introduction to fractional calculus amna al - amri project october 2010
TRANSCRIPT
-
8/8/2019 Introduction to Fractional Calculus Amna Al - Amri Project October 2010
1/29
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
-
8/8/2019 Introduction to Fractional Calculus Amna Al - Amri Project October 2010
2/29
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
-
8/8/2019 Introduction to Fractional Calculus Amna Al - Amri Project October 2010
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
-
8/8/2019 Introduction to Fractional Calculus Amna Al - Amri Project October 2010
4/29
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.
-
8/8/2019 Introduction to Fractional Calculus Amna Al - Amri Project October 2010
5/29
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.
AMNA ABDULSALAM YOUNIS AL-AMRY
-
8/8/2019 Introduction to Fractional Calculus Amna Al - Amri Project October 2010
6/29
-
8/8/2019 Introduction to Fractional Calculus Amna Al - Amri Project October 2010
7/29
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
-
8/8/2019 Introduction to Fractional Calculus Amna Al - Amri Project October 2010
8/29
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)
-
8/8/2019 Introduction to Fractional Calculus Amna Al - Amri Project October 2010
9/29
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)
-
8/8/2019 Introduction to Fractional Calculus Amna Al - Amri Project October 2010
10/29
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
-
8/8/2019 Introduction to Fractional Calculus Amna Al - Amri Project October 2010
11/29
-
8/8/2019 Introduction to Fractional Calculus Amna Al - Amri Project October 2010
12/29
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)
-
8/8/2019 Introduction to Fractional Calculus Amna Al - Amri Project October 2010
13/29
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
= >
-
8/8/2019 Introduction to Fractional Calculus Amna Al - Amri Project October 2010
14/29
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
-
8/8/2019 Introduction to Fractional Calculus Amna Al - Amri Project October 2010
15/29
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)
-
8/8/2019 Introduction to Fractional Calculus Amna Al - Amri Project October 2010
21/29
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)
AMNA ABDULSALAM YOUNIS AL-AMRY
-
8/8/2019 Introduction to Fractional Calculus Amna Al - Amri Project October 2010
22/29
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
+
-
8/8/2019 Introduction to Fractional Calculus Amna Al - Amri Project October 2010
23/29
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
+
-
8/8/2019 Introduction to Fractional Calculus Amna Al - Amri Project October 2010
24/29
-
8/8/2019 Introduction to Fractional Calculus Amna Al - Amri Project October 2010
25/29
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)
-
8/8/2019 Introduction to Fractional Calculus Amna Al - Amri Project October 2010
26/29
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 .
-
8/8/2019 Introduction to Fractional Calculus Amna Al - Amri Project October 2010
27/29
21
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)
AMNA ABDULSALAM YOUNIS AL-AMRY
-
8/8/2019 Introduction to Fractional Calculus Amna Al - Amri Project October 2010
28/29
22
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.
-
8/8/2019 Introduction to Fractional Calculus Amna Al - Amri Project October 2010
29/29
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.
AMNA ABDULSALAM YOUNIS AL-AMRY