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3.6 Semiinfinite Lossless Transmission Line
59
For the distributed feedback zeroorder 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
!
! "
#
#
"
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68
4 Concept of Fractional Divergence and Fractional Curl
J = nv
= nv
J = D
Consider a closed volume, the loss of neutrons from the closed surface is given
as surface integral of neutron current, J.d S. The loss occurring in the volume by
absorption is g
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Electrical Network Analysis
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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
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Electrical Network Analysis
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3.5 Driving Point Impedance of Semiinfinite 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 Semiinfinite Line in Circuits
3.5.1.1 Semiintegrator Circuit
The circuit shown in t
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Electrical Network Analysis
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5.7 Matrix Approach to Discretize Fractional Differintegration and Weights
97
5.7 Matrix Approach to Discretize Fractional Differintegration
and Weights
The weights or the coefficients for approximation of fractional differintegration as
described in the
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2.4 Concluding Comments
( s 2 + a 2 s)v
(v > 1)
s 2 + a2
1
k>0
2
( s + a 2 )k
( s 2 + a 2 s)k , (k > 0)
( s 2 a 2 + s)v
, (v > 1)
s 2 a2
1
, (k > 0)
(s 2 a 2 )k
1
s s+1
1
s + s 2 + a2
1
(s + s 2 + a 2 ) N
1
2
2
s + a (s + s 2 + a 2 )
1
s 2 + a 2 (s + s 2
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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
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Chapter 3
Observation of Fractional Calculus in Physical
System Description
3.1 Introduction
Fractional calculus allows a more compact representation and problem solution
for some spatially distributed systems. Spatially distributed system representation
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Electrical Network Analysis
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3.2 TemperatureHeat Flux Relationship for Heat Flowing in Semiinfinite Conductor
37
Taking Laplace transforms for the above equation gives
2U (s, x)
x2
2
U (s, x) cs
U (s, x) = 0
x2
k
c.sU (s, x) = k
The bounded solution for x tends to is
! "
#
sc
U (s
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5.6 Short Memory Principle: A Moving Start Point Approximation and Its Error
f h(3) (t) =
95
n
h !
(r + 1)(r + 2)h 2 f (t r h) f h(3) (t)
1.2 r=0
n+1
=
h !
r (r + 1)h 2 f (y r h),
1.2 r=1
here also t + h = y is substituted. Expressing the above by rearran
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78
4 Concept of Fractional Divergence and Fractional Curl
H (z) = j E (z)
corresponding
! magnetic field may be obtained. In the above expression
= = . This means each wave field sees media with equivalent constitutive parameters as (+ , + ) and ( , ).
NED University of Engineering & Technology, Karachi
Electrical Network Analysis
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3.6 Semiinfinite Lossless Transmission Line
57
These equations yield the final result by putting i i (t) = i f (t)
1
v o (t) =
RC
!t
c
vi (t)dt + vo (c) =
1
1
c D vi (t) ,
RC t
with (t) = RCvo (t). This is classical integer order calculus, with initial
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Electrical Network Analysis
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3.5 Driving Point Impedance of Semiinfinite 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,
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3.5 Driving Point Impedance of Semiinfinite Lossy Transmission Line
51
Fig. 3.6 Semidifferentiator
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
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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
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5.3 ReimannLiouville Fractional Differintegral
89
In
case
under consideration, voltage stress is finite at all times hence
"
! the
1
D
V
0 t
0 t0 = 0, which leads to the condition of zero initial condition involv!
"
ing fractional differintegral, namely,
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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.
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3.6 Semiinfinite 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
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122
6 Initialized Differintegrals and Generalized Calculus
6.7.2 Side Charging
The case for side charging is less definitive. Criteria for backward compatibility
is the same as the terminal charging case. Relative to zero property the condition
!c
(t ) p1
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5.9 Advance Digital Algorithms Realization for Fractional Controls
103
the backward difference rule gives the digital realization of the GL method with
shortmemory principle. Applying this realization in digital filter realization gives
the Finite Impuls
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3.3 Single Thermocouple Junction Temperature in Measurement of Heat Flux
39
k1
1/2
Q 1 (t) = a Dt Tb (t) ,
1
and
k2
1/2
Q 2 (t) = a Dt Tb (t)
2
where hA is product of convective heat transfer coefficient and surface area, and mc
is product of the mass and
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110
6 Initialized Differintegrals and Generalized Calculus
which is the transfer function of the fundamental fractional differential equation, and
is the fundamental building block for more complicated fractional order systems.
A brief discussion on crite
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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
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5.3 ReimannLiouville Fractional Differintegral
87
Fig. 5.5 Convoluting
function h (t) for several t
t=1 t=2
t=3
t=4
3
4
t=5 t=6
2
1
0
2
1
5
6
In Fig. 5.4f, the point X is
!5
0
f ( )h(5 )d ,
definite value of the integration.
Figure 5.5 demonstrates the se
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74
4 Concept of Fractional Divergence and Fractional Curl
geometric buckling, we get following simple form. The temporal solution is avoided
for simplicity.
(k 1)a + v
D
d 2 (x)
+ B 2 (x) = 0
dx2
B2 =
Here we can apply standard Laplace method with initial
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5.8 Infinitesimal Element Geometrical Interpretation of Fractional Differintegrations
q
a Dt
99
! ta "q
$
$
%
N 1
#
t a
( j q)
f t j
f (t) = lim
N (q)
( j + 1)
N
j =0
N
= lim
T 0
N 1
#
j =0
f (t j T )
( j q)
(q)( j + 1)
T q
T = (t a)/N, N , T 0
The nature
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5.3 ReimannLiouville Fractional Differintegral
0 Dt
f (t) =
83
!t
f ( )dg( )
0
Therefore, the fractional integration of the function is area under the curve for the
plot of f ( ) and g( ), from 0 to t. Let us take three axes , g( ), f ( ), making a
cubic
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5.3 ReimannLiouville Fractional Differintegral
0
1
0
1
2
2
3
85
4
3
5
6
4
7
5
Fig. 5.3 Homogeneous and heterogeneous time
(of the moving object). Fractional integration in time means transformation of the
local time to cosmic time.
5.3.2 Convolution
The R
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7.10 Laplace Transform s w Plane for Fractional Controls Stability
151
or equivalently can be expressed as
y (t) = C
!t
0
"
#
Fq [A, t ] B u( )( ) d + D u(t).
The above solution requires the use of matrix F function, which can be obtained
by the use of it