Post on 09-Feb-2017
transcript
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
FRACTIONAL CALCULUS ANDAPPLICATIONS
V N Krishnachandran
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
International Seminar onRecent Trends in Topology and its Applicationsorganised as part of the Annual Conferenceof Kerala Mathematical Associationheld at St. Joseph’s College, Irinjalakkuda - 680121during 19-21 March 2009.
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Outline
1 Introduction
2 Naive approaches
3 Motivation
4 Definitions
5 Simple examples
6 Geometrical interpretation
7 Applications
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Introduction
Introduction
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Introduction
Leibnitz introduced the notationdny
dxn.
In a letter to L’ Hospital in 1695, Leibniz raised the possibility
definingdny
dxnfor non-integral values of n.
In reply, L’ Hospital wondered : What if n = 12 ?
Leibnitz responded prophetically: “It leads to a paradox, fromwhich one day useful consequences will be drawn”.
That was the beginning of fractional calculus!
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Introduction
What is fractional calculus?
Fractional calculus is the study ofdq
dxq(f (x)) for arbitrary real
or complex values of q.
The term ‘fractional’ is a misnomer. q need not necessarily bea fraction (rational number).
If q > 0 we have a fractional derivative of order q.
If q < 0 we have a fractional integral of order −q.
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Introduction
Compare with the meaning of an.
The situation is similar to the problem of defining, and givinga meaning and an interpretation to, an in the case where n isnot a positive integer.
If n is a positive integer, then an is the result of multiplying aby itself n times. If n is not a positive integer, can we visualisean as multiplication of a by itself n times?
Is a1/2 the result of multiplying a by itself 12 times?
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Naive approaches
Naive approaches
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Naive approaches
Basic ideas
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Naive approaches : Basic ideas
In calculus we have formulas for the n th order derivatives(when n is a positive integer) of certain elementary functionslike the exponential function.
In a naive way these formulas may be generalised to definederivatives of arbitrary order of those elementary functions.
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Naive approaches : Basic ideas
Assuming the following results in a naive way, we can definearbitrary order derivatives of a large class of functions.
Linearity of the operatordq
dxq: For constants c1, c2 and
functions f1(x), f2(x),
dq
dxq(c1f1(x) + c2f2(x)) = c1
dq
dxq(f1(x)) +
dq
dxq(f2(x)).
Composition rule: For arbitrary p, q,(dp
dxp
)(dq
dxq
)(f (x)) =
dp+q
dxp+q(f (x)).
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Naive approaches : Basic ideas
The naive approach yields arbitrary order derivatives of thefollowing classes of functions:
Functions expressible using exponential functions.
Functions expressible as power series.
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Naive approaches : Exponential functions
Functions expressible using exponential functions
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Naive approaches : Exponential functions
For positive integers n we have
dn
dxn(eax) = aneax .
For arbitrary q, real or complex, we define
dq
dxq(eax) = aqeax .
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Naive approaches : Exponential functions
Using linearity we have:
dq
dxq(cos x) = cos
(x + q
π
2
),
dq
dxq(sin x) = sin
(x + q
π
2
).
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Naive approaches : Exponential functions
Extend to functions f (x) having exponential Fourier representationg(α) defined by
f (x) =1√2π
∫ ∞−∞
g(α)e−iαx dα
where
g(α) =1√2π
∫ ∞−∞
f (x)e iαx dx .
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Naive approaches : Exponential functions
Definition
For arbitrary q real or complex, we define
dq
dxq(f (x)) =
1√2π
∫ ∞−∞
g(α)(−iα)qe−iαx dα.
where
g(α) =1√2π
∫ ∞−∞
f (x)e iαx dx .
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Naive approaches : Power functions
Functions expressible as power series
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Naive approaches : Power functions
We require the Gamma function defined by
Γ(p) =
∫ ∞0
tp−1e−t dt.
Note the well-known properties of the Gamma function:
Γ(p + 1) = pΓ(p)
In n is a positive integer Γ(n + 1) = n!.
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Naive approaches : Power functions
For positive integers m, n we have
dn
dxn(xm) = m(m − 1)(m − 2) . . . (m − n + 1)xm−n.
=Γ(m + 1)
Γ(m − n + 1)xm−n.
For arbitrary q real or complex we define
dq
dxq(xm) =
Γ(m + 1)
Γ(m − q + 1)xm−q.
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Naive approaches : Power functions
Definition
If f (x) has a power series expansion
f (x) =∞∑r=0
crx r
then, for arbitrary q real or complex, we define
dq
dxq(f (x)) =
∞∑r=0
crΓ(r + 1)
Γ(r − q + 1)x r−q.
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Naive approaches: Inconsistency
The naive approaches produce inconsistent results.
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Naive approaches: Inconsistency
By exponential functions approach we have
d12
dx12
(1) =d
12
dx12
(e0x) = 012 e0x = 0.
By power functions approach we have
d12
dx12
(1) =d
12
dx12
(x0) =Γ(0 + 1)
Γ(0− 12 + 1)
x0− 12 =
1√πx.
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Motivation for definition
Motivation for definition offractional integral
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Motivation for definition : Formula for differentiation
Generalisation of the formula for differentiation
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Motivation for definition : Formula for differentiation
We change the notations slightly.
We consider a function f (t) of the real variable t.
We consider derivatives of different orders of f (t) at t = x .
We write
D1x (f (t)) =
[d
dtf (t)
]t=x
, D2x (f (t)) =
[d2
dt2f (t)
]t=x
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Motivation for definition : Formula for differentiation
Formulas for derivatives:
D1x (f (t)) = lim
h→0
f (x)− f (x − h)
h.
D2x (f (t)) = lim
h→0
f (x)− 2f (x − h) + f (x − 2h)
h2.
D3x (f (t)) = lim
h→0
f (x)− 3f (x − h) + 3f (x − 2h)− f (x − 3h)
h3
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Motivation for definition : Formula for differentiation
Definition
Let n be a positive integer. Then the derivative of order n of f (t)at t = x is given by
Dnx (f (t)) = lim
h→0
∑nj=0(−1)j
(nj
)f (x − jh)
hn.
This is the original formula for derivatives of order n.
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Motivation for definition : Formula for differentiation
The formula for Dnx (f (t)) in the given form is not suitable for
generalisation. We derive an equivalent formula which is suitablefor generalisation.
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Motivation for definition : Formula for differentiation
Choose fixed a < x .
Choose a positive integer N.
Set h = x−aN . We let N →∞.
Notice that(nj
)= 0 for j > n.
Using gamma functions we have : (−1)n(nj
)= Γ(j−n)
Γ(−n)Γ(j+1) .
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Motivation for definition : Formula for differentiation
The derivative of order n of f (t) at t = x can now be expressed inthe following form.
Dnx (f (t)) = lim
N→∞
1
Γ(−n)
∑N−1j=0
[Γ(j−n)Γ(j+1) f
(x − j
(x−aN
))](x−aN
)−n .
This is the generalised formula for the derivative of order n.
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Motivation for definition : Formula for differentiation
Some observations about the generalised formula for derivativesare in order.
The generalised formula apparently depends on a.
The original formula does not depend on any such constant a.
There is no inconsistency here!
It can be proved that when n is a positive integer, thegeneralised formula is independent of the value of a and thatthe generalised formula and the original formula give the samevalue for Dn
x (f (t)).
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Motivation for definition : Formula for differentiation
Some more observations about the generalised formula forderivatives.
The original formula for Dnx (f (t)) in the original form has no
meaning when n is not an integer.
The generalised formula is meaningful for all values of n. Forall values of n other than positive integers it depends on a. Tosignify this dependence on a explicit we denote the value ofthe generalised formula by aDn
x (f (t)).
When n is a positive integer we have aDnx (f (t)) = Dn
x (f (t)).
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Motivation for definition : Formula for differentiation
Some further observations about the generalised formula forderivatives.
Let us imagine the generalised formula for derivatives as theone true formula for derivatives.
The we consider aDnx (f (t)) as the derivative over the interval
[a, x ], and not as the derivative at t = x .
In this sense, the derivative is not a local property of a givenfunction f (t). It is a local property only when the order ofderivative is a positive integer.
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Motivation for definition : Formula for integration
Generalisation of the formula for integration
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Motivation for definition : Formula for integration
We next derive a general formula for repeated integration. Weconsider a function f (t) defined over the interval [a, x ].The following notations are used :
aJ1x (f (t)) =
∫ x
af (t) dt
aJ2x (f (t)) =
∫ x
adx1
∫ x1
af (t)dt
aJ3x (f (t)) =
∫ x
adx2
∫ x2
adx1
∫ x1
af (t)dt
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Motivation for definition : Formula for integration
Setting h = x−aN and using the definition of integral as the limit of
a sum, we have
aJ1x (f (t)) = lim
N→∞
hN−1∑j=0
f (x − jh)
.
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Motivation for definition : Formula for integration
Application of the formula for integration a second time yields
aJ2x (f (t)) = lim
N→∞
h2N−1∑j=0
(j + 1)f (x − jh)
.
Application of the formula a third time yields
aJ3x (f (t)) = lim
N→∞
h3N−1∑j=0
(j + 1)(j + 2)
2f (x − jh)
Repeated application of the formula yields the general formulagiven in the next frame.
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Motivation for definition : Formula for integration
Definition
Let n be a positive integer. The nth order integral of f (t) over[a, x ] is given by
aJnx (f (t)) = lim
N→∞
hnN−1∑j=0
Γ(j + n)
Γ(n)Γ(j + 1)f (x − jh)
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Generalised formula for differentiation and integration
The nth order derivative :
limN→∞
1
Γ(−n)
[x − a
N
]−n N−1∑j=0
Γ(j − n)
Γ(j + 1)f
(x − j
[x − a
N
]) .
The nth order integral :
limN→∞
1
Γ(n)
[x − a
N
]n N−1∑j=0
Γ(j + n)
Γ(j + 1)f
(x − j
[x − a
N
])
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Unified formula for differentiation and integration
The formula for the n th order derivative and the formula for the nth order integral are special cases of the following unified formula :
limN→∞
1
Γ(−q)
[x − a
N
]−q N−1∑j=0
Γ(j − q)
Γ(j + 1)f
(x − j
[x − a
N
])This gives the n th order derivative when q = n and the n th orderintegral when q = −n. We call this the differintegral of f (t) overthe interval [a, x ]. We denote it by
dq
[d(x − a)]q(f (t)) or aDq
x (f (t)).
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Definition of differintegral
Definition of differintegral
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Definition of differintegral
Definition
The differintegral of order q of f (t) over the interval [a, x ] isdenoted by aDq
x (f (t)) and is given by
limN→∞
1
Γ(−q)
[x − a
N
]−q N−1∑j=0
Γ(j − q)
Γ(j + 1)f
(x − j
[x − a
N
])
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Definition of differintegral
Questions of the existence of the differintegral are not addressed inthis talk.
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Simple examples
Differintegrals of simplefunctions
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Differintegral of unit function
aDqx (1) = (x−a)−q
Γ(1−q)
Special cases
0D12x (1) = (x−0)−
12
Γ(1− 12 )
= 1√πx
−∞D12x (1) = (x−(−∞))−
12
Γ(1− 12 )
= 0
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Differintegral of unit function : Inconsistency resolved
The inconsistency in the naive approaches has now been resolved.
Fractional derivatives using the exponential functions givefractional derivatives over (−∞, x ].
fractional derivatives using the power functin give thefractional derivative over [0, x ].
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Differintegrals of other simple functions
aDqx (0) = 0.
aDqx (t − a) = (x−a)1−q
Γ(2−q) .
aDqx ((t − a)p) = Γ(p+1)
Γ(p−q+1) (x − a)p−q.
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Other definitions of differintegral
Other definitions ofdifferintegrals
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Other definitions of differintegral
The differintegrals can be defined in several different ways. Thenext frame shows a second approach.
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Definition
If q < 0 then
aDqx (f (t)) =
1
Γ(−q)
∫ x
a
f (y)
(x − y)q+1dy .
If q ≥ 0, let n be a positive integer such that n − 1 ≤ q < n.
aDqx (f (t)) =
dn
dxn
[1
Γ(n − q)
∫ x
a
f (y)
(x − y)q−n+1dy
].
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Geometrical interpretation
Geometrical interpretation offractional integrals
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Geometrical interpretation
The non-existence of a geometrical or physical interpretationof the fractional derivatives or integrals was acknowledged inthe first world conference on Fractional Calculus andApplications held in 1974.
F Ben Adda suggested in 1997 a geometrical interpretationusing the idea of a contact of the α th order. But hisinterpretation did not contain any “pictures”.
Igor Podlubny in 2001 discovered an interesting geometricinterpretation of fractional integrals based on the geometricalinterpretation of the Stieltjes integral discoverd by G L bullockin 1988. In this talk we present Podlubny’s geometricinterpretation of the fractional integral.
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Geometrical interpretation of Stieltjes integral
Geometrical interpretation of Stieltjes integral
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Geometrical interpretation of Stieltjes integral
Let g(x) be a monotonically increasing function and let f (x) be anarbitrary function. We consider the geometrical interpretation ofthe Stieltjes integral ∫ b
af (x) dg(x).
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Geometrical interpretation of Stieltjes integral
Choose three mutually perpendicular axes : g -axis, x-axis,f -axis.
Consider the graph of g(x), for x ∈ [a, b], in the (g , x)-plane.Call it the g(x)-curve.
Form a fence along the g(x)-curve by erecting a line segmentof height f (x) at the point (x , g(x)) for every x ∈ [a, b].
Find the shadow of this fence in the (g , f )-plane.
Area of the shadow is the value of the Stieltjes integral∫ ba f (x) dg(x).
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Geometrical interpretation of Stieltjes integral
Area of shadow of fence in (g , f )-plane=∫ ba f (x) dg(x).
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Geometrical interpretation of fractional integral
Geometrical interpretation of fractional integral
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Geometrical interpretation of fractional integral
For q < 0 we have
aDqx f (t) =
1
Γ(−q)
∫ x
a
f (t)
(x − t)q+1dt.
We write
g(t) =1
Γ(−q + 1)
[1
xq− 1
(x − t)q
]We have the Stieltjes integral
aDqx f (t) =
∫ x
af (t) d g(t).
This can be interpreted as the area of the shadow of a fence.
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Geometrical interpretation of fractional integral
In he next few frames we present the visualizations of thefractional integral
0Dqx (f (t))
whenf (t) = t + 0.5 sin(t)
for the following values of q :
q = −0.25, −0.5, −1, −2.5
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
0D−0.25x (t + 0.5 sin(t))
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
0D−0.5x (t + 0.5 sin(t))
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
0D−1x (t + 0.5 sin(t))
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
0D−2.5x (t + 0.5 sin(t))
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Applications
Applications
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Applications : Tautochrone
Tautochrone
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Applications : Tautochrone
The classical problem background
Problem statement :Find the curve x = x(y) passing through the origin, alongwhich a point mass will descend without friction, in the sametime regardless of the point (x(Y ),Y ) at which it starts.
Assumption :We assume a potential of V (y) = gy , where g is accelerationdue to gravity.
Solution : The curve is a cycloid.
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Applications : Tautochrone
We use fractional derivatives to find tautochrone curves underarbitrary potential V (y).
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Applications : Tautochrone
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Applications : Tautochrone
We use the following notations :
Consider a bead of unit mass sliding from rest at (X ,Y ).
Let it slide along a frictionless curve x = x(y).
Let the potential acting on the bead be a function of y only,say V (y).
Let the curve pass through the origin and let the bead reachorigin at time T .
Let s be the arc-length from the origin to (x(y), y).
Let v be the velocity at (x(y), y).
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Applications : Tautochrone
The velocity of the bead is
v = −ds
dt.
By conservation of energy we have
v 2
2= V (Y )− v(y).
This can be written as
− ds√V (Y )− V (y)
=√
2dt.
We also havey = 0 when t = T .
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Applications : Tautochrone
From the observations in the previous frame we have∫ Y
0
ds√V (Y )− V (y)
=√
2T .
This is equivalent to
1
Γ( 12 )
∫ Y
0
dsdV (y) V ′(y)√V (Y )− V (y)
dy =√
2/πT .
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Applications : Tautochrone
The left side of the last equation is a fractional derivative.
0D− 1
2
V (Y )
ds
dV (y)=√
2/πT .
Equivalently
0D− 1
2
V (Y ) 0D1V (Y )s =
√2/πT .
Thus
0D12
V (Y )s =√
2/πT .
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Applications : Tautochrone
Recall that the time T is constant and is independent of thestarting point (x(Y ),Y ).
Hence we replace the constant Y by the variable y .
We get the equation
0D12
V (y)s =√
2/πT .
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Applications : Tautochrone
Solving the fractional-differential equation we get
s = 0D− 1
2
V (y)
(√2/πT
)=√
2/πT 0D− 1
2
V (y)(1)
=√
2/πT 2√
V (y)/π
=2√
2V (Y )
πT
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Other applications
Other applications
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Other applications
The following are some of the areas in which the theory offractional derivatives has been successfully applied :
Signal processing : Application in genetic algorithm
Tensile strength analysis of disorder materials
Electrical circuits with fractance
Viscoelesticity
Fractional-order multipoles in electromagnetism
Electrochemistry and tracer fluid flows
Modelling neurons in biology
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Bibliography
Bibliography
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS
Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications
Bibliography
Igor Podlubny, Geometric and physical interpretation offractional integration and fraction differentiation, FractionalCalculus and Applied Analysis, Vol.5, No. 4 (2002)
Lokenath Debnath, Recent applications of fractional calculusto science and engineering, IJMMS, Vol. 54, pp.3413-3442(2003)
Keith B. Oldham, Jerome Spanier, The Fractional Calculus;Theory and Applications of Differentiation and Integration toArbitrary Order, Academic Press, (1974)
Kenneth S. Miller, Bertram Ross, An Introduction to theFractional Calculus and Fractional Differential Equations,John Wiley & Sons ( 1993)
V N Krishnachandran
FRACTIONAL CALCULUS AND APPLICATIONS