FRACTIONAL CALCULUS for

Applied Science & Engineering

MATHEMATICO-PHYSICS OF GENERALIZED CALCULUS

Module-III

Shantanu DasRRPS

Reactor Control DivisionBARC

2009-2010

Curve fitting-A

System 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-B

Life 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 .3 20

( ) 0 .0 3 3 0 .5 6 2 1 0 .7 2 3( ) 0 .0 4 6 ( ) 1 .2 7 6 0 1 0 .1 9 5 5t

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

Generalization of Newtonian mechanics and differential equations

/ / /0( ) ( ) ( ) ( )m x t b x t k x t f t+ + =

( )x t

/

( 0 ) 0( 0 ) 0

xx

=

=

Mass concentrated at pointMass less springFrictionless springInfinite wall

( )f t m

20( ) ( ) ( )m s b s k X s F s+ + = Spring with friction ( ) ( )

0 1

qq s pk s X s F s

q

=

≤ ≤

1 1

1 1 0

20

20

2

0

2

0

( ) ( ) ( )

( ... ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

n n

n n

n

qq

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 dq X s F s

−

−

=

=

+ + + =

+ + + + + + =

=

⎛ ⎞=⎜ ⎟

⎝ ⎠

∑

∫

Distributed massSpring with massSpring with frictionDamping with spring actionNon conservation systemLeaky wall/termination

System 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 2

q

( )k q

0b

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

m s b s b s k s k X s F s

m q b q b q k q k q s d q X s F s

d x t d x t d x t d x tm b b k k f td t d t d td t

δ δ δ δ δ

+ + + + =

− + − + − + − + =

+ + + + =

Integer Order:

Fractional Order

Continuous Order

0

1

0

( ) ( ) ( )

{ ( ) } * ( ) ( )

q

q

k q s d q X s F s

k q s d q x t f t

∞

∞−

⎛ ⎞=⎜ ⎟

⎝ ⎠⎛ ⎞

=⎜ ⎟⎝ ⎠

∫

∫Lq

( )k q

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 d qs a

∞ =

= =+ ∫

2

02

2

0

( ) ( )( )1 ( ) ( )

( )( )

( ( ) )

q

q

H s G sT sH s G s

k q s d qT s

s a k q s d q

=+

=+ +

∫

∫

Demanded order distribution-

( )k q

q

( ) ( )( )( )

( ) q

F s F sX sP s

k q s dq+∞

−∞

= =

∫( ) ( ) qP s k q s dq

+ ∞

− ∞

= ∫

( )k q = Κ 0 2q≤ ≤

( )2 2

ln( )

0 0

2

( ) ( )

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

1..............ln ( )

q

q q s

P s k q s dq

s dq e dq

ss

∞

− ∞

=

= Κ = Κ

⎡ ⎤−= Κ ⎢ ⎥

⎣ ⎦

∫

∫ ∫

( ) 1F s =

2

12

ln ( )( )1

ln ( )( )1

sX ss

sx ts

−

=⎡ ⎤Κ −⎣ ⎦⎧ ⎫⎪ ⎪= ⎨ ⎬

⎡ ⎤Κ −⎪ ⎪⎣ ⎦⎩ ⎭L

Solving continuous order system

Let the continuous order system be represented as:

where

Let the order distribution be uniform from 0 to 2 with K as order strength ;

Then:

For delta excitation:

Fundamental Response of system is:

A good mathematical research topic!

To other type inputs convolution withthis will give solution

( )k q

0 1 2

Κ

Circuit theory

Fractional order sourceFractional order loadFractional order connectivity

sL

LRB BV

1v 2v 3v

Ci

WWi

BRRBi

( ) ( )L B Bd i tL R i t V

d t+ =

( )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

Circuit equation

Heat flux and temperature for semi infinite heat conductor

12

0( ) [ ( ) ]t s u r fkQ t D T t T

kc

α

αρ

= −

=

0x =x = − ∞

( )surfT t

2

2

0

0

0

( , ) ( , )(0, ) , (0, ) 0( , 0) ( ), ( , 0) ( )

( , 0)( )

surf surf

u 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 ∂ ∂=

∂ ∂= −= =

= = −

∂=

∂

Heat flux measurement with single TC

gTbT

( )iQ t

1 ( )Q t

2 ( )Q t

1 1,k α

2 2,k α

1 2( ) ( ) ( ) bi

d TQ t Q t Q t m cd t

− − =

General equation heat flow relating conducted heat flux through semi infinite heat conductor is , with

( )iQ t1

2( ) ii t b

i

kQ t D Tα

= 2 /i ik cα ρ=

Convective heat input is:( )( ) ( ) ( )i g bQ t h A T t T t= −

1 2

1 2

( ) 1( ) 1 1

b

g

T sT s k km c s s

h A h A α α

=⎛ ⎞⎛ ⎞ + + +⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Impedance RC distributed semi infinite transmission line

12

12

1( ) . ( )

1

1( )

a ti t D v tR

R CRZ sC s

α

α

=

=

⎛ ⎞= ⎜ ⎟

⎝ ⎠

Basic building block for fractional order immittance realization of arbitrary order to make fractional order analogue function generator and fractional order analogue PID controller.

2

2

1

( , ) ( , )

( , ) ( , )

( 0 , ) ( ) , ( , ) 0

v x t i x t Rx

i x t v x tCx t

v i vR R Cx x t

v t v t v t

∂= −

∂∂ ∂

= −∂ ∂

∂ ∂ ∂= =

∂ ∂ ∂= ∞ =

Fractional Divergence

To define non-local flux of material flowing through anisotropic media, lossy volume and heterogeneous ambient.

Non Fickian diffusion phenomena

Anomalous diffusion

( )R E V

2

1d i v l i m . ( )

( ) ( ) 0

11 2

VS

J J J n d S xV

d x B xd x

α α

β

β

β αβ

→

Δ →= ∇ ≡ = Φ

Φ+ Φ =

= +≤ ≤

∫

General relaxation phenomena

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

( 1 ) 20

2

0 2

( , ) ( , )

( , ) ( , )

1( , ) ; 0 1

t F x

t N F

C x t C x t

D C x t C x tx

C x tt

α

α α

∂ = ∂

∂=

∂

≈ < ≤

D

D

( , )C x t

t

Reaction to impulse excitationNon exponential reaction

0( , ) (0, 0) exp( / )1

1( , )

F

C x t C t

D

C x ttα

τ

τ

≠ −

≈

≈

Fractional Curl

In 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)Mapping/modelling in between pure Thevenin and pure Norton circuits.

Electrodynamics

Wave 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 modelling in AC machines, a new field of R&D.

s i n s i n2

D E t E tα α π αω ω ω⎛ ⎞= +⎜ ⎟⎝ ⎠ 2

π α

Electrodynamics

Multi-pole expansion

X

Y

Z

θa rR P

2 2

22

2 3

0

( , )2 c o s

( c o s ) ( 3 c o s 1) . . .2

( c o s )k

kk

q qrR r a a r

q q a q ar r rq 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 ) ( c o s )4 ( )

q a Dr

q a Pr

αα

α

αα

π ε

α θπ ε

− ∞

+

⎛ ⎞ ⎛ ⎞Φ = ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

Γ +Φ =

aa

aFractional Legendre polynomial, Fractionalpoles, dipole, monopoleself similarity-fractal distribution

0

1

2

0 21 22 2

α

α

α

= →

= →

= →

MonoDipoleQuadra

Fractal Geometry & Fractional Calculus0

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

lo g 3 lo g 3 / lo g 2 1 .5 8 51lo g 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.

Relation 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 ( )

F

G ss

G ss

d

λ

τ

τλ

=+

=+

↔

Relation of order to the fractal dimension

Fractional 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

( ) co s( )

30 1, 0 , 12

lo glo g 2

n n

n

f x a b n

a b a b

adb

π

π

∞

=

=

< < > > +

=+

∑

Fractality implies d >1 and it is scale independent, has no smaller scale

Weistrauss

Viscoelasticity

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

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

Viscoelastic Model

Our Experiment – forcing a fluid to spread under a load

Fluid

M

VIDEOCAMERA

Case of Newtonian Fluid

Non-Newtonian caseArea-Time plot

0 5 10 15 20 25 301

0.5

0

0.5

1

1.5

2

ml i

ml1 i

ml2 i

tt i

1.5q =

0.8q =

1.0q =

Viscoelasticity with variable fractional order value

1

1( )( / )

1/

( ) 1

c

q

q

q

q

ss s E

sE s s E

E tt EE

σ σ

σεβ β

σβ

σεβ

−

<

⎡ ⎤= ⎢ ⎥+⎣ ⎦

⎡ ⎤= −⎢ ⎥+⎣ ⎦

⎡ ⎤⎛ ⎞= − −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

( )/( ) 1 Ett eE

βσε −= −

Biology

Muscles and joint tissues in musco-skeletal system seem to behave as visco-elastic material, as 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

Circuit Synthesis

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 n

nk

k k nk

a x x a x

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 .6 6 6 Ω1

0 .7 5 sΩ

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+ + + +⎛ ⎞= = = =⎜ ⎟+ + + +⎝ ⎠

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 & more robustness in control. This scheme also takes lesser controller effort as compared to tuned integer order control system.

+−

Im( )s

1.6k = 022.5±

( )e sℜ

Isodampded line at angle

Gain K=1.00Gain K=2.00

Isodamping lines in complex plane, with gain variation

ObservationsAs the gain is varied the peak overshoot is more or less constant.Obtained lesser controller effort.These are preliminary results on a DC Motor (C) TF tuned with PID (B) and forwarded by a fractional phase shaper (A).OLTF of good control system shows a fractional order integral form of order between 1-2.

Overshoot Independent of Gain Spread-Isodamped systems

ISODAMPING Root-Locus

A B C

0

0

1 .6 , 3 6 , 0 .3 8 , 4 0 .5 % , 1 .7 0 1 3

1, 9 0 , 1, 0 % , 1 .0 0 0p r

p r

k P M M M

k P M M M

ς

ς

= = = = =

= = = = =

The scalar gain, as shown in Figure can be varied by 500% keeping the overshoot constant. The advantage of the phase shaper becomes evident considering the fact that the PID controller alone cannot handle such large variation in gain. The closed loop system, with the PID controller alone becomes unstable with two fold increase in gain.

Iso-damped response with wide parametric spreads

Controller output signal with and without the phase shaper, represented by solid line and dashed line respectively.

Controller effort is lesser with Fractional order PID

Bode plot of an FPP with slope of -20mdB/dec. and its approximation as zigzag straight lines with individual slopes of -20dB/dec. and 0dB/dec.

Fractal Pole-Zero & Constant Phase Element

Realization of Fractional Order Integrator Differentiator and Transfer Functions by Fractal Singularity Structure-Bode’s dream

contd…

Choosing the singularities for approximation by assuminga constant error between the -20 dB/decade line and the zigzag lines.

contd…

Recursive Algorithm Finite range of frequency, can be truncatedto a finite number N, and the approximation becomes, with interlaced poles and zeros:

1

0

0

11( )

11

N

i im N

i iT

sz

H ssspp

−

=

=

⎛ ⎞+⎜ ⎟

⎝ ⎠= ≈⎛ ⎞⎛ ⎞ ++ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

∏

∏

[ / 2 0 ]1 0 y mo Tp p=

[ / 1 0 ( 1 ) ]0 1 0 y m

oz p −=

[ / 1 0 ]1 1 0 y m

op z=

[ / 1 0 ( 1 ) ]1 1 1 0 y mz p −=

[ /10 (1 )]1 110 y m

N Nz p −− −=

the first pole,

the first zero,

the second pole,

the second zero, …………

the Nth zero,

the (N+1)th pole, [ / 1 0 ]

11 0 y mN Np z −=

i Zi Pi CiRfi= Rii Rzi

Ω TP Ω TP

1 2.2537 6.0406 1µ 264.07k 500k 443.71k 500k

2 15.955 42.764 1µ 37.30k 50k 62.67k 100k

3 112.95 302.75 680nf 11.21k 20k 18.83k 20k

4 799.65 2143.3 68nF 10.94k 20k 18.39k 20k

5 5661.1 15173 10nF 10.51k 20k 17.64 20k

6 40078 107420 1nF 14.85k 20k 24.95k 50k

Rfi respectively as shown in Figure-7.Table 1- Calculated values of R-C components

0 1 2 3 4 5

x 10-3

-0.4

-0.2

0

0.2

0.4

Time(Sec)

Amplitu

de

-1 0 1 2 3 4 5 6 70

5

10

15

20

25

30

35

40

45

50

Frequency

Pha

se(d

B)

Fractal real poles and real zeros interlaced to give half order differentiator:

1 1 22

1 2

( ) ( ) . .( ) ( ) . .

s z s zss p s p

− −≅

− −

Fractional Differentiability at Critical Point:

0.5

1 0.5

( ) 2 3 4

3 (2) 4 (1.5)( )(1 1) (0.5 1)

0 0.5(0) 8 (1.5) 0.5

0.5

q qq

q

f x x x

x xf xq q

qf q

q

− −

= + ±

⎡ ⎤Γ Γ= ±⎢ ⎥Γ − + Γ − +⎣ ⎦

<⎧ ⎫⎪ ⎪= Γ =⎨ ⎬⎪ ⎪±∞ >⎩ ⎭

D

D

( )q f xD 0q =

0.25q =

0.5q =0.75q =

The function is continuous at critical pointfrom zero order till less than 0.5, with valuezero.Beyond 0.5 order the function diverges.

Say the function is specific heat of solid.At phase transition point at the derivativeorder 0.5 the value of derivative may beRegarded as ‘Fractional Latent Heat’

Fractal Dimension indicating on-set of Epidemic

140

Mortality/day

20

1.6

10 20 30 40 days

The cause of epidemics exhibited significant change in fractal dimension. Initially behavedas Brownian Motion 1.4-1.6, then dropped to 1.3-1.1 indicating on set of burst between0-16 day (became regular from irregular) and again raised to 1.4 behaves variably.

A normal sensex of stock market

1.6

1.1

SENSEX

The dimension shows normal irregular ‘Brownian’ trading, with dimension slewing towards 1.5-1.6 indicating no bull bear or crash or financial irregularity!Trading is regular with normal irregularity as expected like White-Noise.

Exactly where the signal starts in high noise background

1.6

1.1

Signal buried in 85% white noise, the change in dimension indicates the first arrival time of signal.

50mV

0 Time (s)

-50mV

Identification of singularity by LFD

( )f x a x α= 0 1α< <

At ‘zero’ critical order gives the ‘order of singularity and Local Fractional Derivative gives strength of singularity.

(0 ) ( 1)f aα α= Γ +D

( )f x a x b xα β= + 0 1α β< < <

( 1)(0) ( 1) ( 1)( 1)

f a b x aα β αβα αβ α

−Γ += Γ + + = Γ +

Γ − +D

Write:(0 )( , ) ( ) (0 )

( 1)fG x f x f x b x

αα βα

α⎡ ⎤

= − + =⎢ ⎥Γ +⎣ ⎦

D

( , ) ( 1)( 1)

qq

q

d G x b xdx q

βα ββ

−Γ +=

Γ − +

q β= is critical orderThis way one extracts secondary singularity hidden by primary singularity

1. Single singularity:

2. Multiple singularity:

Fractional Taylor’s series by LFD

Let( )[ ]

00 0

0

( ) ( )( , ; )

( )

q

q

d f x f xF x x x q

d x x

−− =

−it is clear that

0 0( ) ( , 0 , )q f x F x q=D

Using RL Integration, and by integration by parts we get0

00 1

00

( , ; )1( ) ( )( ) ( )

x x

q

F x qf x f x dq x x

ξ ξξ

−

− +− =Γ − −∫

001 0 0

0 0 00

( , ; ) ( )1 1( , ; ) ( )( ) ( )

x x qx xq dF x q x xF x q x x d

q q d qξ ξξ ξ ξ

ξ

−−

− − −⎡ ⎤= − − +⎣ ⎦Γ Γ∫ ∫

Provided the II term exists!

00 0

0 0 00

( ) ( , ; )1( ) ( ) ( ) ( )( 1) ( 1)

x xqq qf x d F x qf x f x x x x x d

q q dξ ξ ξ

ξ

−

− = − + − −Γ + Γ + ∫D

00 0 0

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

qq

qf xf x f x x x R x xq

= + − +Γ +D

More general: ( )0 0

0

0

( ) ( )( ) ( , )( 1 ) ( 1 )0

n qNn q

qn

f x f xf x R xn q

x x=

= Δ + Δ + ΔΓ + Γ +

Δ = − >

∑ D

for 0 1q< <

Usage:

1. LFD provides the coefficient in approximating function

By in vicinity of .The terms are non trivial

For , the critical order

Α ( )f x

0 0( ) ( )( 1)

qf x x xqΑ

+ −Γ + 0x

q α=

For we get equation for tangentThis forms an equivalence class modeled by linear behavior. All curvespassing through a point having same tangent.

2. Analogously all the functions (curves) with same ‘critical order’ and same will form an equivalence class modeled by power law This generalizes definition of tangents.

3 Useful to approximate irregular (non-differentiable) functions by piece-wise smooth (scaling) function; and survey of singularities.

4. Useful as Fractional curve fitting, start point of ‘Fractional Differential Geometry’.5. A useful world of mathematics for “CALCULUS ON FRACTALS”

1q = 10 0 0( ) ( ) ( )[( ) ]qf x f x x x x= + −D

0x

αqD x α

Line/surface/volume integrals of Fractal Distribution:

Fractal Distribution represented by Fractional Continuous Medium and then weperform the integration. The fractional Integrals are considered as an approximate integrals on fractals. This type of new approach is applicable in processes where fractal features of the process or the medium impose thenecessity of using non traditional tools in regular smooth physical equations.Smoothening the microscopic characteristics over physically infinitesimal Volume/surface/line transforms the initial fractal distribution into fractional continuum model. The order of fractional integration is of fractal dimension.

10

0

1( ) ( ) ( )( )

x

xD f x x u f u duα α

α− −= −

Γ ∫3

0 33( ) ( )( / 3)

dd

dV V

rD f r f r dV dVd

−− = ≈

Γ∫ ∫ 3 3( , )dd V K r d d V=3

3 3( , )( / 3)

drK r dd

−

=Γ

2 3d< <

2 2( , )dd S K r d d S= 1 2d< <2

2 2( , )( / 2 )

drK r dd

−

=Γ

1 1( , )dd L K r d d L= 0 1d< <1

1 ( , )( )

drK r dd

−

=Γ

Some laws on Fractal Geometries

Flux through a fractal surface:A flowing quantity trough a fractal surface be represented as:

( )( , )dS d

S

J r t d Sφ = •∫ 2 2( , )dd S K r d d S≡2

22 ( / 2)

d

drdS dS

d

−

=Γ

Gauss’s law on Fractal:

( ) 13 3 2 2( , ) ( ( , ) [ ( , ) ( , )]d d

W W

J r t d S K r d K r d J r t d V−

∂

• =∫ ∫ d iv

( )2 3( , ) [ ( , ) ]W W

J r t d S J r t d V∂

• =∫ ∫ d i v

Stroke’s law on Fractal:

( ) ( ) 12 2 1 1( , [ ( , ) ]d d

L S

E d L K r d K r d E d S−• =∫ ∫ c u r l

( )1 2[ ]L S

E d L E d S• =∫ ∫ c u r l

1 2d< <

3 3( , )dd V K r d d V=2 2( , )dd S K r d d S=

2 2( , )dd S K r d d S=1 1( , )dd L K r d d L=

Existence of Magnetic charges?

In normal cases of smooth geometry indicating no magnetic charges at point exists . Magnetic mono-pole not possible.

Fractional generalization however gives:

0B =d i v

2 2[ ( , ) ] 0K r d B ≠d i v

2 2. ( , )B B K r d=d i v g r a d

For ; indicatingExistence of ‘magnetic monopole charges’ with magnitude of

2 2d ≠ 2 2( , ) 0K r d ≠g r a d 0B ≠d i v

2 2. ( , )me B K r d≈ ∇

For fractal distribution we have thus all sets of conservation laws and set of Maxwell equations and electrodynamics do get modified.

This method perhaps is suitable for dusty plasma cases.

Self similar repeated prolong structure terminal relationand semi-differentiation

( )i t

( )e t C

1

10

11

1

1 1( ) ( ) ( )

( 0 ) 0 ( 0 )

; ;

t

vv

v

de t i d i tC C d t

i t e td d ds s sd t d t d t

ξ ξ−

−

−−

−

= =

≤ = = ≤

↔ ↔ ↔

∫

( )i t

( )e t

0 ( )i t

0 ( )e t1R

0C0R

00

0

00 0

0

1

( )( ) ................................(1)

( )( ) ( ) ................(2)

( ) ( )( ) ........................(3)

e ti tR

de ti t i t Cdt

e t e ti tR

=

− =

−=

Eliminating from (1), (2) & (3)0 0( ) & ( )e t i t

0 1 0 1 0 0 0( ) ( )[ ] ( ) ( )d i t d e tR R i t R R C e t R C

d t d t+ + = +

Continued Fraction Expansion form

0 1 0 1 0 0 0

0 1 0 1 0 0 1 0 0 0 0 0

0 1 0 1 0

0 0

1 0

1

0 0

(0 ) 0 (0 )( ) ( )[ ] ( ) ( )

( )[ ] (0 ) ( )[1 ] (0 )( )( ) 1

1( ) 1 1( )

i ed i t d e tR R i t R R C e t R C

d t d tI s R R R R C s R R C i E s R C s R C e

R R R R C sE sI s R C s

R CE sR I s s

R C

= =

+ + = +

+ + − = + −+ +

=+

= ++

Expand the circuit further( )i t

1 ( )i t0 ( )i t

0 ( )e t1 ( )e t

( )e t2R

1C

1R

0C0R

1

2

11 1

2 1

12

1 1

( ) ( )( )

( )( ) ( )

1( ) 1 ( )( )

( )

e t e ti tR

d e ti t i t Cd t

R CE sI sR I s s

C E s

−=

− =

= ++

1 01

1 1

0 01 1 1

0 0 0 0 1 1 01 1

2 1 1 3 2 1

1( ) 1 1( )

( ) ( ) , ( )

( ) , ( )

R CE sR I s s

R C

R C R C

R C R C

ω τ ω

ω ω

− − −

− −

= ++

≡ = ≡

≡ ≡

3

22

1

0

( ) 1( )

1

E sR I s s

s

ωω

ωω

= ++

++

Generalizing and expanding to infinity

<<

<<

<<

<<

0R0C

iR

iCnR( )e t

( )i t

2 1 2 1 2 3 2 3 02 1

2 2

2 3

1

01 1

2 2 1 1

( ) 1 1 ...( ) 1 1 1

1...

( ) ; ( )

n n n n

nn

n

j j j j j j

E sR I s s s ss

ss

R C R C

ω ω ω ω ωω ωω

ωω

ω

ω ω

− − − −

−

−

− −+ +

= + = ++ + + + ++

++

+

= =

2 1 2 2 2 3 02 1

2 1 2 2 2 3 02 1

0 1 2 1

0 1 2 1

2 1

( ) 1 . . . . . . .( ) 1 1 1

( ) 1 . . . . . . . . . . . . . .( ) 1 1 1 1 1 1

. . . . . . . . . . . . . . . . .1. . . . . . . . . . . . . . . . . . ;2

2

n n n

n

jj

n n n

n

n

n n

n

E sR I s s s s

vs

v v v vv vE sR I sC C C C C

R R R R R R R

vR C s

ω ω ω ωω ω

ω

− − −

− − −

−

−

−

= ++ + + + +

=

++ + + + +

= = = = =

= = = = = =

= 2

2 ( ) 1 2 . . . . . . . . . . . . .( ) 1 1 1 1 1

v

E s v v v v vR I s

=

= ++ + + +

Simplifying CFE

CFE in limit of very large number of stages:

By induction

2 1

2 ( ) 1 2 .. . . . . . . . . .( ) 1 1 1 1

4 1 4 1 1.. . . . . . .1 1 1 1 2 24 1 11

4 1 1

n

E s v v v vR I s

v v v v v v

vv

+

= ++ + +

+ += − −

+ + + ⎡ ⎤+ −+ ⎢ ⎥

+ +⎣ ⎦

…………………………….(1)

……….(2)

From (1) and (2) and dividing by we obtain:2 v

2 1 2 1

2 1 2 1

( ) 4 1 [ 4 1 1 ] [ 4 1 1 ]( ) 4 [ 4 1 1 ] [ 4 1 1 ]

n n

n n

E s C s v v vI s R v v v

+ +

+ +

⎡ ⎤+ + + − + −= ⎢ ⎥

+ + + + −⎢ ⎥⎣ ⎦………(3)Graphically one estimate RHS of (3) to unity as for large ‘n’ and seemingly widespread of “v”; implying, RHS is within 2% of unity for wide frequency/time range

12

12

( ) 1( )

( ) ( )

( ) ( )

E s C sI s R

RE s I sC s

R de t i tC d t

−

−

≈

≈

≈

2

2

2

1661 16

6

166

v n

R C n R Cs

R C t n R C

≤ ≤

≤ ≤

≤ ≤

Dynamics of delay in computer based systems demonstrate the stochastic behavior. The delay of random nature has wide spikes and if a statistics be taken, it is like a power law, with pronounced tail. Effect of network delay in control system is very widely researched topic and has practical relevance to modern computer control industry. The classical method of fluctuation dynamics is by Gaussian assumption of the random behavior, and dynamics of the same applied to fluctuations in financial assets gives integer order differential equation formulations giving Gaussian solutions. We can develop a new extension of fractality concept for dynamics of random delay. We can propose a possible fractional calculus approach to model the evolution of stochastic dynamics of random delay. We consider the fractional form of Langevin type stochastic differential equation, and replace standard ‘white noise’ Gaussian stochastic driving excitation force, by ‘shot-noise’ whose each pulse has randomized amplitude. The proposed fractional dynamic stochastic approach allows obtaining the probability distribution function (pdf) of the modeled random delay. It can be proposed to describe the dynamics of random delay of computer control system along with fractional stationary condition as below:

( ) ( ) ( )q

q

d t t F tdt

τ λτ= + 0 1q< ≤

1

010

( )q

qt

d td t

τ τ−

−=

=( )

0 00

( ; , ) ( )t

t t tt F e dt F t eλ λτ τ τ ′−′ ′= + ∫

Computer delay a case of Fractional Brownian motion!!

For q=1 the process is standard Langevian with solution asPoison’s process as:

1 ,1 ( ) zE z e=

( ) ( )1 10 , 0 ,

0

( ; , ) ( ) ( ) ( )t

q q q qq q q qt F t E t d t F t t t E t tτ τ λ τ λ− −′ ′ ′ ′= + − −∫

Mandelbrot who introduced the term ‘fractal’ observed that in addition to being non-Gaussian, the stochastic process of financial returns show interesting property of ‘self-similarity’. That is the statistical dependencies of ‘random phenomena like financial returns. Brownian motion, have similar functional form for various time increments.

The classical method of fluctuation dynamics is by Gaussian assumption of the random behavior, and dynamics of the same applied to fluctuations in financial assets is widely used in mathematical finance because of simplification it provides in analytical calculations

Similarly dynamics of stock market may too be treated as Brownian Motion & its generalization as Fractional Brownian motion, leading to ‘long ranged correlated’ power law!!

Stock Market & Pricing etc.

And Several More………………………….

Prologue

EXPRESSED DIFFERENTLY WE MAY SAY THATNATURE WORKS WITH FRACTIONAL DERIVATIVES

WE MAY EXPRESS OUR CONCEPTS IN NEWTONIAN TERMS IF WE FIND CONVENIENT, BUT IF WE DO SO, WE MUST REALIZE THAT WE

HAVE MADE A TRANSLATION INTO A LANGUAGE WHICH IS FOREIGN TO THE SYSTEM WE ARE

STUDYING

FRACTIONAL CALCULUS IS THE CALCULUS OF XXI CENTURY

At the end one has to solve

Fractional Differential Equations