Fractional calculus and applications

Introduction Naive approaches Motivation Definitions Simple examples Geometrical interpretation Applications

FRACTIONAL CALCULUS ANDAPPLICATIONS

V N Krishnachandran

V N Krishnachandran

FRACTIONAL CALCULUS AND 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



Outline

1 Introduction

2 Naive approaches

3 Motivation

4 Definitions

5 Simple examples

6 Geometrical interpretation

7 Applications

V N Krishnachandran



Introduction

Introduction

V N Krishnachandran



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!

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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.

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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?

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Naive approaches

Naive approaches

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Naive approaches

Basic ideas

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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.

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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)).

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The naive approach yields arbitrary order derivatives of thefollowing classes of functions:

Functions expressible using exponential functions.

Functions expressible as power series.

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Naive approaches : Exponential functions

Functions expressible using exponential functions

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For positive integers n we have

dn

dxn(eax) = aneax .

For arbitrary q, real or complex, we define

dq

dxq(eax) = aqeax .

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Using linearity we have:

dq

dxq(cos x) = cos

(x + q

π

2

),

dq

dxq(sin x) = sin

(x + q

π

2

).

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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 .

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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 .

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Naive approaches : Power functions

Functions expressible as power series

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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!.

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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.

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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.

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Naive approaches: Inconsistency

The naive approaches produce inconsistent results.

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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.

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Motivation for definition

Motivation for definition offractional integral

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Motivation for definition : Formula for differentiation

Generalisation of the formula for differentiation

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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

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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

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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.

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The formula for Dnx (f (t)) in the given form is not suitable for

generalisation. We derive an equivalent formula which is suitablefor generalisation.

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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) .

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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.

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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)).

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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)).

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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.

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Motivation for definition : Formula for integration

Generalisation of the formula for integration

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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




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)

.

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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.

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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)

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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

])

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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)).

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Definition of differintegral


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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

])

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Questions of the existence of the differintegral are not addressed inthis talk.

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Simple examples

Differintegrals of simplefunctions

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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

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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 ].

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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.

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Other definitions of differintegral

Other definitions ofdifferintegrals

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Other definitions of differintegral

The differintegrals can be defined in several different ways. Thenext frame shows a second approach.

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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

].

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Geometrical interpretation

Geometrical interpretation offractional integrals

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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.

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Geometrical interpretation of Stieltjes integral


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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).

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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).

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Area of shadow of fence in (g , f )-plane=∫ ba f (x) dg(x).

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Geometrical interpretation of fractional integral


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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.

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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

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0D−0.25x (t + 0.5 sin(t))

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0D−0.5x (t + 0.5 sin(t))

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0D−1x (t + 0.5 sin(t))

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0D−2.5x (t + 0.5 sin(t))

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Applications

Applications

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Applications : Tautochrone

Tautochrone

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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.

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We use fractional derivatives to find tautochrone curves underarbitrary potential V (y).

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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




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 .

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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 .

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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 .

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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




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

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Other applications

Other applications

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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

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Bibliography

Bibliography

V N Krishnachandran



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


Education

Fractional calculus and applications