Chapter 5
Fractional Differintegrations: Insight Concepts
5.1 Introduction
This chapter describes the geometric and physical interpretation of fractional integration and fractional differentiation. As a start point, the ReimannLiouville (RL)
fractional in

4.6 Fractional Divergence in Neutron Diffusion Equations
71
elucidated by the fact that at a particular space the neutrons will have spatial longtailed distributions. The effect of this long-tailed statistical probability distribution
will thus get enhanc

4.6 Fractional Divergence in Neutron Diffusion Equations
75
The above is Laplace transformation for LHD definition of the fractional derivative.
In this expression, the second and the third term of the left hand side has fractional derivative of the flux

50
3 Observation of Fractional Calculus in Physical System Description
Here v(t) and i (t) are the voltage and current respectively, at the terminal element, r is the resistance per unit length and is the product of r and c (the capacitance per unit lengt

3.5 Driving Point Impedance of Semi-infinite Lossy Transmission Line
47
and solving for voltage in terms of source current gives
1
R I (0, s)
+ !s
V (0, s) = ! s
"
e
s
v(, 0)d
0
In evaluating the integral on the right, it is now recognized that this ter

70
4 Concept of Fractional Divergence and Fractional Curl
the nature of the curve shown in Fig. 4.2. For a very small observation space area,
the surface flux is the product of neutron current and that area. As the area is made
larger, the neighbouring ne

46
3 Observation of Fractional Calculus in Physical System Description
We are thus left with indeterminate form, and this can be solved by LHopitals rule
as follows (after rearrangement):
lim
2
1
s
x
!x
e+
s
0
e+x
v(, 0)d
s
The LHopital rule says that thi

66
4 Concept of Fractional Divergence and Fractional Curl
Complex systems and their study play a dominant role in exact and life sciences,
embracing a richness of systems such as glasses, liquid crystals, polymers, proteins,
biopolymers, or even ecosystem

3.5 Driving Point Impedance of Semi-infinite Lossy Transmission Line
49
or
i (t) =
2 (t) =
1 d 1/2 v(t)
+ 2 (t)
R dt 1/2
1 d1 (t)
R dt
3.5.1 Practical Application of the Semi-infinite Line in Circuits
3.5.1.1 Semi-integrator Circuit
The circuit shown in t

42
3 Observation of Fractional Calculus in Physical System Description
The exact solution is obtained above, by considering various Biots number
(0.110). However, the first way to approximate this heat transfer phenomena
is by having spatial average of th

64
4 Concept of Fractional Divergence and Fractional Curl
where J is flux vector, V is an arbitrary volume enclosed by surface S, and n is unit
vector normal to the surface.
This is valid only if the flux is indeed a point vector quantity relative to the

44
3 Observation of Fractional Calculus in Physical System Description
In this formulation v is the voltage, i is the current, and v I (t) is a time-dependent
input variable. At x = , the condition is of short circuit. A classical solution using
iterated

48
3 Observation of Fractional Calculus in Physical System Description
Note that in the impedance expression of Z (s), there are two parts, the forced
response due to I (0, s) and the initial condition response due to the initial voltage
distribution in t

88
5 Fractional Differintegrations: Insight Concepts
resistoductance is combination of pure resistance and pure inductor. As integer
order equations, the fractional order requires fractional derivatives (or integrals) as
initial conditions. How does then

3.8 Concluding Comments
61
number of energy/memory storing elements, nor number of initializing constants
nor number of integrations (even fractional) required to solve the system. Thus the
issue of order and the information required together with the fra

3.4 Heat Transfer
41
The initial and boundary conditions are
T (x, t)
= 0 at x = 0
x
k
T (x, t)
+ h (T cfw_x, t T ) = 0, at x = L
x
T (x, t) = Ti , at t = 0
where k is the thermal conductivity of the wall material. With change of variable to
make dimens

54
3 Observation of Fractional Calculus in Physical System Description
This feature is remarkable in the field of control science, indicating that a
system need not be of fractional order to have a fractional order controller. An
integer order system gets

3.5 Driving Point Impedance of Semi-infinite Lossy Transmission Line
51
Fig. 3.6 Semi-differentiator
if
R
_
+
ii
vi
vo
The circuit in Fig. 3.7 is to realize the fractional order PID analog control system.
In this circuit, the offset adjustment parts are n

3.5 Driving Point Impedance of Semi-infinite Lossy Transmission Line
53
J is the payload (inertia). Phase margin of the controlled system is
m = arg |G( j )H ( j )| + ,
the controller characteristics is
H (s) = K 1
K2s + 1
.
s
Here, choose K 2 = T . Note

86
5 Fractional Differintegrations: Insight Concepts
2
a
h ()
1
0
-10
-5
0
5
10
1
b
f ( )
0
-1
-10
-5
0
5
10
2
c
h ( )
1
0
-10
-5
0
5
10
2
d
h (5 )
1
0
-10
-5
0
5
10
2
Y = f (t)
e
0
f () . h (5 )
-2
-10
-5
0
5
10
1
X
f
d 1/2 f (t)
0
-1
-10
-5
0
5
10
TIME

3.5 Driving Point Impedance of Semi-infinite Lossy Transmission Line
45
standard transform pairs.
V (x, s) =
Equivalently,
V (x, s) =
!x
0
$
s # 1
1
" s e+(x) v(, 0) d
0 2
x
$
!
s # 1
1
" s e(x) v(, 0) d
2
0
% & '
% & '
V (0, s)
s
s
+ " s sinh x
+V (0,

52
3 Observation of Fractional Calculus in Physical System Description
Table 3.1 Practical results from semi-integrator circuit measurement
V0
Vi
Input
Input
Phase
frequency frequency angle
(Hz) f
(radian) (degree)
Vi (Volt)
V0 (Volt)
G=
50
100
150
200
25

56
3 Observation of Fractional Calculus in Physical System Description
l
l
i(t)
v(t)
l
c
c
l
c
Fig. 3.10 Semi-infinite lossless transmission line
In Fig. 3.6, the input element is a lumped resistor R and the feedback element is a
lumped capacitor C. Then

58
3 Observation of Fractional Calculus in Physical System Description
which is charge on the capacitor C f , but also carried in the remainder of the (t)
function, which accounts for the distributed charge along the semi-infinite line. It is
also observe

3.6 Semi-infinite Lossless Transmission Line
59
For the distributed feedback zero-order elements, the expression in the circuit is
0 vo (t) =
!
L
i f (t) + (t) =
C
!
L
i f (t) +
C
!
L
(t)
C
Putting i i (t) = i f (t), yields the final result as
!
! "
#
#
"

4.3 Fractional Kinetic Equation
65
(a)
(b)
J.dS
1
J .dS
V S
S
0
Volume (V)
0
Volume (V)
Fig. 4.2 Effect of dipersive flux and neutron velocity fluctuation with macroscopic scale of observation extension
there is a long-standing and growing body of eviden

3.6 Semi-infinite Lossless Transmission Line
55
the driving point impedance is obtained as follows:
!
1
L
L [I (0, 0)]
I (0, s)
+
V (0, s) =
[V ( p, 0)] p=s LC
C
C
s
LC
#
1 "
+
V ( p, 0) p=s LC
LCs
!
Notice that the voltage is composed of two parts: the

4.5 Classical Constitutive Neutron Diffusion Equation
69
energy/temperature) that arise only as the observation space grows larger, invalidating the limit. Also the neutrons are no longer in homogeneous medium. The
dispersive fluxes for a given volume are

60
3 Observation of Fractional Calculus in Physical System Description
This zero-order operation in general returns the input function vi (t) (with amplification or attenuation), also provides the extra time function (associated with the
memorized charges

72
4 Concept of Fractional Divergence and Fractional Curl
d 1+
a + S = 0, . . . 0 < < 1
d x 1+
d
D a + S = 0, . . . 1 < < 1
dx
D
One may interpret the simplified form of .J is that a fractional divergence operator is applied to Fickian dispersion term.