Introduction to Fractional Calculus Amna Al - Amri Project October 2010

8/8/2019 Introduction to Fractional Calculus Amna Al - Amri Project October 2010

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Introduction to Fractional Calculus

A Project

Submitted to the University of Nizwa in Partial Fulfillment of the

Requirements for the Degree of Bachelor of Education in Mathematics

By

AMNA ABDULSALAM YOUNIS AL-AMRY

Supervised by

Professor Dr. Ahmed S. El-Karamany

2010


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Supervisor Certification

We certify that this thesis entitled" Introduction of Fractional

Calculus" have been made by the Bachelor student

AMNA ABDULSALAM YOUNIS AL-AMRYunder my supervision at the

University of Nizwa as Partial Fulfillment of the Requirements for the

Degree of Bachelor of Education in Mathematics.

Signature

Professor Dr. Ahmed Sadek El-Karamany

In view of the available of the recommendations I forwarded this

thesis for debate the examining committee.

Signature


3/29

Committee Certification

We certify that, we have read this project entitled" Introduction of Fractional

Calculus " and examining committee examined the student AMNA

ABDULSALAM YOUNIS AL-AMRY in its contents and what it connected with it,

and that our opinion it meets the standard of project for degree bachelor of Education

in Mathematics.

1) Signature

Name: Professor Dr. Ahmed S. El-Karamany

Chairman and Supervisor. Date: 27/5/2010

2) Signature

Name: Professor Dr. Mohammad Elatrash

Member Date: 27/5/2010

3) Signature

Name: Dr. Mahmood Khalid Jasim

Associate Professor

Member Date: 27/5/2010

Approved by:

Professor Dr. Ahmed S. El-Karamany

Head of Department of Mathematical and Physical Sciences.

Date: 27/5/2010


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Abstract

i

AAbbssttrraacctt

This work is devoted to exploreFractional Calculus whichhas been used

successfully to modify many existing models of Physical processes.

The Work consists of 5 chapters. In chapter 1; the introduction, some historical notes

and an approach to fractional calculus using generalization of Cauchy formula for

n folde integral. Laplace transform of the convolution and its properties are given.

Integrals and derivatives of fractional order are defined in chapter 2. In this Chapter

integrals and derivatives of fractional order of power and exponential functions are

calculated. Semi-integrals and semi-derivatives of power function, exponential function,

sine and cosine functions are given in chapter 3. In chapter 4 some illustrating solved

problems are given. The Tautochrone problemisdiscussed in chapter 5 as an applicationof semi-derivatives in solving Abels integral equation. A list of references is given at the

end of this project.


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ACKNOWLEDGEMENT

Thanks firstly and finally to God who deserves all thanks.

I would like to express my deep appreciation to my supervisor

Professor Dr. Ahmed Sadek El-Karamany the Head of department of

Mathematical and Physical Sciences for his encouragement, Support,

guidance and valuable suggestions. His enthusiasm and interest for

science are contagious.

I was also lucky to be able to associate myself with the talented

and hard working members of the DMPS.

My special thanks are due to the University of Nizwa, College

of Arts & Sciences and to all staff members for their assistance and

support during my study.

My thanks also go to my friends and colleagues for their support

and help.



6/29


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1

Chapter 1

Introduction1.1 Introduction

Fractional calculus is three centuries old as the conventional calculus, but not very

popular among science and/or engineering community. The beauty of this subject

is that fractional derivatives (and integrals) are not a local (or point) property (or

quantity). Thereby this considers the history and non-local distributed effects. In

other words, perhaps this subject translates the reality of nature better! Therefore

to make this subject available as popular subject to science and engineering community,

it adds another dimension to understand or describe basic nature in a better

way. Perhaps fractional calculus is what nature understands, and to talk with nature

in this language is therefore efficient. For the past three centuries, this subject was with

mathematicians, and only in last few years, this was pulled to several (applied) fields

of engineering , science and economics. However, recent attempt is on to have

the definition of fractional derivative as local operator specifically to fractal science

theory. Next decade will see several applications based on this 300 years (old) new

subject, which can be thought of as superset of fractional differintegral calculus, the

conventional integer order calculus being a part of it. Differintegration is an operator

doing differentiation and sometimes integrations, in a general sense. In this project,

fractional order is limited to only real numbers; the complex order differentigrations

are not touched. Also the applications and discussions are limited to fixed fractional

order differintegrals, and the variable order of differintegation is kept as a future

research subject. Perhaps the fractional calculus will be the calculus of twenty-first

century. In this project, attempt is made to make this topic application oriented forregular science and engineering applications. Therefore, rigorous mathematics is

kept minimal. In this introductory chapter.

1.2 Birth of Fractional Calculus

In a letter dated 30th September 1695, LHopital wrote to Leibniz asking him a

particular notation that he had used in his publication for the nth derivative of a

function


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2

n

n

D f

Dx

i.e., what would the result be ifn = 1/2. Leibnizs response an apparent paradox

from which one day useful consequences will be drawn. (Debnath 2004) In these

words, fractional calculus was born. Studies over the intervening 300 years have proved

at least half right. It is clear that within the twentieth century especially numerous

applications have been found. However, these applications and mathematical background

surrounding fractional calculus are far from paradoxical. While the physical meaning is

difficult to grasp, the definitions are no more rigorous than integer order counterpart.

1.3 Fractional Calculus as Generalization of Integer Order Calculus

Let us consider n an integer and when we sayxn we quickly visualizex multiply n

times will give the result. Now we still get a result ifn is not an integer but fail to

visualize how. Like to visualize 2is hard to visualize, but it exists.

An approach to fractional calculus may be given using the generalization of Cauchy

formula for n folder integral , the Laplace Transform of the convolution and its

properties (Gorenflo and Mainardi 1997).

The Laplace Transform of the function ( )f t is defined by

0

{ ( )} ( ) ( )stL f t e f t dt F s

= = (1.1)

The convolution of two functions ( )f t and ( )g t is denoted by ( )( ) f g t and is

defined by the relation

0 0

( )( ) ( ) ( ) ( ) ( )

t t

f g t f t x g x dx f t y g y dy = = (1.2)

It is easy to prove that

0 0

( )( ) ( ) ( ) ( )( ) ( ) ( )

t t

f g t f t x g x dx g f t g t x f x dx = = = (1.3)


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3

We prove the following theorem

Theorem 1.1:

{( )} { ( )} { ( )} ( ) ( )L f g L f t L g t F s G s = = (1.4)

where

( ) { ( )}, ( ) { ( )}F s L f t G s L g t = = (1.5)

Proof: From the definitions of Laplace transform and the convolution we get

0 0 0 0

{ } ( ( ) ( ) ) ( ) ( )

t t

st st L f g e f y g t y dy dt e f y g t y dydt

= = (1.6)

The region of integration is a simple vertically region in the tOy plane where Oy is the

vertical axis and Otis the horizontal axis.

It is the triangle in the first coordinate quarter whose base in the infinity and whose sides

are the axis Ot and the line y t= : {( , ) : 0 0 }V R t y t y t = < .

Reversing the order of integration we convert the region to simple horizontally:

{( , ) : 0 }h R t y y t y= < < and the integral (1.6) takes the form

0 0 0

0

{ } ( ) ( ) ( ) ( )

( )( ( ) )

tst st

y

st

y

L f g e f y g t y dydt e f y g t y dtdy

f y e g t y dt dy

= =

=

Substituting t y x = into the internal integral, where y is kept fixed, we get

( )

0 0 0

0 0

{ } ( )( ( ) ) ( )( ( ) )

( ( ) )( ( ) ) ( ) ( )

st s x y

y t y y x

sy sx

L f g f y e g t y dt dy f y e g x dx dy

e f y dy e g x dx F s G s

+

= = = =

== =

= =

This proves the theorem. As a consequence we have

1

0

{ ( ) ( )} ( ) ( ) ( )

t

L F s G s f g f t x g x dx = = (1.7)


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4

Below is the Laplace transforms and inverse Laplace transforms of some functions that

play an essential role in the generalization we are going to present

The Laplace transform and the inverse Laplace transform of some functions:

Laplace Transform The Inverse Laplace Transform

11 ( 1)!{ }n

n

nL t

s

=1

1 1{ }( )

n

n

tL

ns

=

2

0

( ){ ( ) }

tF s

L f x dxs

=1

0

( ){ } ( )

tF s

L f x dxs

=

3

2

0 0

( ){ ( ) }

t t

F s L f x dxdxs

= 1

2

0 0

( ){ } ( )

t t

F s L f x dxdxs

=

4

0 0

( ){ ( ) }

t t t

n

o

n

F s L f x dxdx dx

s=

1

0 0 0

( ){ } ( )

( )

t t t

nn

n

n

F s L f x dxdx dx

s

D f t

=

=

Where

21 2 1

0 0

( ) 2 p x p x p x e dx x e dx

= = (1.8)

Using the convolution theorem with

11( ) , ( )

( )

n

n

tG s g t

ns

= =

we obtain

11 1 1

0 0

( ) 1 1{ } ( )( ) ( )( ) ( )

( ) ( ) ( )

t tnn n

n

F s t L f t f x t x dx x f t x dx

n n ns

= = =

Therefore, we get from the last row

1 1

0 0

1 1( ) ( ) ( ) ( )

( ) ( )

t t

n n n D f t t x f x dx x f t x dx

n n

= = (1.9)

Here we have used that if two continuous functions have the same Laplace transform they

are identical (Learch Theorem). This means that we restrict our derivation to include only

continuous functions


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6

Chapter 2

Fractional integrals and derivatives

Definition (1):Let 0 > and let be continuous on (0, ) and integrable on any

finite subinterval[ , ]a b . Then we shall say

that is an element in the class C, and for 0t > , we call

1

0

1( ) ( )

( )

t

D f t x f x dx

=

(2.1a)

the Riemann-Liouville fractional integral of ( )t of order 0 > .The symbol

( )J f is also used for the expression (2.1) . We can add to the definition (1) that

0 0( ) ( ) ( )J f t D f t f t = = , 1 1

0

( ) ( ) ( )t

J f t D f t f t dt = = (2.1b)

Using the linearity property of the definite integral we obtain

11 2 1 2

0

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

( )

t

D C f t C g t t x C f x C g x dx

= (2.2)

1 11 2

1 20 0

( ) ( ) ( ) ( ) ( ) ( )( ) ( )

t tC C

t x f x dx t x g x dx C D f t C D g t

= =

Where 1C and 2C are constants.

Because we now have a well-defined class of functions to which our definition of

the fractional integral applies, it is worthwhile to see the fractional integrals of

the power function . The fractional integration of even the simplest functions can lead

to complicated higher transcendental functions.

Example (1): The fractional integral of the power function.

Let ( ) , 1 f t t = > . Then,

1

0

1( ) ( )

( )

t

D t t x x dx

= (2.3)

Substituting x t u= into Eq. (2.3) we get

1

1

0

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

( ) ( ) ( 1)

t t t D t u u du B

+ + + += = + =

+ +

(2.4)


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7

Example (2): The fractional integral of the constant function

Let ( ) f t C = . Then, using the preceding expression (2.4) and the linearity

property (2.2) one obtains

( )D C =

0 0

( 1)lim[ ( )] lim( 1) ( 1)

Ct C t D Ct

+

+= =

+ + + (2.5)

As special case we get the fraction integral of ( ) 1f t =

(1) 0( 1)

tD

= >+

(2.6)

Example (3): The fractional integral of the exponential function

( ) , 0.tt e =

1 ( ) 1

0 0

1( )

( ) ( )

t ttt t x xe D e x e dx x e dx

= =

Substitution x u = leads to:

1

0

( ) ( , ) ( , )( ) ( )

tt tt u

t

e e D e u e du t E

= = = (2.7)

Where: the incomplete gamma function ( , )x is defined by the relation:

1

0

( , )

x

t x t e dt = (2.8)

and the Et- function ( , )tE is defined by the relation:

( , )( , )

( )

t

t

e tE

= (2.9)

Definition (2):: The fractional derivative of ( )t of order 0> (if exists) can be

defined in terms of fractional integral ( )D f t as:

( )( ) ( )m mD f t D D f t = (2.10)

Where m is an integer [ ] , and [ ] , [ , 1)x n x n n= + is the ceiling function

(the greatest integer that equal or less thanx ).

Example (4): The fractional derivative of the power function

( ) , 1t t

= >


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8

Applying the definition (2.10) Using Eq. (2.4) with m = we get

( ) ( 1)( ) [ ( )]

(1 )

mm m m t

D t D D t D

m

+ += =

+ +

(2.11)

Sincem is integer, Eq. (2.11) yields

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

(1 )

m m tD t

m

+ + + +=

+ +

(2.12)

From the properties of the Gamma function we have

( 1 ) ( )( 1) ( 1) ( 1)m m m + + = + + + + (2.13)

Substituting from (2.13) into Eq. (2.12) we get

( 1)( ) ( )

( 1 ) D t t

+=+

(2.14)

For integer m and n we have

( 1)( ) ( ),

( 1)

n m m nm D t t m n

m n

+= > +

(2.15)

Example (5):The fractional derivative of a constant function

Eq. (2.14) when 0 yields

(1) ,(1 )

tD

=

( is a noninteger ) (2.16)

Before proving some properties of the fractional derivatives we introduce the Leibniz

rule for the fractional derivative of product of two functions.

Leibniz rule: For integer n the Leibniz rule for the product of two n times

differentiable functions can be written in the form

( ) ( ) ( ) ( )

0

1 2 2

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

( 1) ( 1 )

( 1)( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

2

k n

n n k k

k

n n n n

nD f t g t D f t D g t

k n k

n nD f t g t n D f t Dg t D f t D g t f t D g t

=

=

+= + +

= + + + +

0

( 1)( ) ( )

( 1) ( 1 )

k nk n k

k

nD f t D g t

k n k

=

=

+=

+ + (2.17)

The analogy between the two expressions (2.14) and (2.15) suggests to generalize

(2.17) for the fractional derivative of order 0 > as follows


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9

( ) ( ) ( )0

1

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

( 1) ( 1 )

( ) ( ) ( ) ( ) ( ) ( )

k k

k

D f t g t D f t D g t k k

D f t g t D f t Dg t f t D g t

=

+=

+ +

= + + + +

0

( 1) ( ) ( )( 1) ( 1 )

k k

k

D f t D g t k k

=

+= + + (2.18)

Since is not integer the upper limit of the sum in (2.18) is infinite.

Now, we are able to prove the following properties of the fractional derivative.

Property (1):Let ( )g t C= is the constant function, then

[ ( )] [ ( )]D Cf t CD f t = (2.19)

Proof:since 0 ,D C C = and 0, 1,2,m D C m= = we get upon substituting

( )g t C= into (2.18):

( ) ( )0

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

( 1) ( 1 )

k k

k

D f t C D f t D C D f t C C D f t k k

=

+= = =

+ +

Property (2):

( ) ( )[ ( ) ( )] ( ) ( )D f t h t D f t D h t + = + (2.20)

Proof: Let0

( ) 1g t t= = then, ( )0

[ ( ) ( )] [ ( ) ( ) ]D f t h t D t f t h t

+ = + and

( )

( ) ( )

0 0

0

0

0

( 1)[ ( ) ( ) ] ( )

( 1) ( 1 )

( 1)( ) ( ) ( )

( 1) ( 1 )

k k

k

k k

k

D t f t h t D f t D t k k

D h t D t D f t D g t k k

=

=

++ =

+ +

++ = +

+ +

Combining properties (1) and (2) we obtain the following linearity property of the

fractional derivative

( ) ( )1 2 1 2[ ( ) ( )] ( ) ( )D C f t C h t C D f t C D h t + = + (2.21)

Where 1C and 2C are constants.

Theorem (1): The Index law for fractional integrals

(i)( )

( ( )) ( )D D f t D f t += (2.25a)

(ii) ( ( )) ( ( ))D D f t D D f t = (2.25b)

For C

, 0 1< the substitution 2x u = leads to:

1 22

0

2( )( ) erf ( )

tt tt ue e D e e du t

= = (3.5)

Where2

0

2erf ( )

t

xt e dx

= is the error function.

Setting lna = where 0, 1a a> in Eq. (3.5)we get

12 ( ) erf ( ln )

ln

tt a

D a t a

a

= (3.6)

The semi- derivative of the exponential function can be obtained from (3.5) using the

definition (3.2):

12

1( ) erf ( )t t D e e t

t

= + (3.7)

Where used the Leibnizs formula

( )

( )

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

v t

u t

d dv duf x dx f v t f u t

dt dt dt =

(3.8)


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15

From Eqs. (3.3b) , (3.5) and (3.7) we get:1 12 2

1( ) ( 1)t t D e D e

= ,

4): The functionssint andcost :

12

0

1 sin( ) 2(sin ) [sin ( ) cos ( )]

t

t x dx D t t C t t S t x

= = (3.9)

12

2(cos ) [cos ( ) sin ( )] D t t C t t S t

= + (3.10)

Where ( )C t and ( )S t are the Fresnel integrals defined by the relations

2 2

0 0

( ) cos( ) , ( ) sin( ) ,

t t

C t y dy S t y dy= = and therefore, we have

2 2

0 0

( ) cos( ) , ( ) sin( ) ,

t t

C t y dy S t y dy= = (3.11)

The semi -derivatives can be obtained from the definition (3.2) using Leibniz formula

(3.8) as:

1 12 2

2(sin ) { (sin )} { [sin ( ) cos ( )]}

dD t D D t t C t t S t

dt

= =

2 1[cos ( ) sin ( )] [sin cos cos sin ]t C t t S t t t t t

t = + + . Therefore,

12

2(sin ) [cos ( ) sin ( )] D t t C t t S t

= + (3.12)

Taking into consideration Eq. (3.10) we get

1 12 2(sin ) (cos ) D t D t

= (3.13)

Similarly, we get1 12 2

1(cos ) (sin ) D t D t

t

= (3.14)

Taking into consideration Eq. (3.3a) the preceding equation can be written in the

form:

1 12 2(sin ) (1 cos ) D t D t

= (3.15)



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16

Chapter 4

Examples

(Semi- Integrals and Semi-Derivatives of Some Algebraic Functions)

( )f x 1/ 2

0 0

1 ( ) 1 ( )( )

t t f x dx f t x dx

D f xt x x

= =

1/2

1/2

( )

[ ( )]

D f x

D D f x

=

1 1( )

x

0

1x

y x z

dy x dz

dy

y x y

=

=

1

1/ 2 1/ 2

0

1(1 )

1 1 1( , )2 2

z z dz

B

=

= =

zero

2( )x

0

1

3 1( , ) ( )2 2 2 2

x

y x z

dy x dz

x y dy

y

x x xB

=

=

= = =

2

3 1

1 x+

0

2(1 )sin ,

2(1 )sin cos

1

1

x

y x

dy x d

dy

x y y

= +

= ++

1tan

1

0

2 (1 ) sin cos 2tan ( )

(1 ) sin cos

xx d

xx

+= =

+

1

(1 )x x+

41 x+

0

2(1 )sin ,

2(1 )sin cos

11x

y x

dy x d

x y dy

y

= +

= +

+

1(1 )tan ( )

x xx

+= +

11 tan ( )x

x

+

5

3/ 2

1

(1 )x+

3

0

2(1 )sin ,

2(1 )sin cos

1

(1 )

x

y x

dy x d

dy

x y y

= +

= ++

2

(1 )

x

x =

+

2

1

(1 )

x

x x

+


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17

6 1

1 x

0

2(1 )sinh

2(1 )sinh cosh

1

1

x

y x

dy x d

dy

x y y

=

= +

12tanh ( )x

=

1

(1 )x x

71 x

2(1 )sinh

2(1 )sinh cosh

0

11x

y x

dy x d

x y dy

y

=

=

+

1(1 )tanh ( )

x xx

+

1

1[

tanh ( )]

x

x

8

3/ 2

1

(1 )x

2(1 )sinh

2(1 )sinh cosh30

1

(1 )

x

y x

dy x d dy

x y y

= =

+

2

(1 )

x

x =

21(1 )

xx x

+

9 1( )1 x

0

2

2

(1 ) tan

2(1 )tan sec

1

(1 )

x

y x

dy x d

dy

x y y

=

= +

12sin ( )

(1 )

x

x

=

3/ 2

1

3/ 2

1

(1 )

sin ( )

(1 )

x

x x

x x

x x

+

10 1( )1 x+

0

2

2

(1 ) tanh

2(1 )tanh sech

1

(1 )

x

y x

dy x d

dy

x y y

= +

= ++

12sinh ( )

(1 )

x

x

=+

3 / 2

1

3 / 2

1

(1 )

sinh ( )

(1 )

x

x x

x x

x x

+

+

+

11

(1 )

x

x+

0

2sin

2 sin cos

1

(1 )

x

y x t

dy x t tdt

x y dy

x y y

=

=

+

1 x

=

+

3/ 22(1 )x

+

12 1

(1 )x x+

0

2sin

2 sin cos

1

(1 )

x

y x t

dy x t tdt

dy

x y x y y

=

=+ 3/ 22(1 )x

+


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19

Cha pte r 5

Applications

The Tautochrone Curve

A tautochrone curve or isochrone curve is the curve for which the time taken by an object

sliding without friction in uniform gravity to its lowest point independent of its starting

point. Abel was interested in the tautochrone problem; that is, determining a curve in the

( ),x y plane such that the time required for a particle to slide down the curve to its

lowest point is independent of its initial placement on the curve.

Firstly, we shall consider the time required for descent as a function of initial height.

Let us fix the lowest point of the curve at the origin and position the curve in the positive

quadrant of the plane, denoting by ( )0 0, M x y the initial point and ( ),P x y any point

on the curve between (0,0)O and ( )0 0,x y . We denote the curve by ( )S S y= ( at

P )and then, (0) 0S = . we also have the following conditions: at point M we have

00,t y y= = , 0v = ( the initial velocity) and at point O we have

0( ), 0, 0t T y y S= = = . Assuming no frictional losses we may apply the

conservation of energy law (the sum of kinetic energy and potential energy is constant) :

2 20

1 1

2 2 M M p P E mv mgy E mv mgy= + = = + . Since 0Mv = we get:

20

1( ) ( )

2

dSm mg y y

dt= (5.1)


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20

Since the distance S is decreasing as the time increases we have 0dS

dt< .

Therefore, Eq. (5.1) leads to 0( ) 2 ( )dS dS dy

g y ydt dy dt

= = . Thus, separating the

variables one obtains

0

( )

2( )

dSdy

dyg dt

y y=

(5.2)

Integrating from 0,t = to t T= which corresponds from 0y y= to 0y = we get

0

0

00

( / )2

( )

T

y

dS dyg dt dy

y y=

. Therefore,

0

0

00

( / )2 ( )

( )

ydS dy

g T y dyy y

= (5.3)

Equation (5.3) can be written in the form

0

( / )2 ( )

( )

ydS dz

g T y dzy z

=

(5.4)

Equation (5.4) is the Abels integral equation. We shall use the fractional calculus to

obtain the solution of this equation (Samko et all 1993). Recalling the definition of the

semi integral:1/2

0

1 ( )( )

yf z

D f y dzy z

= we get

1/2 1/2 1/22 ( ) (( / )) ( ( )) ( ( ))gT y D dS dy D DS y D S y = = = . (5.5)

Therefore, we get

1/22( ) { ( )}g

S y D T y

= (5.6)

Thus, we have obtained the solution of Abels integral equation in terms of the semi-

integral of ( )T y .


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Now, to get the required curve we must set ( ) .T y T Const = = and use

1/2

{ } 2y

D C C

= and Eq. (5.6) becomes

2 22( ) 2 2 , 2 / y g

S y T ay a gT

= = = (5.7)

Differentiating both sides with respect to y one obtains

2( ) 1 ( )dS y dx a

dy dy y= + = , from which

2 21 ( ) ( ) 1dx a dx a

dy y dy y+ = = (5.8)

Let

2sin ( / 2)y a = . (5.9)

Then from (5.8) we get

cot( / 2) tan( / 2)dx dy

dy dx = = (5.10)

From (5.9) and (5.10) we get sin( / 2)cos( / 2) tan( / 2)dy dadx dx

= = which leads to

2cos ( / 2)

dxa

d

= , and we get (1 cos )

2

dx a

d

= +

From the preceding equation we obtain ( sin )2

ax C = + + . The curve is passing

through the origin so, 0x = at 0y = and therefore, 0C = . Thus we obtain the

parametric equations of the tautochrone curve (a cycloid)

( sin ) (1 cos )2 2

a ax y = + = (5.11)



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References

[1] L. Debnath, (2004), A brief historical introduction to fractional calculus, INT. J.

MATH. EDUC. SCI., TECHNOL., vol. 35, No 4, 487-501.

[2] Gorenflo, R. and F. Mainardi (1997) , Fractional calculus : integral and differential

equations of fractional order, in Fractals and Fractional Calculus in Continuum

Mechanics (Ed. A. Carpinteri and F. Mainardi), Springer Verlag , Wien.

[3] Oldham Keith B. and Spanier Jerome.( 2002), The fractional Calculus, 2nd

Ed,

Dover Publications , INC, Mineola, New York.

[4] I. Padlubny (1999), Fractional Differential Equations, Academic Press, N.Y.

[5] Samko S. G., Kilbas, A. A. and O. I. Marichev (1993), Fractional Integrals and

Derivatives, Theory and Applications, Gordon and Breach, Amsterdam.


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Learning Outcomes

1. I knew about modern branches in Mathematics and theirapplications.

2. I reviewed many subjects such as Laplace Transforms, Gammaand Beta functions, double integrals and techniques ofintegration. I learned how to use many mathematical skills.

3. I learned new subjects like the error function, Fresnel integralsDawsons function, incomplete Gamma function .and Etfunction which were not taught in the undergraduate program.

4. This Project helped me to gain much knowledge in physics andgeometry.

5. Helped me in gaining critical thinking.


Documents

Introduction to Fractional Calculus Amna Al - Amri Project October 2010