aw+ + Is characteristic
equation in w-plane. 1arg arg ,mod( ) mod( )2
w s w s= =045+
045−
STABLEUnder damped
UNSTABLE
Im( )w
Re( )w
w-plane
2a<− 0arg( ) 45w <± Unstable0arg( ) 90s <±
0 090 180± −±0 045 90± −±0 090 180± −± 0 0180 360± −±
0 2a> >−0 2a< <
2a> 0arg( ) 180w >± 0arg( ) 360s >±
Stable
HyperdampedUltradamped
A first order system with fractional term may become unstablecan have oscillatory behavior and can behave as stable second orderstable under damped systemsClassical order definition with number of energy storage element and ornumber of initial condition can give misleading information about the responseIn presence of fractional order terms.
Comment regarding system order
On contrary to widely accepted opinion in integer order theory, the first order system cannot go into instability or oscillations, the presence of fractional order elements in the first order system can give a counterintuitive result.
On contrary to widely accepted opinion that chaos cannot occur in continuous-time system of order less than three (in presence of non-linearity as feed back), fractional order system of order less than three can display chaotic behavior, with non linear feed back.
Order definition in classical theory saying the order is number of energy storage elements, or number of initialization constants required or the nature of output of damped nature, is not therefore valid in the presence of fractional order element.
Power series functions used in fractional calculus
Exponential function forms basis in the integer order calculus so is MITTAG LEFFLER function for the fractional calculus
1
0
( )( 1) ( )
n nq q qq
q q qn
a t s sE atnq s s a s a
−∞
=
= ↔ =Γ + − −∑
1
,0
( )( ) ( 1) 1
bna a ba
qa b b a a
n
t s sE tna b s s s
−⎛ ⎞+⎜ ⎟ −∞ ⎝ ⎠
=
= ↔ =Γ + − −∑
, 0 10
( 1)( )( ) ( )
n
a b n mn
t mE tan b na b s
∞∞= +
=
Γ += ↔∑
Γ + Γ +∑( 1) 1
0
1( , )({ 1} ) ( )
n n q
q qn
a tF a tn q s a
+ −∞
=
= ↔Γ + −∑
Mittag-Leffler
Agarwal
Erdelyi
Robotnov-Hartley
Many more like Miller-Ross, Generalized G, Generalized R, Fox function1 ,1
1 , 0
22 , 0
, 1
( ) e x p ( )( , 0 , ) e x p ( )
( , 0 , ) s in
( ) ( , 0 , )qq a q
E t tR a t a t
a R a t a t
E a t R a t−
=
=
− =
− = −
Solution of fractional differential equation (in ML function)Fractional differential equation of broacher (tracking filter)
0 .2 5
0 .2 5
0 .2 5
0 .2 5
( ) ( ) ( )
( 0 ) 0( ) ( ) ( )
1( ) ( )1
d y t y t x td tys Y s Y s X s
Y s X ss
+ =
=
+ =
=+ For step excitation 1( )X s s=
0.25 0.25 0.25 1
0.25 0.25 0.25 0.25
1 1 1 1 1( ) 11 1 ( 1) 1
s s sY ss s s s s s s s s
−⎛ ⎞⎛ ⎞= = − = − = −⎜ ⎟⎜ ⎟+ + + +⎝ ⎠ ⎝ ⎠
0.25 11 1 1
0.25
0.250.25
1( ) ( )1
( ) 1 ( )
sy t Y ss s
y t E t
−− − − ⎛ ⎞⎛ ⎞= ℑ = ℑ −ℑ ⎜ ⎟⎜ ⎟ +⎝ ⎠ ⎝ ⎠
= − −
1( ) 1 ( ) 1 exp( )y t E t t= − − = − −
022.5−
09 0−
5−20−Gain
For first order solution is: Phase
logω
Salient points observed in the discussion:
The distributed effect of parameters distributed over large space gives half order of derivative or integration.
Can this be taken as general rule that semi infinite distributed self similar structures behave with half order of calculus?
If the distribution in space gives order of derivative as fractional order suggesting non-local behavior, can we say event distributed in time (historical behavior hereditary character temporal memory behavior be represented with fractional differ-integration of time?
The solution seems to have self similar pattern, time/space power series with fractional power real power.
Reality of systems are naturally not point quantity thus fractional calculus is the language what nature understands the best.
End of part-I
Essence of FRACTIONAL CALCULUS in applied sciences
Part-II
WORK SHOP ONFRACTIONAL ORDER SYSTEM
28-29 March, 2008
IEEE KOLKATA CHAPTERDRDL HYDERABAD
BRNS(DAE) MUMBAI
Shantanu DasReactor Control Division
BARC
Reimann Liouvelli (RL) fractional integration:Repeated n-fold integration generalization to arbitrary order
1
0
2
0 0 0
3 2
0 0 0 0
1
0 0 0 0
( ) ( )
( ) ( ) ( ) ( )
1( ) ( ) ( ) ( )2
1( ) ........................ ( ) ( ) ( )( 1)!
1( )(
t
t
t t t
t
t t t t
t
t t t tn n
t
n
t
d f t f d
d f t f d d t f d
d f t f d d d t f d
d f t f d t f dn
d f tα
τ τ
τ τ τ τ τ τ
τ τ τ τ τ τ τ
τ τ τ τ τ
α
−
−
−
− −
−
=
= = −
= = −
= = −−
=Γ
∫
∫ ∫ ∫
∫ ∫ ∫ ∫
∫ ∫ ∫ ∫
1
0
( ) ( ))
t
t f dατ τ τ−−∫
Convolution with power function RL fractional integration:
[ ]1 1
0
1
( )( ) ( ) ( ) * ( )* ( )( ) ( )
( )( )
t
tt td f t f d f t f t t
tt
α αα
α
α
α
τ τ τα α
α
− −−
−
⎛ ⎞−= = = Φ⎜ ⎟Γ Γ⎝ ⎠
Φ =Γ
∫
( )f t
( )tαΦ
ℑ
ℑ
1−ℑ ( )td f tα−
Fractional derivative the Euler (1730) formula for monomial
{ }
{ }
{ }0.5
1 0.50.5
( ) .................... ( )
( 1)( 2)............( 1)
( 1) ( 1)( 2)........( 1) ( 1)( 1)
( 1)
(1 1)(1 0.5 1) (1
n
n
nn
m m nn
nm m n
n
d f x d d d f xdx dx dx dx
d x m m m m n xdx
m m m m m n m nd mx xdx m n
d xx xdx
−
−
−
=
= − − − +
Γ + = − − − + Γ − +
Γ +=Γ − +
Γ += =Γ − + Γ
20.5) 0.5 (0.5)
x xπ
= =+ Γ
For positive index the process is differentiation For negative index the process is integration
Reimann Liouvelli (RL) Fractional derivative Left Hand Definition (LHD)
0f (1)f (2)f (3)f (4)f( 1)f −( 2)f −( 3)f −
0.7(1)f d f−=
3 (2.3)(2) (1)f d f f= =
2.3
( )f x ( )mxd α− − m
xd ( )xd f xα
1
0
1( ) ( ) ( )( )
tmm
t m
dd f t t f ddt m
α ατ τ τα
− − +⎡ ⎤= −⎢ ⎥Γ −⎣ ⎦
∫
Here ‘m’ is the integer just greater than fractional order of derivative
Caputo (1967) Fractional derivative Right Hand Definition (RHD)
0f (1)f (2)f (3)f (4)f( 1)f −( 2)f −( 3)f −
0.7(2) (1)f d f−=
3 (3)(1)f d f f= =
2.3
( )f x mxd ( )m
xd α− − ( )xd f xα
1
0
1 ( )( ) ( )( )
t mm
t m
d fd f t t dm d
α α ττ τα τ
− − +⎡ ⎤= −⎢ ⎥Γ −⎣ ⎦
∫
Here ‘m’ is the integer just greater than the fractional order derivative
DualityFor LHD fractional derivative of constant is not zeroThis fact lead to RL or LHD approach to consider “limit of differentiation” (lower terminal) to minus infinity. The physical significance of this minus infinity is starting the physical processes at time immemorial!! However lower limit to minus infinity is necessary abstraction for steady state (sinusoidal) response. For LHD are required. This posses physical interpretability.
For RHD the fractional derivative of the constant is zero. But this requires also with in mathematical world this posses a problem.
Our mathematical tools go far beyond our physical understanding
[ ] 10 (1 )xd C C xα αα − −≠ = Γ −
(0 ) 0 ,f = (1) (2) ( )... 0mf f f= = =
1 2(0), (0)x xd f d fα α− −
Standardization of symbols for fractional differintegrals
Initialized differintegration
Uninitialized differitegrations
Initialization function
For a function born at time (space) and the differintegration starts at time (space)
qc tD±
qc td ±
( ) ( , , , , )t f q a c tψ ψ= ±a=( )f t
c=
( )f t
a c0 t
( , , , , )q qc t c tD d f q a c tψ± ±= + ±
Initialized fractional integration0
( )0 0t t
f tt≥⎧
= ⎨ <⎩
10 t
( )f t
{ } { }3
21 1 0.5 12 20 0
0
1 4( ) 0 ( )(0.5) 3
t
t ttD t d t t dψ τ τ τπ
− − −= + = = − =Γ ∫
{ }1 1
2 21 1 ( )
1( ) { ( ) , , 0,1, }2
t tD t d t
t f t t t
ψ
ψ ψ
− −= +
= = −
{ }1
21 0.5 121
1
1 2( 1) (2 1)( )(0.5) 3
t
tt td t t dτ τ τ
π− − − +
= − =Γ ∫1
3 10.5 1 2 2
0
1 2( ) ( ) 2 ( 1) (2 1)(0.5) 3
t t d t t tψ τ τ τπ
− ⎡ ⎤= − = − − +⎢ ⎥⎣ ⎦Γ ∫
1 t
12 ( )c tD f t−
( )tψ
{ }12
1 td t−
{ }12
1 tD t−
( )tψ Is the history of the functional process since birth and the history effect decays with time, memory is lost!!
( ) ( )q qc t a tD f t D f tt c a
− −=
≥ ≥
{ } { } { }1 1 12 2 2
0 1 1 ( )t t tD t D t d t tψ− − −= = +
0
Initialization function fractional integration
a c t
( )f t
∑
( )qa td f ta t c
−
≤ ≤
( )qc td f tt c
−
≥
( )f t
( )tψ
( )qc tD f t−
Solution of FDE
( )
( )
12
012
0 @ 0
( ) 1 2 (1)
1 1 1 1 112 2 2 2 2
1 1 12 2 2
0 @ 0
12
12
0.5,0.5
( ) ( ) 0
0,[ ( )]
( ) ( ) (0) (0) ...
( ) ( ) (0) ( ) (0)
( ) ( ) [ ( )]
( )
( ) (
t
t t
n n n n
t t
D f t bf t
t D f t C
f t s F s s f s f
f t s F s s f s F s s f
f t s F s D f t
CF ss b
f t Ct E
−=
− −
− −
−
=
−
+ =
> =
⎡ ⎤ℑ = − − −⎣ ⎦⎡ ⎤ℑ = − = −⎢ ⎥⎣ ⎦⎡ ⎤ℑ = −⎢ ⎥⎣ ⎦
=+
= − )
1( ) exp( ) ( ) , 1
b t
f t C t erfc t btπ
⎛ ⎞= − =⎜ ⎟⎝ ⎠
Solution of FDE with initialization function1
20
12
0
12
12
12
111 , 022
1 , 020
( ) ( ) 0
( ) ( ) ( ) 0
( ) ( ) ( ) 0( )( )
( ) ( )
( )
1 ( , 0 , )
( ) ( , 0 , ) ( )
t
t
t
D f t b f t
d f t t b f t
s F s s b F ssF s
s bt C t
CF ss b
R b ts b
f t R b t d
ψ
ψψ
ψ δ
τ ψ τ τ
−
+ =
+ + =
+ + =
= −+
= −
=+
⎡ ⎤ℑ ↔ −⎢ ⎥
+⎣ ⎦
= − − −∫
for 0t>
For
1 1,2 2
( ) ( )Cf t E b tt
= −
General solution
Formal methods to solve fractional differential equation
1. Laplace Transforms
1. Fractional Greens function.
1. Mellin Transforms
1. Power Series Method.
1. Babenko’s Symbolic calculus method.
1. Orthogonal Polynomial decomposition.
1. Adomian Decomposition.
1. Numerical
Synthesis of fractional order immittancesNewton method of root evaluation
1
0
11
1
0
13
1 3
1 5 4 3 23
2 5 4 3 23
( ) , 1
( 1)( ) ( 1)( 1)( ) ( 1)13, , 1,
1 1 22 1
1 1 24 80 92 42 44 42 92 80 24 1
n
nk
k k nk
x a x a
n x n ax xn x n a
n a xs
sxs ss
s s s s sxs s s s s ss
−−
−
= =
− + +=
+ + −
= = =
+⎛ ⎞= = =⎜ ⎟ +⎝ ⎠
+ + + + +⎛ ⎞= = =⎜ ⎟ + + + + +⎝ ⎠
1Ω
1Ω
0.666Ω1
0.75sΩ
0x
1x
0.50.5 0.25 0.15
3 1 3 1 1 3 5 1 1 1.35, , ,3 1 3 5 3 1.35s s s sss s s s s s s s+ + + +⎛ ⎞= = = =⎜ ⎟+ + + +⎝ ⎠
Initialization of fractional derivativeRiemann-Liouvelli derivative
∑ ∑pc td − m
c td( )q
c tD f t( )f t
1( , , , , )f p a c tψ −
( )h t
2( , , , , )h m a c tψ
{ }{ }
2
1 2
( )1 2
( ) ( ) ( , , , , )
( ) ( ) ( , , , , ) ( , , , , )
( ) ( ) ( ) ( )
( ) ( ) ( , , , , )
q mc t c t
q m pc t c t c t
q q mc t c t
q qc t c t
D f t d h t h m a c t
D f t d d f t f p a c t h m a c t
D f t d f t t t
D f t d f t f q a c t
ψ
ψ ψ
ψ ψ
ψ
−
= +
= + − +
= + +
= +
( )( ) ( )q q
c t a t
q m pD f t D f t= −
=
For terminal initializationFor side initialization is arbitrary
2 0ψ =2ψ
Integer order calculus in fractional contextRL derivative
2 21 2 1 1
2 2
1( ) ( ) ( ) ( ) ( )(1)
t t
a ta a
d d dD f t f t t f d f ddt dt dt
τ τ τ τ τ− −= = − =Γ ∫ ∫
Integrate the function from a to t and then obtain second derivative.
Obtaining the differentiation in fractional context imbibes history (hereditary) of the function from start of the differentiation process.
This also describes the ‘non-local’ behavior in space or time.
Forward and backward differentiation integer order derivative in fractional contextRL derivative
{ }
21
2
2
2
/ //
/
( ) ( ) ( )
1 ( ) ( ) ( )2
1( ) ( ) ( )2
( )
t
a ta
d dD f t f t f ddt dt
d f t f a t adt
f t t a f t
f t
τ τ⎡ ⎤
= = ⎢ ⎥⎣ ⎦
⎡ ⎤= + −⎢ ⎥⎣ ⎦
= + −
→
∫
21 2 2 1 1
2
1
1( ) ( 1) ( ) ( )(1)
( )
a
t at
a t
dD f t t f ddt
D f t
τ τ τ− −= − −Γ
= −
∫
Forward RL
Backward RL
{ } 11( ) ( ) ( )( )
tmq m q
a t ma
dD f t t f dm q dt
τ τ τ− −= −Γ − ∫
11( ) ( 1) ( ) ( )( )
amq m m q
t a mt
dD f t t f dm q dt
τ τ τ− −= − −Γ − ∫
a t a
If forward and backward derivatives are equal (with sign) then fractional derivative at a POINT exist, meaning to get fractional derivative at point entire character of function be known!
Grunwald-Letnikov(GL) fractional differintegration
2 2
1
/
0
1 2 1 10 0// 2 2
01
//20
( ) ( )( ) lim
( ) ( ) ( ) ( )lim lim( ) lim
( 2 ) 2 ( ) ( )( ) lim
h
h h
h
h
f x h f xf xh
f x h h f x h f x h f xh hf x
hf x h f x h f xf x
h
→
→ →
→
→
+ −=
+ + − + + −−
=
+ − + −=
0 0
0 0
1 ( 1)( ) lim ( 1) ( )! ( 1)
! ( 1)!( )! ! ( 1)
( )( ) lim ( 1) ( )! ( )
( 1)..( 1) (( 1)!
x ah
ma x h m
x ah
ma x h m
m
D f x f x mhh m m
m m m m m
mD f x h f x mhm
mm m m
αα
α α
αα
α α αα α
αα
α α α α α α
−⎡ ⎤⎢ ⎥⎣ ⎦
→=
−⎡ ⎤⎢ ⎥⎣ ⎦
−
→=
Γ += − −
Γ − +
⎛ ⎞ Γ += ↔⎜ ⎟ − Γ − +⎝ ⎠
Γ += − −
Γ
− −⎡ ⎤ ⎛ ⎞ − − − − − += = = −⎜ ⎟⎢ ⎥
⎣ ⎦ ⎝ ⎠
∑
∑1)! ( )( 1)
!( 1)! ! ( )mm m
m mα
α α+ − Γ +
↔ −− Γ
GL differintegration as digital filter structure:
∑
( )f tD D D
( )f t D− ( 2 )f t D− ( 3 )f t D−
4t aD −⎡ ⎤= ⎢ ⎥⎣ ⎦
0w1w 2w 3w [ ] qD −
( )qa tD f t
[ ]1 1
0 00 0
( ) ( )( ) lim ( ) lim ( )( ) ( 1)
q N Nqq
a t kT Dk k
T k qD f t f t k T D w f t kDq k
− − −−
Δ → →= =
Δ Γ −= − Δ = −
Γ − Γ +∑ ∑Digital filter FIR/IIRTustin Discretization with Generating FunctionMatrix approach FFT for weightsShort Memory Principle
About weights of GL in fractional differintegration
0 0
1( ) lim ( )( )
( )( ) ( 1)
qkqx k
k
D f x w f x k xx
k qwq k
∞
+ Δ →=
= − ΔΔ
Γ −=Γ − Γ +
∑
Is apparent that fractional derivative is limit of a weighted average of the values over the function from minus infinity to point of interest (x), these weights corresponds (in limit) to a power function defined by the order of the fractional derivative (q). This averaging is for forward derivative. For backward derivative, this is limit of a average of values over the function from point of interest (x) to plus infinity. Therefore the forward fractional derivative operator has memory of the function from minus infinity to x, and backward derivative has memory of the function from x to plus infinity.
Thus point fractional derivative at a point x has a unique power law ‘memory’ both forward and backward on function Local fractional Derivative at a point depends on the character of entire function. Integer order derivative depends only on local behavior meaning slope of function at point. Fractional derivative is non-local phenomena
1 12 2
q q qD D D+ −= +
Strength of weights and power law exponents of fractional derivative.
510−
410−
310−
210−
110−
010
1 10 100
kw
Number of cells Past
Slope-0.1Slope-0.5
Slope-0.9
0.1q=0.5q=
0.9q=
Log-log plot demonstrating power law decay in weights placed on the 100 closest cells in calculating q-th derivative. Weights depending on fractional derivative for 0.1, 0.5, 0.9. The larger order derivative place more weights on proximal cells and dependence on distal cells decrease very quickly as distance x increases.The lower order derivatives place relatively less weight on proximal cell and dependence on distal cell decrease very slowly as x increases.
Curve fitting-ASystem identification
⊗⊗
⊗
⊗ ⊗
⊗ ⊗⊗
⊗
( )y t
t
Step input ( )u t
⊗ Set of measured values ,average error margin*( 0, )iy i M=
*
0( )
1
M
i ii
y yQ
M=
−=
+
∑2
2
3
2.571 0.83
2.571 0.83
4
1.0315
1.0315
4
1.8675 ( ) 5.518 ( ) 0.0063 ( ) ( )
3 10
0.7943 ( ) 5.2385 ( ) 1.5960 ( ) ( )
10
6.288 ( ) 1.8508 ( ) ( )
4 10
d dy t y t y t u tdt dt
Qd dy t y t y t u tdt dt
Qd y t y t u tdt
Q
−
−
−
+ + =
×
+ + =
+ =
×
∼
∼
∼
Curve fitting-BLife span estimation, Predictive Maintenance, Reliability analysis…
. During a certain period, after installation of a wire on load, an enhancementof its properties is observed. Say yield point.
. Then properties of wires become worse and worse until it breaks down.
. The period of enhancement is shorter than the period of decrease ofProperty and the general shape of the process curve is not symmetric.
Set of experimental measurements is fitted with fractional differential equation with .
initial values of fitted function and (m-1) derivatives.The fractional integration and its fractional order represents the cumulative impact of the previous history loading on the present state of wire. The order of fractional integration is related to shape of memory function of wire material.
1 2, , ..., ny y y2
0 1 2 0( ) ... ( )m ty t a a t a t a D y tα−= + + + −(0 )mα< ≤
0 1 2 1, , ,... ma a a a −
Experimental fit quadratic and fractional order regression
×××× × × ×
××
×× × ×
×
2
1.320
( ) 0.033 0.562 10.723( ) 0.046 ( ) 1.2760 10.1955t
y t t ty t D y t t−
= + +
= − + +
A
B
AB
It is obvious that the order of fractional integration would be different for different wires because they work in different conditions. Thus it is necessary to apply this regression in each case separately. Main problem is that each particular wire changes its property due to certain very peculiar causes(heredity/history). The order 1.32 is for this particular wire of 2.4mm diameter at this loading, a 2.8mm diameter wire will have different order
time
Yield point
Infinitesimal element fractional integration
( )f t
N-1,N-2………2 1 0 j=t
( )f t j T− Δ
( )f t j Tα − Δ
[ ]{ }[ ]{ }
1
0
0
( ) lim .......... ( ) ( ( 1) )..........
( ) lim .......... ( ) ( ( 1) )..........
( ) ( 1 ) ( 1)0 1, ,( ) ( 1) ( ) ( 2)
a t T
q qa t T
D f t T f t j T f t j T
D f t T f t j T f t j T
j q j q jqq j j q j
α β
α β
−
Δ →
−
Δ →
= +Δ − Δ + − + Δ
= +Δ − Δ + − + Δ
Γ − Γ + − Γ +< < = =
Γ − Γ + Γ − Γ +
Fractional integration can be viewed a area under the curveMultiplied byIn between volume and area
( )f t j Tα − Δ1qT −Δ[ ( ). ].f t j T T Tα − Δ Δ Δ { }( ).f t j T Tα − Δ Δ
Infinitesimal element fractional differentiation
0
1
0
( ) ( )( ) lim ......
( ) ( )( ) lim
qa t qT
a t T
f t qf t TD f tT
f t f t TD f tT
Δ →
Δ →
− − Δ= +
Δ− − Δ
=Δ
Fractional derivative can be viewed as fractional slope, fractional rate of change. Fractional derivative is slope between andi.e. equal to multiplied by
( )f t ( )qf t T− Δ
( ) ( )f t qf t TT
− − ΔΔ 1
1qT −
⎛ ⎞⎜ ⎟Δ⎝ ⎠
TΔ
[ ( ) ( )]f t qf t T− −Δ
( )( )
f tf t T−Δ
( )qf t T− Δ ( )qf t
1q =
1q <
( )f t
A
BSlope between A & Bmultiplied byIs fractional slope of fractional differentiation
1( )qT − +Δ
A practical challenging instrumentation problem(Dr U Paul NPD/BARC)
Total Absorption Gamma Calorimeter International Project
Observation:Energy resolution of the detector with long pencil (1cmX2cmX20cm) crystal depends on interaction point of incident Gamma photon. Crystal defects inhomogenity (along the length of 1D crystal) is responsible for observed behavior. Scintillating light photons propagates through inhomogeneous medium before being collected by read out device PMT
RequirementEnergy resolution independent of interaction point in crystalDevelopment of technique and instrument which can compensate the resolution by fractal technique.
New Science application in fractional calculusApplication of flow of matter/energy through fractal defected porous path.
……………….……………………………….
…………………………………….This is the beginning