SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY...

85
SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor Dr. Khaled Jaber This Thesis was Submitted in Partial Fulfillment of the Requirements for the Masterโ€™s Degree of Science in Mathematics Faculty of Graduate Studies Zarqa University May, 2016

Transcript of SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY...

Page 1: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

i

SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING

CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION

By

Shadi Ahmad Al-Tarawneh

Supervisor

Dr. Khaled Jaber

This Thesis was Submitted in Partial Fulfillment of the Requirements for

the Masterโ€™s Degree of Science in Mathematics

Faculty of Graduate Studies

Zarqa University

May, 2016

Page 2: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

ii

COMMITTEE DECISION

This Thesis/Dissertation (Solving Fractional Differential Equations by Using Conformable

Fractional Derivatives Definition) was Successfully Defended and Approved on

โ€ฆโ€ฆโ€ฆโ€ฆโ€ฆโ€ฆโ€ฆโ€ฆโ€ฆ..

Examination Committee

Signature

Dr. Khaled Jaber (Supervisor)

Assoc. Prof. of Mathematics ------------------------------

------------------------------

Dr. (Member)

------------------------------

Dr. (Member)

------------------------------

Dr. (Member)

------------------------------

Page 3: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

iii

ACKNOWLEDGEMENT

In the name of Allah, the most Gracious, most Merciful.

First and foremost, I thank ALLAH for bestowing me with health, patience, and

knowledge to complete this thesis and without ALLAHโ€™s grace, we couldn't have done it.

So to ALLAH returns all the praise and gratitude.

I would like to express my gratitude to Dr. Khaled Jaber, the supervisor of my thesis,

who was a generous and instructor. I was blessed to be supervised by him. Thanks go to

him for his guidance, suggestions and invaluable encouragement throughout the

development of this research.

Also, I should thank with great respect and honor all my professors, doctors and

instructors to be taught by them.

My great gratitude is due to my parents, beloved brothers, sisters and all friends for

their encouragement, support, prayers and being always there for me.

Last, but not least, I would like to thank my friend Omar Al Nasaan and my beloved

wife Ghosoun Al Hindi for their help, support, effort and encouragement was in the end

what made this thesis possible.

Page 4: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

iv

Table of Contents

COMMITTEE DECISION ............................................................................................................... ii

ACKNOWLEDGEMENT ................................................................................................................ iii

Table of Contents ................................................................................................................................ iv

List of Symbols .................................................................................................................................. vi

List of Abbreviations ....................................................................................................................... vii

List of Figures and Tables .............................................................................................................. viii

ABSTRACT ........................................................................................................................................ ix

INTRODUCTION .............................................................................................................................. 1

Chapter one: Basic Concepts and Preliminaries ................................................................................. 3

1.1 History of Fractional Calculus ............................................................................................... 3

1.2 Some Special Functions .......................................................................................................... 3

1.2.1. Gamma Function ........................................................................................................... 4

1.2.2. The Beta Function ......................................................................................................... 8

1.2.3 Mittag-Leffler Function .............................................................................................. 10

1.3 The Popular Definitions of Fractional Derivatives/Integrals in Fractional Calculus .............. 12

1.3.1. Riemann-Liouville (RL) ................................................................................................ 13

1.3.2. M.Caputo (1967) .......................................................................................................... 13

1.3.3. Oldham and Spainer (1974) ....................................................................................... 13

1.3.4. Kolwanker and Gangel (1994) .................................................................................. 13

1.3.5. Conformable Fractional Derivative (2014) ............................................................. 13

1.4 Riemann-Liouville (R-L) Fractional Derivative and Integration ....................................... 14

1.4.1. Riemann-Liouville Fractional Integration .............................................................. 14

1.4.2. Riemann-Liouville Fractional Derivative ............................................................... 19

1.5 Caputo Fractional Operator .................................................................................................. 25

1.6 Comparison between Riemann-Liouville and Caputo Fractional Derivative Operators . 36

1.7 Ordinary Differential Equations ........................................................................................... 39

1.7.1. Bernoulli Differential Equation ................................................................................ 39

1.7.2 Second-Order Linear Differential Equations .......................................................... 39

Chapter Two: Conformable Fractional Definition ............................................................................ 41

2.1 Conformable Fractional Derivative ..................................................................................... 41

2.2. Conformable Fractional Integrals ....................................................................................... 52

2.3 Applications ............................................................................................................................ 54

Page 5: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

v

2.4. Abelโ€™s Formula and Wronskain for Conformable Fractional Differential Equation ...... 55

2.4.1. The Wronskain ............................................................................................................. 56

2.4.2. Abelโ€™s Formula ............................................................................................................ 57

Chapter 3: Exact Solution of Riccati Fractional Differential Equation ............................................. 59

3.1 Fractional Riccati Differential Equation (FRDE) ............................................................... 59

3.2 Applications: .......................................................................................................................... 67

Future Work ...................................................................................................................................... 70

Conclusions ....................................................................................................................................... 71

REFERENCES.................................................................................................................................. 72

Abstract in Arabic ............................................................................................................................. 76

Page 6: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

vi

List of Symbols

Symbol Denoted

โ„• The set of Natural Numbers

โ„ The set of Real Numbers

๐›พ(๐‘ , ๐‘ฅ) The Lower Incomplete Gamma Function

๐œ“(๐‘ฅ) The Digamma Function

๐ต๐‘ฅ(๐‘Ž, ๐‘) The Incomplete Beta Function

๐ธ๐›ผ,๐›ฝ(๐‘ง) The Two-Parameters Mittage-Leffler Function

๐ธ๐›ผ,๐›ฝ(๐‘˜)(๐‘ฅ) The k-th Derivative of Mittage-Leffler Function

๐ท๐‘โˆ’๐‘๐‘“(๐‘ฅ) The Riemann-Liouville Fractional Integral

๐ท๐‘๐‘๐‘“(๐‘ฅ) The Riemann-Liouville Fractional Derivative

๐ท๐‘ ๐‘Žโˆ’๐‘๐‘“(๐‘ฅ) The Caputo Fractional Derivative

๐‘‡๐›ผ๐‘“(๐‘ฅ) The Conformable Fractional Derivative

๐ฝ๐›ผ๐‘“(๐‘ฅ) The Conformable Fractional Integral

ฮ“(๐‘ฅ) Gamma Function

Page 7: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

vii

List of Abbreviations

Abbreviation Denoted

R-L Riemann-Liouville

FDEs Fractional Differential Equations

FRDE Fractional Riccati Differential Equation

Page 8: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

viii

List of Figures and Tables

Figure/Table Page

Figure 1 : Graph of Gamma Function ฮ“(๐‘ฅ) 5

Table 1: Comparison between Riemann-Liouville and Caputo

38

Page 9: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

ix

SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING

CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION

By

Shadi Ahmad Al-Tarawneh

Supervisor

Dr. Khaled Jaber

ABSTRACT

Ordinary and partial fractional differential equations are very important in many fields

like Fluid Mechanics, Biology, Physics, Optics, Electrochemistry of Corrosion,

Engineering, Viscoelasticity, Electrical Networks and Control Theory of Dynamic Systems.

The fractional Ricatti equation is studied by many researchers by using different

numerical methods. Our interest in solving fractional differential equations began when

Prof. Khalil, et al., presented a new simpler and more efficient definition of fractional

derivative. The new definition reflects a nature extension of normal derivative which is

called โ€œconformable fractional derivativeโ€.

In this thesis, we found an exact solution to the fractional Ricatti differential equation,

and we introduced some theorems which lead us to find a second solution when we have a

given particular solution.

Page 10: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

1

INTRODUCTION

The sense of differentiation operator ๐ท = ๐‘‘ ๐‘‘๐‘ฅโ„ is known to all who went through

ordinary calculus. And for proper function ๐‘“, the ๐‘›โ€™th derivative of ๐‘“, namely ๐ท๐‘›๐‘“(๐‘ฅ) =

๐‘‘๐‘›๐‘“(๐‘ฅ)๐‘‘๐‘ฅ๐‘›โ„ is well defined where ๐‘› is positive integer.

The beginning of derivative theory of non-integers order dates back to leibnizโ€™s note in

his letter to Lโ€™Hopital, dated 30 September 1695 [4, 5]. He questioned that what would it

mean if the derivative of one half is discussed [4, 5, 10]. Ever after the fractional calculus

has got the interest, such as Euler, Laplace, Fourier, Abel, Liouville, Rieman, and Lauraut.

Since three centuries, fractional calculus became the traditional calculus but not very

common amongst science and engineering community. This field of applied mathematics

translates the reality of nature better! Therefore, to make this field ready as prevalent

subject to science and engineering community, add another dimension to understand or

describe basic nature in accessible way. Possibly factional calculus is what nature

comprehend and to talk with nature in this language is more effective [4].Fractional

calculus was a theoretical since till some economies and engineering applications involve

fractional differential equations [4].

Most fractional differential equations (FDEs) donโ€™t have exact solution, so

approximate and numerical techniques [6, 24, 25] must be used. Various numerical and

approximate methods to solve the FDEs have been discussed as variational iteration

method [9], homotopy perturbation method [24], Adomainโ€™s decomposition method [32],

Page 11: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

2

homotopy analysis method [31], collocation method [12, 13, 28] and finite difference

method [26, 27, 29].

Riccati differential equation refers to the Italian Nobleman Count Jacopo Francesco

Riccati (1676-1754). The fractional Riccati equation is studied by many researchers using

different numerical methods [15, 18, 20].

Recently, Khalil, et al. [14] introduced a new definition of fractional calculus which is

simpler and more efficient. The new definition reflects a nature extension of normal

derivative which is called โ€œconformable fractional derivativeโ€.

The objective of the present thesis is to use conformable fractional derivative to solve

fractional differential equation, specifically, fractional Riccate differential equation.

The thesis is organized as follows, chapter one contains seven sections, and each

handles a preliminary concept of some important special functions and some basic

information about linear differential equation. Also this chapter gives the two familiar

operators of fractional calculus which are: Rieman-Liouville (R-L) and Caputo operators

and study several important rules, as well as, the differences between these operators.

Chapter two focuses on a new definition of โ€œconformable fractional derivativesโ€ and

studies the rules of differentiation and integration.

In chapter three we found an exact solution of fractional Riccati differential equation

and introduced some theorems which lead us to find a second solution when we have a

given particular solution.

Page 12: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

3

Chapter one: Basic Concepts and Preliminaries

This chapter shows popular fractional derivatives presented by Riemann-liouville

(R-L) and Caputoโ€™s fractional differential operators and their properties. At first, it is

needed to introduce some special functions like Gamma function, Beta function and

Mittage-Leffler and their properties, then we will introduce some basic differential

equations of first order.

1.1 History of Fractional Calculus

The history of โ€œfractional derivativeโ€ started in 1695 by Lโ€™Hopetal, when he

questioned Leibniz what would it mean ๐ท๐‘›๐‘ฅ

๐ท๐‘ฅ๐‘› if ๐‘› =

1

2 in his letter, Leibniz answered that

would be a paradox. This was the beginning of โ€œfractional derivativeโ€ and influence on

this new concept to a number of mathematicians like Laplace, Euler, Fourier, Lacroix,

Riemann, Abel and Liouville. Lacroix was the first mathematician who released a

paper mentioning fractional derivatives in it. He began with the polynomial ๐‘“(๐‘ฅ) =

๐‘ฅ๐‘š, where m is a positive integer, and differentiated it n times where ๐‘š โ‰ฅ ๐‘› to get

๐ท๐‘“

๐ท๐‘ฅ=

๐‘š!

(๐‘šโˆ’๐‘›)! ๐‘ฅ๐‘šโˆ’๐‘› ,then he used Legendres symbol ฮ“ to have

๐ท๐‘›๐‘“

๐ท๐‘ฅ๐‘›=

ฮ“(๐‘š+1)

ฮ“(๐‘šโˆ’๐‘›+1)๐‘ฅ๐‘šโˆ’๐‘› .

Using this formula when ๐‘š = 1 ๐‘Ž๐‘›๐‘‘ ๐‘› =1

2 he obtained ๐ท

1

2๐‘“ =2โˆš๐‘ฅ

โˆš๐œ‹.

1.2 Some Special Functions

In this section we are going to introduce the basic definitions and properties of

the upcoming special functions: Gamma, Beta and Mittag-Leffler which are the corner

stone in fractional calculus.

Page 13: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

4

1.2.1. Gamma Function

The Gamma function is considered as an extension to the factorial function to

real and complex numbers not only integers. It plays an important role in many fields

of applied science. It has many equivalent definitions, from those, one can prove that

the Gamma function is defined for all real numbers except at ๐‘ฅ = 0,โˆ’1 ,โˆ’2 ,โ€ฆ, also

ฮ“(๐‘ฅ) has an integral representation for complex number ๐‘, where the real part of the

complex number Z is positive[17], and it can be presented in many formulas as we

will discuss below.

Definition 1.2.1. [23] (Euler, 1730) Let ๐‘ฅ > 0 The Gamma function is defined by

ฮ“(๐‘ฅ) = โˆซ(โˆ’ log(๐‘ก))๐‘ฅโˆ’1๐‘‘๐‘ก

1

0

, (1.1)

by elementary changes of variables these historical definitions take the more usual

forms:

Theorem 1.2.1. [17, 23] For ๐‘ฅ > 0,

ฮ“(๐‘ฅ) = โˆซ ๐‘ก๐‘ฅโˆ’1๐‘’โˆ’๐‘ก๐‘‘๐‘ก ,

โˆž

0

(1.2)

or sometimes

ฮ“(๐‘ฅ) = 2โˆซ ๐‘ก2๐‘ฅโˆ’1๐‘’โˆ’๐‘ก2๐‘‘๐‘ก

โˆž

0

. (1.3)

Proof: Use respectively the changes of variable ๐‘ข = โˆ’log (๐‘ก) and ๐‘ข2 = โˆ’ log(๐‘ก) in

(1.1)

Page 14: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

5

Figure 1 : Graph of Gamma Function ฮ“(๐‘ฅ)

Another definition of the Gamma function was written in a letter from Euler to

his friend Gold bach in October 13, 1729 is shown below.

Definition 1.2.2. [23] (Euler, 1729 and Gauss, 1811) Let ๐‘ฅ > 0 , ๐‘ โˆˆ ๐‘, define:

ฮ“๐‘(๐‘ฅ) =

๐‘!. ๐‘๐‘ฅ

๐‘ฅ(๐‘ฅ + 1)โ€ฆ (๐‘ฅ + ๐‘)

= ๐‘๐‘ฅ

๐‘ฅ(1 + ๐‘ฅ 1โ„ )โ€ฆ(1 + ๐‘ฅ ๐‘โ„ )

(1.4)

Theorem 1.2.2. [23] (Weierstrass) For any real number, except the non-positive

integers {0,-1 โ€ฆ} we have the infinite product

Page 15: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

6

1

ฮ“(๐‘ฅ)= ๐‘ฅ๐‘’๐›พ๐‘ฅโˆ (1 +

๐‘ฅ

๐‘) ๐‘’โˆ’๐‘ฅ ๐‘โ„โˆž

๐‘=1 . (1.5)

where ฮณ is the Eulerโ€™s constant ฮณ =0.5772156649015328606065120900824024310421...

which is defined by: ๐›พ = ๐‘™๐‘–๐‘š๐‘โ†’โˆž (1 +1

2+โ‹ฏ+

1

๐‘โˆ’ ๐‘™๐‘œ๐‘” (๐‘)) .

Below are two important properties of Gamma function.

Theorem 1.2.3. [16, 17, 23] let ๐‘ฅ โ‰  0, ๐‘› โˆˆ โ„•, then:

1. ฮ“(n + 1) = n! (1.6)

2. ฮ“(๐‘ฅ) = ฮ“(๐‘ฅ+1)

๐‘ฅ, for negative value of x . (1.7)

3. ฮ“(๐‘ฅ)ฮ“(1 โˆ’ ๐‘ฅ) = ๐œ‹

๐‘ ๐‘–๐‘›(๐œ‹๐‘ฅ) . (1.8)

4. ๐‘‘๐‘›

๐œ•๐‘ฅ๐‘›ฮ“(๐‘ฅ) = โˆซ ๐‘ก๐‘ฅโˆ’1๐‘’โˆ’๐‘ก(๐‘™๐‘› ๐‘ก)๐‘›๐‘‘๐‘ก

โˆž

0

, ๐‘ฅ > 0 . (1.9)

5. ฮ“(๐‘ฅ) = ๐‘ฅโˆ’1โˆ (1 +1

๐‘›)๐‘ฅ

(1 +๐‘ฅ

๐‘›)โˆ’1

โˆž๐‘›=1 . (1.10)

6. ฮ“ (1

2+ ๐‘ง)ฮ“ (

1

2โˆ’ ๐‘ง) = ๐œ‹ sec ๐œ‹๐‘ง. (1.11)

7. 1

ฮ“(๐‘ง)= ๐‘ง lim

๐‘›โ†’โˆž{๐‘›โˆ’๐‘งโˆ (1 +

๐‘ง

๐‘˜)

๐‘›

๐‘˜=1

} (1.12)

From the above we can get:

(a) ฮ“ (1

2) = โˆš๐œ‹

(b) ฮ“ (5

2) =

3

2ฮ“ (3

2) =

3

2.1

2ฮ“ (1

2) =

3

4โˆš๐œ‹

Page 16: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

7

(c) ฮ“ (โˆ’3

2) =

ฮ“(โˆ’3 2โ„ + 1)

โˆ’32โ„

=ฮ“(โˆ’12 )

โˆ’32โ„=

ฮ“(12)

โˆ’32 โˆ™

โˆ’12

=4

3โˆš๐œ‹

Definition 1.2.3. The lower incomplete Gamma function is defined by [17, 19]:

๐›พ(๐‘ , ๐‘ฅ) = โˆซ๐‘ก๐‘ โˆ’1๐‘’โˆ’๐‘ก๐‘ฅ

0

. ๐‘‘๐‘ก (1.13)

and the upper incomplete Gamma function

๐›ค(๐‘ , ๐‘ฅ) = โˆซ ๐‘ก๐‘ โˆ’1๐‘’โˆ’๐‘ก. ๐‘‘๐‘ก

โˆž

๐‘ฅ

(1.14)

The Relation between Gamma function and incomplete Gamma function is given by

[17].

(a) ๐›พ(๐‘ , ๐‘ฅ) =โˆ‘

๐‘ฅ๐‘ ๐‘’โˆ’๐‘ฅ๐‘ฅ๐‘˜

๐‘ (๐‘  + 1)โ€ฆ(๐‘  + ๐‘˜)

โˆž

๐‘˜=0

= ๐‘ฅ๐‘ ๐›ค(๐‘ )๐‘’โˆ’๐‘ฅโˆ‘๐‘ฅ๐‘˜

๐›ค(๐‘  + ๐‘˜ + 1)

โˆž

๐‘˜=0

(1.15)

(b) ๐‘™๐‘–๐‘š๐‘ฅโ†’โˆž

๐›พ(๐‘ , ๐‘ฅ) = ฮ“(๐‘ ) (1.16)

(c) ๐›พ(๐‘ , ๐‘ฅ) + ฮ“(๐‘ , ๐‘ฅ) = ฮ“(๐‘ ) (1.17)

Definition 1.2.4. The Digamma function ฯˆ(x) is defined by [17]

๐œ“(๐‘ฅ) =

๐‘‘

๐‘‘๐‘ฅ๐‘™๐‘› ฮ“(๐‘ฅ) =

๐›คโ€ฒ(๐‘ฅ)

๐›ค(๐‘ฅ) (1.18)

Here are some properties of Digamma functions:

Page 17: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

8

1) ๐œ“(๐‘ง + ๐‘›) = ๐œ“(๐‘ง) +1

๐‘ง+

1

๐‘ง + 1+โ‹ฏ+

1

๐‘ง + ๐‘› โˆ’ 1 (1.19)

2) ๐œ“(๐‘ง) โˆ’ ๐œ“(1 โˆ’ ๐‘ง) =โˆ’๐œ‹

tan(๐œ‹๐‘ง) (1.20)

1.2.2. The Beta Function

The Beta function is useful function related to the Gamma functions. It is defined

for ๐‘ฅ > 0 and ๐‘ฆ > 0 by the two equivalent identities:

Definition 1.2.5. [23] The Beta function (or Eulerian integral of the first kind) is given

by

๐ต(๐‘ฅ, ๐‘ฆ) = โˆซ ๐‘ก๐‘ฅโˆ’1(1 โˆ’ ๐‘ก)๐‘ฆโˆ’1๐‘‘๐‘ก1

0; 0 โ‰ค ๐‘ก โ‰ค 1 (1.21)

= 2 โˆซ ๐‘ ๐‘–๐‘›(๐‘ก)2๐‘ฅโˆ’1 ๐‘๐‘œ๐‘ (๐‘ก)2๐‘ฆโˆ’1 ๐‘‘๐‘ก

๐œ‹ 2โ„

0

; 0 โ‰ค ๐‘ก โ‰ค๐œ‹

2

This definition is also applicable for complex numbers ๐‘ฅ and ๐‘ฆ such as ๐‘…๐‘’(๐‘ฅ) > 0

and ๐‘…๐‘’(๐‘ฆ) > 0, and Euler gave (1.22) in 1730. The name of Beta function was

introduced for the first time by Jacques Binet (1786-1856) in (1839) [23] and he

provided many achievements on the subject.

The Beta function is symmetric as will be shown in the next theorem:

Theorem 1.2.5. let ๐‘…๐‘’(๐‘ฅ) > 0 and ๐‘…๐‘’(๐‘ฆ) > 0 , Then

๐ต(๐‘ฅ, ๐‘ฆ) =

ฮ“(๐‘ฅ)ฮ“(๐‘ฆ)

ฮ“(๐‘ฅ + ๐‘ฆ)= ๐ต(๐‘ฆ, ๐‘ฅ) (1.22)

Proof: by using the definite integral (1.3)

Page 18: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

9

ฮ“(๐‘ฅ)ฮ“(๐‘ฆ) = 4โˆซ ๐‘ข2๐‘ฅโˆ’1๐‘’โˆ’๐‘ข2๐‘‘๐‘ขโˆซ ๐‘ฃ2๐‘ฆโˆ’1๐‘’โˆ’๐‘ฃ

2๐‘‘๐‘ฃ

โˆž

0

โˆž

0

= 4โˆซ โˆซ ๐‘’โˆ’(๐‘ข2+๐‘ฃ2) ๐‘ข2๐‘ฅโˆ’1๐‘ฃ2๐‘ฆโˆ’1

โˆž

0

๐‘‘๐‘ข๐‘‘๐‘ฃ

โˆž

0

Now by using the polar variables ๐‘ข = ๐‘Ÿ cos๐œƒ and ๐‘ฃ = ๐‘Ÿ sin ๐œƒ so that,

ฮ“(๐‘ฅ)ฮ“(๐‘ฆ) = 4โˆซ โˆซ ๐‘’โˆ’๐‘Ÿ2

๐œ‹ 2โ„

0

๐‘Ÿ2(๐‘ฅ+๐‘ฆ)โˆ’1 cos2xโˆ’1 ๐œƒ sin2yโˆ’1 ๐œƒ ๐‘‘๐‘Ÿ๐‘‘๐œƒ

โˆž

0

= 2โˆซ ๐‘’โˆ’๐‘Ÿ2

โˆž

0

๐‘Ÿ2(๐‘ฅ+๐‘ฆ)โˆ’1๐‘‘๐‘Ÿ. 2 โˆซ cos2xโˆ’1 ๐œƒ sin2yโˆ’1 ๐œƒ๐‘‘๐œƒ

๐œ‹ 2โ„

0

= ฮ“(x + y)B(x, y) โˆŽ

From relation (1.23) follows

๐ต(๐‘ฅ + 1, ๐‘ฆ) =ฮ“(๐‘ฅ + 1)ฮ“(๐‘ฆ)

ฮ“(๐‘ฅ + ๐‘ฆ + 1)=

xฮ“(๐‘ฅ)ฮ“(๐‘ฆ)

(x + y)ฮ“(๐‘ฅ + ๐‘ฆ)=

๐‘ฅ

๐‘ฅ + ๐‘ฆ๐ต(๐‘ฅ, ๐‘ฆ)

This is the beta function functional equation

๐ต(๐‘ฅ + 1, ๐‘ฆ) =๐‘ฅ

๐‘ฅ + ๐‘ฆ๐ต(๐‘ฅ, ๐‘ฆ) (1.23)

Definition 1.2.6. The incomplete Beta function ๐ต๐œ(๐‘ฅ, ๐‘ฆ)is defined by:

๐ต๐œ(๐‘ฅ, ๐‘ฆ) = โˆซ ๐‘ก๐‘ฅโˆ’1(1 โˆ’ ๐‘ก)๐‘ฆโˆ’1. ๐‘‘๐‘ก ,

๐œ

0

0 < ๐œ < 1 (1.24)

Note that from the above:

๐ต (1

2,1

2) = ๐œ‹

Page 19: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

10

๐ต (1

3,2

3) =

2 โˆš3

3๐œ‹

๐ต (1

4,3

4) = ๐œ‹ โˆš2

๐ต(๐‘ฅ, 1 โˆ’ ๐‘ฅ) = ๐œ‹

sin (๐œ‹๐‘ฅ)

๐ต(๐‘ฅ, 1) = 1

๐‘ฅ

๐ต(๐‘ฅ, ๐‘›) = (๐‘› โˆ’ 1)!

๐‘ฅ. (๐‘ฅ + 1)โ€ฆ (๐‘ฅ + ๐‘› โˆ’ 1) ๐‘› โ‰ฅ 1

๐ต(๐‘š, ๐‘›) = (๐‘š โˆ’ 1)! (๐‘› โˆ’ 1)!

(๐‘š + ๐‘› โˆ’ 1)! ๐‘š โ‰ฅ 1 , ๐‘› โ‰ฅ 1

1.2.3 Mittag-Leffler Function

The Mittag-Leffler function is a generalization of the exponential function and it is

one of the most important functions that are related to fractional differential equations.

Definition 1.2.7. [3, 5, 17] The one and two-parameter Mittag-Leffler functions are

defined, respectively, by:

๐ธ๐‘Ž(๐‘ฅ) = โˆ‘

๐‘ฅ๐‘›

ฮ“(๐‘Ž๐‘› + 1) , ๐‘Ž > 0

โˆž

๐‘›=0

(1.25)

๐ธ๐‘Ž,๐‘(๐‘ฅ) = โˆ‘

๐‘ฅ๐‘›

ฮ“(๐‘Ž๐‘› + ๐‘) , ๐‘Ž > 0, ๐‘ > 0

โˆž

๐‘›=0

(1.26)

If ๐‘Ž = 1 and ๐‘ โˆˆ โ„•

Page 20: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

11

๐ธ1,1(๐‘ฅ) = โˆ‘

๐‘ฅ๐‘›

ฮ“(๐‘› + 1)=

โˆž

๐‘›=0

โˆ‘๐‘ฅ๐‘›

n!

โˆž

๐‘›=0

= ๐‘’๐‘ฅ (1.27)

๐ธ1,2(๐‘ฅ) = โˆ‘

๐‘ฅ๐‘›

ฮ“(๐‘› + 2)

โˆž

๐‘›=0

=โˆ‘๐‘ฅ๐‘›

(๐‘› + 1)!

โˆž

๐‘›=0

=1

๐‘ฅโˆ‘

๐‘ฅ๐‘›+1

(n + 1)!

โˆž

๐‘›=0

=๐‘’๐‘ฅ โˆ’ 1

๐‘ฅ

(1.28)

๐ธ1,3(๐‘ฅ) = โˆ‘

๐‘ฅ๐‘›

ฮ“(๐‘› + 3)

โˆž

๐‘›=0

=โˆ‘๐‘ฅ๐‘›

(๐‘› + 2)!

โˆž

๐‘›=0

=1

๐‘ฅ2โˆ‘

๐‘ฅ๐‘›+2

(n + 2)!

โˆž

๐‘›=0

=๐‘’๐‘ฅ โˆ’ 1โˆ’ ๐‘ฅ

๐‘ฅ2

(1.29)

In general,

๐ธ1,๐‘š =1

๐‘ฅ๐‘šโˆ’1{๐‘’๐‘ฅ โˆ’ โˆ‘

๐‘ฅ๐‘›

n!

๐‘šโˆ’2

๐‘›=0

} (1.30)

Easily, we can obtain the following:

(a) ๐ธ2,1(๐‘ฅ2) = โˆ‘

๐‘ฅ2๐‘›

๐›ค(2๐‘› + 1)

โˆž

๐‘›=0

=โˆ‘๐‘ฅ2๐‘›

(2๐‘›)!

โˆž

๐‘›=0

= ๐‘๐‘œ๐‘ โ„Ž(๐‘ฅ) (1.31)

(b) ๐ธ2,2(๐‘ฅ2) = โˆ‘

๐‘ฅ2๐‘›

๐›ค(2๐‘› + 2)

โˆž

๐‘›=0

=โˆ‘๐‘ฅ2๐‘›+1

๐‘ฅ(2๐‘› + 1)!

โˆž

๐‘›=0

=๐‘ ๐‘–๐‘›โ„Ž(๐‘ฅ)

๐‘ฅ (1.32)

(c) ๐ธ2,1(โˆ’๐‘ฅ2) = โˆ‘

(โˆ’๐‘ฅ2)๐‘›

ฮ“(2๐‘› + 1)

โˆž

๐‘›=0

=โˆ‘(โˆ’1)๐‘›๐‘ฅ2๐‘›

(2๐‘›)!

โˆž

๐‘›=0

= ๐‘๐‘œ๐‘ (๐‘ฅ) (1.33)

(d) ๐ธ2,2(โˆ’๐‘ฅ2) = โˆ‘

(โˆ’๐‘ฅ2)๐‘›

ฮ“(2๐‘› + 2)

โˆž

๐‘›=0

=โˆ‘(โˆ’1)๐‘›๐‘ฅ2๐‘›+1

๐‘ฅ(2๐‘› + 1)!

โˆž

๐‘›=0

=๐‘ ๐‘–๐‘›(๐‘ฅ)

๐‘ฅ (1.34)

The Mittage-Leffler function has the following relations :

Page 21: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

12

๐ธ๐‘Ž,๐‘(๐‘ฅ) = ๐‘ฅ ๐ธ๐‘Ž,๐‘Ž+๐‘(๐‘ฅ) +

1

๐›ค(๐‘) (1.35)

๐ธ๐‘Ž,๐‘(๐‘ฅ) = ๐‘๐ธ๐‘Ž,๐‘+1(๐‘ฅ) + ๐‘Ž๐‘ฅ

๐‘‘

๐‘‘๐‘ฅ ๐ธ๐‘Ž,๐‘+1(๐‘ฅ) (1.36)

Obviously, from (1.36) we have

๐‘‘

๐‘‘๐‘ฅ ๐ธ๐‘Ž,๐‘(๐‘ฅ) =

1

๐‘Ž๐‘ฅ [๐ธ๐‘Ž,๐‘โˆ’1(๐‘ฅ) โˆ’ (๐‘ โˆ’ 1) ๐ธ๐‘Ž,๐‘(๐‘ฅ) ] (1.37)

The ๐‘š-th derivative of Mittage-Leffler function is given as follows:

๐‘‘๐‘š

๐‘‘๐‘ฅ๐‘š[๐‘ฅ๐‘โˆ’1 ๐ธ๐‘Ž,๐‘(๐‘ฅ

๐‘Ž)] = ๐‘ฅ๐‘โˆ’๐‘šโˆ’1๐ธ๐‘Ž,๐‘โˆ’๐‘š(๐‘ฅ๐‘Ž) , ๐‘ โˆ’ ๐‘š > 0 , ๐‘š = 0, 1 ,โ‹ฏ (1.38)

The integration of the Mittage-Leffler function is given as follows:

โˆซ ๐ธ๐‘Ž,๐‘(๐œ† ๐‘ก

๐‘Ž)๐‘ก๐‘โˆ’1๐‘ฅ

0

๐‘‘๐‘ก = ๐‘ฅ๐‘๐ธ๐‘Ž,๐‘+1(๐œ† ๐‘ฅ๐‘Ž) (1.39)

The relation (1.39) is a special case and the following relation is more general:

1

ฮ“(๐‘ฃ)โˆซ (๐‘ฅ โˆ’ ๐‘ก)๐‘ฃโˆ’1๐‘ฅ

0

๐ธ๐‘Ž,๐‘(๐œ† ๐‘ก๐‘Ž)๐‘ก๐‘โˆ’1๐‘‘๐‘ก = ๐‘ฅ๐‘+๐‘ฃโˆ’1 ๐ธ๐‘Ž,๐‘+๐‘ฃ(๐œ† ๐‘ฅ

๐‘Ž) , ๐‘ฃ > 0 (1.40)

From (1.40) we obtain the following important formulas:

1

ฮ“(๐‘)โˆซ (๐‘ฅ โˆ’ ๐‘ก)๐‘โˆ’1๐‘’๐‘Ž๐‘ก ๐‘‘๐‘ก = ๐‘ฅ๐‘ ๐ธ1,๐‘+1(๐‘Ž๐‘ฅ) , ๐‘ > 0 ๐‘ฅ

0

(1.41)

1

ฮ“(๐‘)โˆซ (๐‘ฅ โˆ’ ๐‘ก)๐‘โˆ’1 cosh(๐‘Ž๐‘ก) ๐‘‘๐‘ก = ๐‘ฅ๐‘ ๐ธ2,๐‘+1((๐‘Ž๐‘ฅ)

2) , ๐‘ > 0 ๐‘ฅ

0

(1.42)

1

๐›ค(๐‘)โˆซ (๐‘ฅ โˆ’ ๐‘ก)๐‘โˆ’1 ๐‘ ๐‘–๐‘›โ„Ž(๐‘Ž๐‘ก) ๐‘‘๐‘ก = ๐‘Ž ๐‘ฅ๐‘+1 ๐ธ2,๐‘+2((๐‘Ž๐‘ฅ)

2) , ๐‘ > 0 ๐‘ฅ

0

(1.43)

1.3 The Popular Definitions of Fractional Derivatives/Integrals in

Fractional Calculus

In this section we listed the popular definition of fractional calculus [3]:

Page 22: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

13

1.3.1. Riemann-Liouville (RL) [3, 4, 5]:

๐ท๐‘ก๐›ผ

๐‘Ž ๐‘“(๐‘ก) =1

๐›ค(๐‘› โˆ’ ๐›ผ)(๐‘‘

๐‘‘๐‘ก)๐‘›

โˆซ๐‘“(๐‘ฅ)

(๐‘ก โˆ’ ๐‘ฅ)๐›ผโˆ’๐‘›+1๐‘‘๐‘ฅ

๐‘ก

๐‘Ž

(1.44)

(๐‘› โˆ’ 1) โ‰ค ๐›ผ < ๐‘› ,where ๐›ผ is a real number, ๐‘› is integer.

1.3.2. M.Caputo (1967) [3,4]:

๐ท๐‘ก๐›ผ

๐‘Ž๐‘ ๐‘“(๐‘ก) =

1

๐›ค(๐‘› โˆ’ ๐›ผ)โˆซ

๐‘“(๐‘›)(๐‘ฅ)

(๐‘ก โˆ’ ๐‘ฅ)๐›ผ+1โˆ’๐‘›๐‘‘๐‘ฅ

๐‘ก

๐‘Ž

(1.45)

(๐‘› โˆ’ 1) โ‰ค ๐›ผ < ๐‘› , where ๐›ผ is a real number and ๐‘› is integer

1.3.3. Oldham and Spainer (1974) [4]:

The scaling property for fractional derivatives

๐‘‘๐‘ž๐‘“(๐›ฝ๐‘ฅ)

๐‘‘๐‘ฅ๐‘ž= ๐›ฝ๐‘ž

๐‘‘๐‘ž๐‘“(๐›ฝ๐‘ฅ)

๐‘‘(๐›ฝ๐‘ฅ)๐‘ž (1.46)

1.3.4. Kolwanker and Gangel (1994) [4]:

Kolwanker and Gangel (KG) defined a local fractional derivative to explain the

behavior of โ€œcontinuous but nowhere differentiableโ€ function for 0 < ๐‘ < 1 , the local

fractional derivative at point ๐‘ฅ = ๐‘ฆ , for ๐‘“: [0,1] โ†’ โ„ is:

๐ท๐‘๐‘“(๐‘ฆ) =

๐‘‘๐‘(๐‘“(๐‘ฅ) โˆ’ ๐‘“(๐‘ฆ))

๐‘‘(๐‘ฅ โˆ’ ๐‘ฆ)๐‘ (1.47)

1.3.5. Conformable Fractional Derivative (2014) [4]:

let ๐‘“: [0,โˆž) โ†’ ๐‘… , ๐‘ก > 0 , then the Conformable fractional derivative of ๐‘“ of order ๐›ผ is

defined by

Page 23: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

14

๐‘‡๐›ผ(๐‘“)(๐‘ก) = ๐‘™๐‘–๐‘š

๐œ€โ†’0

๐‘“(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ) โˆ’ ๐‘“(๐‘ก)

๐œ€ (1.48)

for all ๐‘ก > 0 , ๐›ผ โˆˆ (0,1)

1.4 Riemann-Liouville (R-L) Fractional Derivative and Integration

In this section, we listed some presentations, rules and properties of Riemann-

Liouville integration and differentiation and their proofs.

1.4.1. Riemann-Liouville Fractional Integration

We need to use the following fact to define the fractional integration of Riemann-

Liouville:

If ๐‘“ is an integrable function on[๐‘Ž, ๐‘], then for ๐‘› โˆˆ โ„• and for ๐‘ฅ โˆˆ [๐‘Ž, ๐‘], we have

๐ท๐›ผโˆ’๐‘›๐‘“(๐‘ฅ) =

1

(๐‘› โˆ’ 1)!โˆซ(๐‘ฅ โˆ’ ๐‘ก)๐‘›โˆ’1๐‘“(๐‘ก)

๐‘ฅ

๐‘Ž

๐‘‘๐‘ก. (1.49)

By using (๐‘› โˆ’ 1)! = ฮ“(๐‘›) and if we replace the order (๐‘›) by the order(๐‘),

where ๐‘ โˆˆ ๐‘…, then we get the following definition:-

Definition 1.4.1. [3, 4, 19, 22] let ๐‘“(๐‘ฅ) be a piecewise continuous on ๐œ‡ = (0,โˆž) and

intergrable on any finite subinterval of ๐œ‡โ€ฒ = [0,โˆž) and for ๐‘ > 0 , ๐‘ฅ > 0 we call

๐ทโˆ’๐‘๐‘“(๐‘ฅ) =1

ฮ“(๐‘)โˆซ(๐‘ฅ โˆ’ ๐‘ก)๐‘โˆ’1๐‘“(๐‘ก) ๐‘‘๐‘ก

๐‘ฅ

0

(1.50)

The Riemann-Liouville fractional integral of order ๐‘ of ๐‘“

Remark 1.4.1. [19, 22] The Riemannโ€™s definition is given by:

Page 24: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

15

๐ท๐‘โˆ’๐‘๐‘“(๐‘ฅ) =

1

ฮ“(๐‘)โˆซ(๐‘ฅ โˆ’ ๐‘ก)๐‘โˆ’1๐‘“(๐‘ก) ๐‘‘๐‘ก.

๐‘ฅ

๐‘

(1.51)

where ๐‘ > 0 , ๐‘ฅ > ๐‘

The Liouvilleโ€™s definition is given by:

๐ทโˆ’โˆžโˆ’๐‘ ๐‘“(๐‘ฅ) =

1

ฮ“(๐‘) โˆซ(๐‘ฅ โˆ’ ๐‘ก)๐‘โˆ’1๐‘“(๐‘ก)๐‘‘๐‘ก.

๐‘ฅ

โˆ’โˆž

where p > 0

Note that we use the symbol ๐ทโˆ’๐‘๐‘“(๐‘ฅ) instead of ๐ท0โˆ’๐‘๐‘“(๐‘ฅ) when the lower limit of

the integral equals zero.

Properties 1.4.1. [19,22] If ๐‘“(๐‘ฅ) and โ„Ž(๐‘ฅ) are continuous functions a, ๐‘ โˆˆ ๐‘… , and

๐‘›,๐‘š > 0 , then:

๐ท๐‘Žโˆ’๐‘›(๐ท๐‘Ž

โˆ’๐‘š๐‘“(๐‘ฅ)) = ๐ท๐‘Žโˆ’๐‘š(๐ท๐‘Ž

โˆ’๐‘›๐‘“(๐‘ฅ)) = ๐ท๐‘Žโˆ’(๐‘›+๐‘š)๐‘“(๐‘ฅ) (1.52)

๐ท๐›ผโˆ’๐‘›(๐‘Ž๐‘“(๐‘ฅ) + ๐‘โ„Ž(๐‘ฅ)) = ๐‘Ž๐ท๐›ผ

โˆ’๐‘›๐‘“(๐‘ฅ) + ๐‘๐ท๐›ผโˆ’๐‘›โ„Ž(๐‘ฅ) (1.53)

Theorem 1.4.1. [19,22] (Basic rules of Riemann-Liouville fractional integral)

Let ๐‘ > 0 , ๐‘ฅ > 0, then

1. ๐ทโˆ’๐‘๐‘ฅ๐œ‡ =ฮ“(๐œ‡ + 1)

ฮ“(๐‘ + ๐œ‡ + 1)๐‘ฅ๐‘+๐œ‡ , ๐œ‡ > โˆ’1 (1.54)

2. ๐ทโˆ’๐‘๐‘ =๐‘

ฮ“(๐‘ + 1)๐‘ฅ๐‘ , ๐‘ is a constant (1.55)

Page 25: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

16

3. ๐ทโˆ’๐‘๐‘’๐‘๐‘ฅ =

๐‘’๐‘๐‘ฅ

๐‘๐‘๐›ค(๐‘)๐›พ(๐‘, ๐‘๐‘ฅ) , ๐‘Ž > 0

where ฮณ(p,cx) is the lower incomplete Gamma functions

(1.56)

4. ๐ทโˆ’๐‘(sin ๐‘๐‘ฅ) = ๐‘๐‘ฅ๐‘+1๐ธ2,๐‘+2 (โˆ’(๐‘๐‘ฅ )2) (1.57)

5. ๐ทโˆ’๐‘(cos ๐‘๐‘ฅ) = ๐‘ฅ๐‘+1๐ธ2,๐‘+1 (โˆ’(๐‘๐‘ฅ )2) (1.58)

6. ๐ทโˆ’๐‘(cosh ๐‘๐‘ฅ) = ๐‘ฅ๐‘๐ธ2,๐‘+1 ((๐‘๐‘ฅ )2) (1.59)

7. ๐ทโˆ’๐‘(sinh ๐‘๐‘ฅ) = ๐‘๐‘ฅ๐‘+1๐ธ2,๐‘+2 ((๐‘๐‘ฅ )2) (1.60)

8. ๐ทโˆ’๐‘ ๐‘™๐‘› ๐‘ฅ =๐‘ฅ๐‘

ฮ“(๐‘ + 1)[๐‘™๐‘› ๐‘ฅ โˆ’ ๐›พ โˆ’ ๐œ“(๐‘ + 1)] (1.61)

where ๐œ“ is the digamma function and ๐›พ = โˆ’๐œ“(1) = โˆ’ฮ“โ€ฒ(1) โ‰ˆ

0.5772157 is Euler constant.

Proof:

(1) ๐ทโˆ’๐‘๐‘ฅ๐œ‡ = 1

ฮ“(๐‘)โˆซ (๐‘ฅ โˆ’ ๐‘ก)๐‘โˆ’1๐‘ก๐œ‡๐‘ฅ

0๐‘‘๐‘ก

=1

ฮ“(๐‘)โˆซ(1 โˆ’

๐‘ก

๐‘ฅ)๐‘โˆ’1

๐‘ฅ๐‘โˆ’1๐‘ก๐œ‡๐‘ฅ

0

๐‘‘๐‘ก

By substituting ๐‘ข = ๐‘ก ๐‘ฅโ„

= 1

ฮ“(๐‘)โˆซ(1 โˆ’ ๐‘ข)๐‘โˆ’1๐‘ฅ๐‘โˆ’1(๐‘ข๐‘ฅ)๐œ‡1

0

๐‘ฅ๐‘‘๐‘ข

=1

ฮ“(๐‘)โˆซ(1 โˆ’ ๐‘ข)๐‘โˆ’1๐‘ข๐œ‡๐‘ฅ๐œ‡+๐‘1

0

๐‘‘๐‘ข

Page 26: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

17

= 1

ฮ“(๐‘)๐‘ฅ๐œ‡+๐‘๐›ฃ(๐œ‡ + 1, ๐‘)

=ฮ“(๐œ‡ + 1)

ฮ“(๐‘ + ๐œ‡ + 1)๐‘ฅ๐‘+๐œ‡ โˆŽ

(2) If we set ๐œ‡ = 0 in (1.54), then the proof is complete

(3)

๐ทโˆ’๐‘๐‘’๐‘๐‘ฅ =1

ฮ“(๐‘)โˆซ(๐‘ฅ โˆ’ ๐‘ก)๐‘โˆ’1๐‘ฅ

0

๐‘’๐‘๐‘ก๐‘‘๐‘ก

=1

ฮ“(๐‘)โˆซ(

๐‘(๐‘ฅ โˆ’ ๐‘ก)

๐‘)

๐‘โˆ’1๐‘ฅ

0

๐‘’๐‘๐‘ก๐‘‘๐‘ก

=1

ฮ“(๐‘)โˆซ๐‘ข๐‘โˆ’1

๐‘๐‘โˆ’1

๐‘๐‘ฅ

0

๐‘’๐‘๐‘ฅโˆ’๐‘ข๐‘‘๐‘ข

๐‘=

๐‘’๐‘๐‘ฅ

๐‘๐‘๐›ค(๐‘)โˆซ ๐‘ข๐‘โˆ’1๐‘๐‘ฅ

0

๐‘’โˆ’๐‘ข๐‘‘๐‘ข ,

By substituting ๐‘ข = ๐‘(๐‘ฅ โˆ’ ๐‘ก)

=๐‘’๐‘๐‘ฅ

๐‘๐‘ฮ“(๐‘)๐›พ(๐‘, ๐‘๐‘ฅ) โˆŽ

(4) ๐ทโˆ’๐‘(๐‘ ๐‘–๐‘› ๐‘๐‘ฅ) =1

ฮ“(๐‘)โˆซ(๐‘ฅ โˆ’ ๐‘ก)๐‘โˆ’1๐‘ฅ

0

๐‘ ๐‘–๐‘›(๐‘๐‘ก) ๐‘‘๐‘ก

=1

ฮ“(๐‘)โˆซ(๐‘ฅ โˆ’ ๐‘ก)๐‘โˆ’1๐‘ฅ

0

๐‘ ๐‘–๐‘›(๐‘๐‘ก)

๐‘๐‘ก๐‘๐‘ก ๐‘‘๐‘ก

Simply by using (1.34) and (1.40),

๐ทโˆ’๐‘(๐‘ ๐‘–๐‘› ๐‘๐‘ฅ) = ๐‘๐‘ฅ๐‘+1๐ธ2,๐‘+2(โˆ’(๐‘๐‘ฅ)2) โˆŽ

Page 27: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

18

(5) Follows by using (1.33) and (1.40) and the same as (4).

(6) By using (1.42), we get:

๐ทโˆ’๐‘(๐‘๐‘œ๐‘ โ„Ž(๐‘๐‘ฅ)) =1

๐›ค(๐‘)โˆซ(๐‘ฅ โˆ’ ๐‘ก)๐‘โˆ’1 ๐‘๐‘œ๐‘ โ„Ž(๐‘๐‘ก) . ๐‘‘๐‘ก

๐‘ฅ

0

= ๐‘ฅ๐‘๐ธ2,๐‘+1((๐‘๐‘ฅ)2) โˆŽ

(7) Follows by using (1.43) and the same as (6).

(8) The proof can be found in [19].

Remark 1.4.2. [19,22] The fractional integral of ๐‘๐‘œ๐‘ (๐‘๐‘ฅ) , ๐‘ ๐‘–๐‘›(๐‘๐‘ฅ) can be expressed in

generalized ๐‘ ๐‘–๐‘› and ๐‘๐‘œ๐‘  functions as:

๐ทโˆ’๐‘ cos(๐‘๐‘ฅ) =1

ฮ“(๐‘)โˆซ(๐‘ฅ โˆ’ ๐‘ก)๐‘โˆ’1๐‘ฅ

0

cos(๐‘๐‘ก) ๐‘‘๐‘ก = ๐ถ๐‘ฅ(๐‘, ๐‘Ž) (1.62)

๐ทโˆ’๐‘ sin(๐‘๐‘ฅ) =1

ฮ“(๐‘)โˆซ(๐‘ฅ โˆ’ ๐‘ก)

๐‘ฅ

0

sin(๐‘๐‘ก) ๐‘‘๐‘ก = ๐‘†๐‘ฅ(๐‘, ๐‘Ž) (1.63)

Remark 1.4.3 [19, 22] We can express the fractional integral function ๐‘’๐‘๐‘ฅ by using

Mittage-Leffler function as

๐ทโˆ’๐‘๐‘’๐‘๐‘ฅ = ๐‘ฅ๐‘๐ธ1,๐‘+1(๐‘๐‘ฅ) (1.61)

Proof:

By using (1.15) and (1.27), then

Dโˆ’p๐‘’๐‘๐‘ฅ =๐‘’๐‘๐‘ฅ

๐‘๐‘ฮ“(๐‘)๐›พ(๐‘, ๐‘๐‘ฅ) =

๐‘’๐‘๐‘ฅ

๐‘๐‘ฮ“(๐‘)(๐‘๐‘ฅ)๐‘ฮ“(๐‘)๐‘’โˆ’๐‘๐‘ฅ๐ธ1,๐‘+1(๐‘๐‘ฅ)

Page 28: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

19

= ๐‘ฅ๐‘๐ธ1,๐‘+1(๐‘๐‘ฅ)

1.4.2. Riemann-Liouville Fractional Derivative

The most important approaches to define the fractional derivative is using the

integration of fractional order in the same as the following fact:

๐ท๐›ผ๐‘๐‘“ = ๐ท๐›ผ

๐‘ž(๐ท๐‘โˆ’๐‘ž๐‘“) , ๐‘, ๐‘ž โˆˆ โ„•, ๐‘ž > ๐‘

Riemann-Liouville use the later fact to introduce the following definition:

Definition 1.4.2. [3, 5, 19,22] (Riemann-Liouville fractional derivative)

The Riemann-Liouville fractional derivative of ๐‘“(๐‘ฅ) of order ๐›ผ, ๐‘› โˆ’ 1 < ๐›ผ < ๐‘›,

๐‘› โˆˆ โ„• is defined by:

๐ท๐‘Ž๐›ผ๐‘“(๐‘ฅ) = ๐ท๐‘› (๐ท๐‘Ž

โˆ’(๐‘›โˆ’๐›ผ)๐‘“(๐‘ฅ))

=๐‘‘๐‘›

๐‘‘๐‘ฅ๐‘›1

ฮ“(๐‘› โˆ’ ๐›ผ)โˆซ(๐‘ฅ โˆ’ ๐‘ก)๐‘›โˆ’๐›ผ+1๐‘ฅ

๐‘Ž

๐‘“(๐‘ก). ๐‘‘๐‘ก

(1.62)

Definition 1.4.3. [3, 5, 19, 22] Let ๐‘“(๐‘ฅ) be a function defined on the closed interval

[๐‘Ž, ๐‘] and let ๐›ผ โˆˆ [0,1), then the left Riemann-Liouville ๐›ผ derivative of ๐‘“(๐‘ฅ) is:

๐ท๐‘Ž๐›ผ๐‘“(๐‘ฅ) =

1

ฮ“(1 โˆ’ ๐›ผ)

๐‘‘

๐‘‘๐‘ฅโˆซ

๐‘“(๐‘ก)

(๐‘ฅ โˆ’ ๐‘ก)๐›ผ

๐‘ฅ

๐‘Ž

. ๐‘‘๐‘ก (1.63)

The right Riemann-Liouville ๐›ผ derivative of ๐‘“(๐‘ฅ) is

๐ท๐‘๐›ผ๐‘“(๐‘ฅ) =

โˆ’1

ฮ“(1 โˆ’ ๐›ผ)

๐‘‘

๐‘‘๐‘ฅโˆซ

๐‘“(๐‘ก)

(๐‘ก โˆ’ ๐‘ฅ)๐›ผ

๐‘

๐‘ฅ

. ๐‘‘๐‘ก (1.64)

Page 29: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

20

But when ๐›ผ is any number greater than 1. Then the definition will be as the

following

Definition 1.4.4. [3, 5, 19, 22]

Let ๐‘“(๐‘ฅ) be a function defined on the closed interval [๐‘Ž, ๐‘] and let ๐›ผ โˆˆ

[๐‘› โˆ’ 1, ๐‘›), ๐‘› โˆˆ โ„•. Then the left Riemann-Liouville ๐›ผ derivative of ๐‘“(๐‘ฅ) is:

๐ท๐‘Ž๐›ผ๐‘“(๐‘ฅ) =

1

ฮ“(๐‘› โˆ’ ๐›ผ)

๐‘‘๐‘›

๐‘‘๐‘ฅ๐‘›โˆซ

๐‘“(๐‘ก)

(๐‘ฅ โˆ’ ๐‘ก)๐›ผโˆ’๐‘›+1

๐‘ฅ

๐‘Ž

. ๐‘‘๐‘ก (1.65)

and the right Riemann-Liouville ๐›ผ derivative of ๐‘“(๐‘ฅ) is

๐ท๐‘๐›ผ๐‘“(๐‘ฅ) =

(โˆ’1)๐‘›

ฮ“(๐‘› โˆ’ ๐›ผ)

๐‘‘๐‘›

๐‘‘๐‘ฅ๐‘›โˆซ

๐‘“(๐‘ก)

(๐‘ก โˆ’ ๐‘ฅ)๐›ผโˆ’๐‘›+1

๐‘

๐‘ฅ

. ๐‘‘๐‘ก (1.66)

Note that the required condition required in the definitions is to be ๐‘›-times

continuously differentiable.

The relationship between integration and differentiation of Riemann-Liouville

operators for the arbitrary order ๐‘ are shown as follows:

The Derivative of fractional integral could be shown as:

๐ท๐›ผ๐‘ (๐ท๐›ผ

โˆ’๐‘ž๐‘“(๐‘ฅ)) = ๐ท๐‘โˆ’๐‘ž๐‘“(๐‘ฅ), (1.67)

where ๐‘“(๐‘ฅ) is continuous also ๐‘ โ‰ฅ ๐‘ž โ‰ฅ 0

precisely, when ๐‘ž โ‰ฅ 0 then ๐ท๐›ผ๐‘ž (๐ท๐›ผ

โˆ’๐‘ž๐‘“(๐‘ฅ)) = ๐‘“(๐‘ฅ) (1.68)

Page 30: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

21

Preposition 1.4.2. [19,22] Let ๐‘“1(๐‘ฅ), ๐‘“2(๐‘ฅ) be two functions defined on[๐‘Ž, ๐‘], and let

๐›ผ โˆˆ [๐‘› โˆ’ 1, ๐‘›), ๐‘› โˆˆ โ„•, ๐œ†, ๐›ฝ โˆˆ โ„‚ and ๐ท๐‘Ž๐›ผ๐‘“1(๐‘ฅ),๐ท๐‘Ž

๐›ผ๐‘“2(๐‘ฅ) exist, then

๐ท๐‘Ž๐›ผ[๐œ†๐‘“1(๐‘ฅ) + ๐›ฝ๐‘“2(๐‘ฅ)] = ๐œ†๐ท๐‘Ž

๐›ผ๐‘“1(๐‘ฅ) + ๐›ฝ๐ท๐‘Ž๐›ผ๐‘“2(๐‘ฅ) (1.69)

Proof:

๐ท๐‘Ž๐›ผ[๐œ†๐‘“1(๐‘ฅ) + ๐›ฝ๐‘“2(๐‘ฅ)] =

1

ฮ“(๐‘› โˆ’ ๐›ผ)

๐‘‘๐‘›

๐‘‘๐‘ฅ๐‘›โˆซ[๐œ†๐‘“1(๐‘ฅ) + ๐›ฝ๐‘“2(๐‘ฅ)]

(๐‘ฅ โˆ’ ๐‘ก)๐›ผโˆ’๐‘›+1

๐‘ฅ

๐‘Ž

๐‘‘๐‘ก

=๐œ†

ฮ“(๐‘› โˆ’ ๐›ผ)

๐‘‘๐‘›

๐‘‘๐‘ฅ๐‘›โˆซ

๐‘“1(๐‘ฅ)

(๐‘ฅ โˆ’ ๐‘ก)๐›ผโˆ’๐‘›+1

๐‘ฅ

๐‘Ž

๐‘‘๐‘ก +๐›ฝ

ฮ“(๐‘› โˆ’ ๐›ผ)

๐‘‘๐‘›

๐‘‘๐‘ฅ๐‘›โˆซ

๐‘“2(๐‘ฅ)

(๐‘ฅ โˆ’ ๐‘ก)๐›ผโˆ’๐‘›+1

๐‘ฅ

๐‘Ž

๐‘‘๐‘ก

= ๐œ†๐ท๐‘Ž๐›ผ๐‘“1(๐‘ฅ) + ๐›ฝ๐ท๐‘Ž

๐›ผ๐‘“2(๐‘ฅ) โˆŽ

Preposition 1.4.3. (Interpolation Property)

Let ๐‘”(๐‘ฅ) be a function defined on [๐‘Ž, ๐‘] and let ๐›ผ โˆˆ [0,1). Let ๐‘”(๐‘ฅ) have a

continuous derivative of sufficient order and ๐ท๐‘Ž๐›ผ๐‘”(๐‘ฅ) exists, then

lim๐›ผโ†’1

๐ท๐‘Ž๐›ผ๐‘”(๐‘ฅ) = ๐‘”โ€ฒ(๐‘ฅ) (1.70)

and lim๐›ผโ†’0

๐ท๐‘Ž๐›ผ๐‘” (๐‘ฅ) = ๐‘”(๐‘ฅ) (1.71)

Proof: see [22]

We can generalize the above equalities in preposition 1.4.3 for any positive

number ๐›ผ to be

lim๐›ผโ†’๐‘›

๐ท๐‘Ž๐›ผ๐‘”(๐‘ฅ) = ๐‘”(๐‘›)(๐‘ฅ) (1.72)

Page 31: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

22

and lim๐›ผโ†’๐‘›โˆ’1

๐ท๐‘Ž๐›ผ๐‘” (๐‘ฅ) = ๐‘”(๐‘›โˆ’1)(๐‘ฅ) (1.73)

where ๐›ผ โˆˆ [๐‘› โˆ’ 1, ๐‘›) , ๐‘› โˆˆ โ„• and with the same condition of the preposition (1.4.3).

Preposition1.4.4. (Some properties of Riemann-Liouville fractional derivative)

1) The integral of (Riemann-Liouville) derivative is given by

๐ท๐‘Žโˆ’๐‘(๐ท๐‘Ž

๐‘ž๐‘“(๐‘ฅ)) = ๐ท๐‘Ž

๐‘žโˆ’๐‘๐‘“(๐‘ฅ) โˆ’โˆ‘[๐ท๐›ผ

๐‘žโˆ’๐‘˜โˆ’1๐‘“(๐‘ฅ)]

๐‘ฅ=๐‘Ž

(๐‘ฅ โˆ’ ๐‘Ž)๐‘โˆ’๐‘˜โˆ’1

ฮ“(๐‘ โˆ’ ๐‘˜)

๐‘›โˆ’1

๐‘˜=0

(1.74)

where ๐‘› โˆ’ 1 < ๐‘ž < ๐‘› , ๐‘› โˆˆ โ„•

2) ๐ท๐‘Žโˆ’๐›ผ(๐ท๐‘Ž

๐›ผ๐‘“(๐‘ฅ)) = ๐‘“(๐‘ฅ) โˆ’โˆ‘[๐ท๐›ผ๐›ผโˆ’๐‘˜โˆ’1๐‘“(๐‘ฅ)]๐‘ฅ=๐‘Ž

(๐‘ฅ โˆ’ ๐‘Ž)๐›ผโˆ’๐‘˜โˆ’1

ฮ“(๐›ผ โˆ’ ๐‘˜)

๐‘›โˆ’1

๐‘˜=0

(1.75)

3) The fractional derivative of fractional derivative is shown as:-

๐ท๐‘Ž๐‘(๐ท๐‘Ž

๐›ผ๐‘“(๐‘ฅ)) = ๐ท๐‘+๐›ผ๐‘“(๐‘ฅ) โˆ’ โˆ‘[๐ท๐›ผ๐›ผโˆ’๐‘˜โˆ’1๐‘“(๐‘ฅ)]๐‘ฅ=๐‘Ž

(๐‘ฅ โˆ’ ๐‘Ž)โˆ’๐‘โˆ’๐‘˜โˆ’1

ฮ“(โˆ’p โˆ’ k)

๐‘šโˆ’1

๐‘˜=0

where ๐‘› โˆ’ 1 < ๐‘ < ๐‘› , ๐‘š โˆ’ 1 < ๐›ผ < ๐‘š, ๐‘›,๐‘š โˆˆ โ„•

(1.76)

Remark 1.4.4.

๐ท๐‘๐ท๐‘ž๐‘“(๐‘ฅ) = ๐ท๐‘+๐‘ž๐‘“(๐‘ฅ) = ๐ท๐‘ž๐ท๐‘๐‘“(๐‘ฅ) (1.77)

if and only if

๐‘“(๐‘˜)(0) = 0 , ๐‘˜ = 0,1,โ€ฆ , ๐‘Ÿ where ๐‘Ÿ = max(๐‘›,๐‘š),

where ๐‘šโˆ’ 1 โ‰ค ๐‘ < ๐‘š and ๐‘› โˆ’ 1 โ‰ค ๐‘ž < ๐‘›

Page 32: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

23

Theorem 1.4.2. [19,22] The Riemann-Liouville ๐‘ derivative does not satisfy the

following

1) ๐ท๐›ผ๐‘(๐‘“โ„Ž) = ๐‘“๐ท๐‘Ž

๐‘(โ„Ž) + โ„Ž๐ท๐‘Ž๐‘(๐‘“)

2) ๐ท๐‘Ž๐‘(๐‘“ โˆ˜ โ„Ž) = ๐‘“(๐‘)(โ„Ž(๐‘ฅ))โ„Ž(๐›ผ)(๐‘ฅ)

3) ๐ท๐›ผ๐‘ (๐‘“โ„Žโ„ ) =

โ„Ž๐ท๐‘Ž๐‘(๐‘“) โˆ’ ๐‘“๐ท๐‘Ž

๐‘(โ„Ž)

โ„Ž2

Theorem 1.4.2. [19,22] The Riemann-Liouville ๐‘ derivative of known functions:

Let ๐‘ > 0 , ๐‘ฅ > 0 , ๐‘˜, ๐‘ โˆˆ โ„ , then

1) ๐ท๐‘๐‘ฅ๐œ‡ =ฮ“(๐œ‡ + 1)

ฮ“(๐œ‡ โˆ’ ๐‘ + 1)๐‘ฅ๐œ‡โˆ’๐‘ , ๐œ‡ > โˆ’1 (1.78)

2) ๐ท๐‘๐‘ =๐‘

ฮ“(1 โˆ’ ๐‘)๐‘ฅโˆ’๐‘ (1.79)

3) ๐ท๐‘๐‘’๐‘๐‘ฅ = ๐‘ฅโˆ’๐‘๐ธ1,1โˆ’๐‘(๐‘๐‘ฅ) (1.80)

4) ๐ท๐‘ cos(๐‘๐‘ฅ) = ๐‘ฅโˆ’๐‘๐ธ2,1โˆ’๐‘(โˆ’(๐‘๐‘ฅ)2) (1.81)

5) ๐ท๐‘ sin(๐‘๐‘ฅ) = ๐‘๐‘ฅ1โˆ’๐‘๐ธ2,2โˆ’๐‘((๐‘๐‘ฅ)2) (1.82)

6) ๐ท๐‘ cosh(๐‘๐‘ฅ) = ๐‘ฅโˆ’๐‘๐ธ2,1โˆ’๐‘((๐‘๐‘ฅ)2) (1.83)

7) ๐ท๐‘ sinh(๐‘๐‘ฅ) = ๐‘๐‘ฅ1โˆ’๐‘๐ธ2,2โˆ’๐‘((๐‘๐‘ฅ)2) (1.84)

8) ๐ท๐‘ ln(๐‘ฅ) =๐‘ฅโˆ’๐‘

ฮ“(1 โˆ’ ๐‘)[ln(๐‘ฅ) โˆ’ ๐›พ โˆ’ ๐œ“(1 โˆ’ ๐‘)] (1.85)

Proof:

(1) Let ๐‘› โˆ’ 1 < ๐‘ < ๐‘› , ๐‘› โˆˆ โ„•, then

๐ท๐‘๐‘ฅ๐œ‡ = ๐ท๐‘›[๐ทโˆ’(๐‘›โˆ’๐‘)๐‘ฅ๐œ‡]

Page 33: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

24

= ๐ท๐‘› [ฮ“(๐œ‡ + 1)

ฮ“(๐œ‡ + ๐‘› โˆ’ ๐‘ + 1)๐‘ฅ๐œ‡+๐‘›โˆ’๐‘]

=ฮ“(๐œ‡ + 1)

ฮ“(๐œ‡ + ๐‘› โˆ’ ๐‘ + 1).ฮ“(๐œ‡ + ๐‘› โˆ’ ๐‘ + 1)

ฮ“(๐œ‡ + ๐‘› โˆ’ ๐‘ โˆ’ ๐‘› + 1)๐‘ฅ๐œ‡+๐‘›โˆ’๐‘โˆ’๐‘›

=ฮ“(ฮผ + 1)

ฮ“(ฮผ โˆ’ p + 1)๐‘ฅ๐œ‡โˆ’๐‘ โˆŽ

(2) It follows by substituting ๐œ‡ = 0 in (1.78)

(3) Using ๐ทโˆ’๐‘๐‘’๐‘๐‘ฅ = ๐‘ฅ๐‘๐ธ1,๐‘+1(๐‘๐‘ฅ)

= ๐‘ฅ๐‘โˆ‘(๐‘๐‘ฅ)๐‘˜

ฮ“(๐‘˜ + ๐‘ + 1)

โˆž

๐‘˜=0

=โˆ‘๐‘๐‘˜๐‘ฅ๐‘˜+๐‘

ฮ“(๐‘˜ + ๐‘ + 1)

โˆž

๐‘˜=0

=โˆ‘๐‘๐‘˜

ฮ“(๐‘˜ + 1)

โˆž

๐‘˜=0

.ฮ“(๐‘˜ + 1)

ฮ“(๐‘˜ + ๐‘ + 1)๐‘ฅ๐‘˜+๐‘

=โˆ‘๐‘๐‘˜

ฮ“(๐‘˜ + 1)

โˆž

๐‘˜=0

๐ทโˆ’๐‘๐‘ฅ๐‘˜

(1.86)

Now, by using (1.86) we have

๐ท๐‘๐‘’๐‘๐‘ฅ = ๐ท๐‘›[๐ทโˆ’(๐‘›โˆ’๐‘)๐‘’๐‘๐‘ฅ] = ๐ท๐‘› [โˆ‘๐‘๐‘˜

ฮ“(k + 1)

โˆž

๐‘˜=0

๐ทโˆ’(๐‘›โˆ’๐‘)๐‘ฅ๐‘˜]

=โˆ‘๐‘๐‘˜

ฮ“(k + 1)

โˆž

๐‘˜=0

๐ท๐‘๐‘ฅ๐‘˜ =โˆ‘๐‘๐‘˜

ฮ“(k + 1).ฮ“(๐‘˜ + 1)๐‘ฅ๐‘˜โˆ’๐‘

ฮ“(๐‘˜ โˆ’ ๐‘ + 1)

โˆž

๐‘˜=0

= ๐‘ฅโˆ’๐‘โˆ‘(๐‘๐‘ฅ)๐‘˜

ฮ“(๐‘˜ โˆ’ ๐‘ + 1)

โˆž

๐‘˜=0

= ๐‘ฅโˆ’๐‘๐ธ1,1โˆ’๐‘(๐‘๐‘ฅ) โˆŽ

(4) ๐ท๐‘ cos(๐‘๐‘ฅ) = ๐ท๐‘›[๐ทโˆ’(๐‘›โˆ’๐‘) cos(๐‘๐‘ฅ)]

Page 34: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

25

= ๐ท๐‘›[๐‘ฅ๐‘›โˆ’๐‘๐ธ2,๐‘›โˆ’๐‘+1(โˆ’(๐‘๐‘ฅ)2)]

= ๐‘ฅ๐‘›โˆ’๐‘+1โˆ’๐‘›+1๐ธ2,๐‘›โˆ’๐‘+1โˆ’๐‘›(โˆ’(๐‘๐‘ฅ)2)

= ๐‘ฅโˆ’๐‘๐ธ2,1โˆ’๐‘(โˆ’(๐‘๐‘ฅ)2) โˆŽ

(5) Similarly of (4).

(6) ๐ท๐‘ cosh(๐‘๐‘ฅ) = ๐ท๐‘›[๐ทโˆ’(๐‘›โˆ’๐‘) cosh(๐‘๐‘ฅ)]

= ๐ท๐‘›[๐‘ฅ๐‘›โˆ’๐‘๐ธ2,๐‘›โˆ’๐‘+1((๐‘๐‘ฅ)2)]

= ๐‘ฅ๐‘›โˆ’๐‘+1โˆ’๐‘›โˆ’1๐ธ2,๐‘›โˆ’๐‘+1โˆ’๐‘›(โˆ’(๐‘๐‘ฅ)2)

= ๐‘ฅโˆ’๐‘๐ธ2,1โˆ’๐‘((๐‘๐‘ฅ)2) โˆŽ

(7) Similarly of (6)

(8) To proof see [19]

1.5 Caputo Fractional Operator

In 1967 M.Caputo published a paper[11]. He put a new definition of fractional

derivative. In this section we introduced Caputo fractional derivative and some

properties of this definition.

Definition 1.5.1. [3, 4, 5, 11, 19, 22] let ๐‘“ be ๐‘› โˆ’times differentiable function,

๐‘ฅ, ๐‘Ž โˆˆ โ„ , ๐‘ฅ > ๐‘Ž and ๐›ผ โˆˆ [0,1). Then the Caputo fractional differential operator of

order ๐›ผ of ๐‘“ is defined by:

๐ท๐‘Ž๐›ผ๐‘ ๐‘“(๐‘ฅ) =

1

ฮ“(1 โˆ’ ๐›ผ)โˆซ

๐‘“โ€ฒ(๐‘ก)

(๐‘ฅ โˆ’ ๐‘ก)๐›ผ

๐‘ฅ

๐‘Ž

๐‘‘๐‘ก

Page 35: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

26

Definition 1.5.2. [3, 4, 5, 11, 19,22] let ๐‘“ be ๐‘›-times differentiable function, ๐‘ฅ, ๐‘Ž โˆˆ

โ„ , ๐‘ฅ > ๐‘Ž and ๐›ผ โˆˆ (๐‘›, ๐‘› โˆ’ 1). Then the caputo fractional differential operator of ๐›ผ is

defined as:

๐ท๐‘Ž๐›ผ๐‘“(๐‘ฅ)๐‘ =

1

ฮ“(๐‘› โˆ’ ๐›ผ)โˆซ

๐‘“(๐‘›)(๐‘ก)

(๐‘ฅ โˆ’ ๐‘ก)๐›ผโˆ’๐‘›+1

๐‘ฅ

๐‘Ž

๐‘‘๐‘ก

Remark 1.5.1.

Because of similarity between (R-L) and Caputo fractional integration, the symbol

๐ท๐‘Žโˆ’๐›ผ๐‘“(๐‘ฅ) will be indicated to (R-L) and Caputo fractional integral.

Remark 1.5.2

The symbol ๐ท๐‘ ๐‘Ž๐›ผ๐‘“(๐‘ฅ)is used to denote Caputo fractional derivative of order ๐›ผ

with lower limit ๐‘Ž and the symbol ๐ท๐›ผ๐‘“(๐‘ฅ)๐‘ is used to denote caputo fractional

derivative of order ๐›ผ with lower limit 0.

Preposition 1.5.1. [11, 19,22] let ๐‘“(๐‘ฅ), ๐‘”(๐‘ฅ) be two functions such that both

๐ท๐‘ ๐‘Ž๐›ผ๐‘“(๐‘ฅ), ๐ท๐‘Ž

๐›ผ๐‘ ๐‘”(๐‘ฅ) exist for ๐›ผ โˆˆ [0,1) and let ๐‘Ž, ๐‘ โˆˆ โ„‚.

Then

๐ท๐‘Ž๐›ผ(๐‘Ž๐‘“(๐‘ฅ) + ๐‘๐‘”(๐‘ฅ)) = ๐‘Ž ๐ท๐‘Ž

๐›ผ๐‘ ๐‘“(๐‘ฅ) +๐‘ ๐‘ ๐ท๐‘Ž๐›ผ๐‘ ๐‘”(๐‘ฅ) (1.87)

Proof: using the definition of Caputo fractional ๐›ผ derivative

๐ท๐‘Ž๐›ผ๐‘ (๐‘Ž๐‘“(๐‘ฅ) + ๐‘๐‘”(๐‘ฅ)) =

1

ฮ“(1 โˆ’ ๐›ผ)โˆซ(๐‘Ž๐‘“(๐‘ก) + ๐‘๐‘”(๐‘ก))โ€ฒ

(๐‘ฅ โˆ’ ๐‘ก)๐›ผ

๐‘ฅ

๐‘Ž

๐‘‘๐‘ก

Page 36: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

27

=1

ฮ“(1 โˆ’ ๐›ผ)[๐‘Žโˆซ

๐‘“โ€ฒ(๐‘ฅ)

(๐‘ฅ โˆ’ ๐‘ก)๐›ผ

๐‘ฅ

๐‘Ž

๐‘‘๐‘ก + ๐‘โˆซ๐‘”โ€ฒ(๐‘ฅ)

(๐‘ฅ โˆ’ ๐‘ก)๐›ผ๐‘‘๐‘ก

๐‘ฅ

๐‘Ž

]

=1

ฮ“(1โˆ’๐›ผ)๐‘Ž โˆซ

๐‘“โ€ฒ(๐‘ฅ)

(๐‘ฅโˆ’๐‘ก)๐›ผ

๐‘ฅ

๐‘Ž๐‘‘๐‘ก +

1

ฮ“(1โˆ’๐›ผ)๐‘ โˆซ

๐‘”โ€ฒ(๐‘ฅ)

(๐‘ฅโˆ’๐‘ก)๐›ผ๐‘‘๐‘ก

๐‘ฅ

๐‘Ž

= ๐‘Ž ๐ท๐‘Ž๐›ผ๐‘ ๐‘“(๐‘ฅ) + ๐‘ ๐ท๐‘Ž

๐›ผ๐‘ ๐‘”(๐‘ฅ) โˆŽ

We can generalize the previous result for any ๐›ผ โˆˆ [๐‘› โˆ’ 1, ๐‘›)

The Relation between integration and differentiation of Caputo operator of order

๐›ผ are given as shown:

The Caputo derivative of fractional integral is

๐ท๐‘Ž๐›ผ๐‘ (๐ท๐‘Ž

โˆ’๐›ผ๐‘“(๐‘ฅ)) = ๐‘“(๐‘ฅ) (1.88)

The fractional integral of Caputo derivative is

๐ท๐‘Žโˆ’๐›ผ( ๐ท๐‘ ๐‘Ž

๐›ผ๐‘“(๐‘ฅ)) = ๐‘“(๐‘ฅ) โˆ’ โˆ‘(๐‘ฅ โˆ’ ๐‘Ž)๐‘š

๐‘š!

๐‘›โˆ’1

๐‘š=0

๐‘“(๐‘š)(๐‘Ž) (1.89)

From (1.88) and (1.89) we have

๐ท๐‘Ž๐›ผ๐‘ (๐ท๐‘Ž

โˆ’๐›ผ๐‘“(๐‘ฅ)) โ‰  ๐ท๐‘Žโˆ’๐›ผ( ๐ท๐‘ ๐‘Ž

๐›ผ๐‘“(๐‘ฅ)) (1.90)

Generally, we can conclude:

๐ท๐‘›[๐ทโˆ’(๐‘›โˆ’๐›ผ)๐‘“(๐‘ฅ)] โ‰  ๐ทโˆ’(๐‘›โˆ’๐›ผ)[๐ท๐‘›๐‘“(๐‘ฅ)]

Page 37: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

28

Thus

๐ท๐‘Ž๐›ผ๐‘“(๐‘ฅ) โ‰  ๐ท๐‘Ž

๐›ผ๐‘ ๐‘“(๐‘ฅ) (1.91)

which implies that the Caputo derivative is not equivalent with (Riemann-

Liouville) derivative.

Preposition 1.5.2. [11, 19,22] let ๐‘› โˆˆ โ„• , ๐›ผ โˆˆ [๐‘› โˆ’ 1, ๐‘›). Let the function ๐‘“(๐‘ฅ) be an n-

times differentiable function. Then the representation of the Caputo ๐›ผ derivative:

๐ท๐‘Ž๐›ผ๐‘ ๐‘“(๐‘ฅ) = ๐ท๐‘Ž

โˆ’(๐‘›โˆ’๐›ผ) ๐ท๐‘Ž๐‘›๐‘ ๐‘“(๐‘ฅ) (1.92)

where ๐ท๐‘Žโˆ’๐›ผ๐‘“(๐‘ฅ) =

1

ฮ“(๐›ผ)โˆซ

๐‘“(๐‘ก)

(๐‘ฅโˆ’๐‘ก)1โˆ’๐›ผ

๐›ผ

๐‘Ž๐‘‘๐‘ก

is the Riemann-Liouville ๐›ผ integral

Theorem1.5.1. [11, 19,22] (Relation between Caputo ๐›ผ derivative and Riemann-

Liouville ๐›ผ derivative).

Let ๐‘› โˆˆ โ„•,๐›ผ โˆˆ [๐‘› โˆ’ 1, ๐‘›). And let ๐‘“(๐‘ฅ) be a function such that ๐ท๐‘Ž๐›ผ๐‘ ๐‘“(๐‘ฅ) and

๐ท๐‘Ž๐›ผ๐‘“(๐‘ฅ) exist. Then the relation between the (R-L) and the Caputo derivatives is given

by:

๐ท๐‘Ž๐›ผ๐‘ ๐‘“(๐‘ฅ) = ๐ท๐‘Ž

๐›ผ๐‘“(๐‘ฅ) โˆ’โˆ‘(๐‘ฅ โˆ’ ๐‘Ž)๐‘˜โˆ’๐›ผ

ฮ“(๐‘˜ + 1 โˆ’ ๐›ผ)

๐‘›โˆ’1

๐‘˜=0

๐‘“(๐‘˜)(๐‘Ž) (1.93)

Proof: The well-known Taylor Series expansion of ๐‘“ about ๐‘ฅ = 0 is

Page 38: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

29

๐‘“(๐‘ฅ) = ๐‘“(0) + ๐‘ฅ๐‘“โ€ฒ(0) +๐‘ฅ2

๐‘ฅ!๐‘“โ€ฒโ€ฒ(0) +โ‹ฏ+

๐‘ฅ๐‘›โˆ’1

(๐‘› โˆ’ 1)!๐‘“(๐‘›โˆ’1)(0) + ๐‘…๐‘›โˆ’1

=โˆ‘๐‘ฅ๐‘˜

ฮ“(๐‘˜ + 1)

๐‘›โˆ’1

๐‘˜=0

๐‘“(๐‘˜)(0) + ๐‘…๐‘›โˆ’1

(1.94)

where, considering the following

๐ทโˆ’๐‘›๐‘“(๐‘ก) = โˆซโˆซ โ€ฆ

๐‘ก1

๐‘Ž

๐‘ก

๐‘Ž

โˆซ ๐‘“(๐œ†)

๐‘ก๐‘›โˆ’1

๐‘Ž

๐‘‘๐œ†โ€ฆ๐‘‘๐œ†2 ๐‘‘๐œ†1

=1

(๐‘› โˆ’ 1)!โˆซ๐‘“(๐œ†)

๐‘ก

๐‘Ž

(๐‘ก โˆ’ ๐œ†)(๐‘›โˆ’1)๐‘‘๐œ†

(1.95)

The previous formula is called cauchyโ€™s formula for repeated integration.

๐‘…๐‘›โˆ’1 = โˆซ๐‘“(๐‘›)(๐‘ก)(๐‘ฅ โˆ’ ๐‘ก)๐‘›โˆ’1

(๐‘› โˆ’ 1)!

๐‘ฅ

0

๐‘‘๐‘ก =1

ฮ“(๐‘›)โˆซ๐‘“(๐‘›)(๐‘ก)

๐‘ฅ

0

(๐‘ฅ โˆ’ 1)๐‘›โˆ’1๐‘‘๐‘ก (1.96)

Now, by using linearity of Riemann-Liouville, the (Riemann-Liouville) derivative

of power function, the properties of Riemann-Liouville integrals and the representation

formula.

๐ท๐‘Ž๐›ผ๐‘“(๐‘ก) = ๐ท๐‘Ž

๐›ผ [โˆ‘๐‘ฅ๐‘˜

ฮ“(๐‘˜ + 1)๐‘“(๐‘˜)(0) + ๐‘…๐‘›โˆ’1

๐‘›โˆ’1

๐‘˜=0

] = โˆ‘๐ท๐‘Ž๐›ผ

๐‘›โˆ’1

๐‘˜=0

๐‘ฅ๐‘˜

ฮ“(๐‘˜ + 1)๐‘“(๐‘˜)(0) + ๐ท๐‘Ž

๐›ผ๐‘…๐‘›โˆ’1

=โˆ‘ฮ“(๐‘˜ + 1)

ฮ“(๐‘˜ โˆ’ ๐›ผ + 1)

๐‘›โˆ’1

๐‘˜=0

๐‘ฅ๐‘˜โˆ’๐›ผ

ฮ“(๐‘˜ + 1)๐‘“(๐‘˜)(0) + ๐ท๐‘Ž

๐›ผ๐ทโˆ’๐‘›๐‘“(๐‘›)(๐‘ฅ)

Page 39: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

30

=โˆ‘๐‘ฅ๐‘˜โˆ’๐›ผ

ฮ“(๐‘˜ โˆ’ ๐›ผ + 1)

๐‘›โˆ’1

๐‘˜=0

๐‘“(๐‘˜)(0) + ๐ทโˆ’(๐‘›โˆ’๐›ผ)๐‘“(๐‘›)(๐‘ฅ)

= โˆ‘๐‘ฅ๐‘˜โˆ’๐›ผ

ฮ“(๐‘˜ โˆ’ ๐›ผ + 1)

๐‘›โˆ’1

๐‘˜=0

๐‘“(๐‘˜)(0) + ๐ท๐‘Ž๐›ผ๐‘ ๐‘“(๐‘ฅ)

โˆด ๐ท๐‘Ž๐›ผ๐‘ ๐‘“(๐‘ฅ) = ๐ท๐‘Ž

๐›ผ๐‘“(๐‘ก) โˆ’โˆ‘๐‘ฅ๐‘˜โˆ’๐›ผ

ฮ“(๐‘˜ โˆ’ ๐›ผ + 1)

๐‘›โˆ’1

๐‘˜=0

๐‘“(๐‘˜)(0)

Preposition 1.5.3. [11, 19, 22] Let ๐›ผ โˆˆ [0,1], let ๐‘“(๐‘ฅ) be a function with second

continuous bounded derivative in [๐‘Ž, ๐‘‡] for every ๐‘‡ > ๐‘Ž and ๐ท๐‘Ž๐›ผ๐‘ ๐‘“(๐‘ฅ) exist, then:

1) lim๐‘Žโ†’1

๐ท๐‘Ž๐›ผ๐‘ ๐‘“(๐‘ฅ) = ๐‘“โ€ฒ(๐‘ฅ) (1.97)

2) lim๐‘Žโ†’0

๐ท๐‘ ๐‘Ž๐›ผ๐‘“(๐‘ฅ) = ๐‘“(๐‘ฅ) โˆ’ ๐‘“(๐‘Ž) (1.98)

To proof see [11].

We can generalize the above equations in preposition 1.5.3 for any positive ๐›ผ to be:

lim๐‘Žโ†’๐‘›

๐ท๐‘Ž๐›ผ๐‘ ๐‘“(๐‘ฅ) = ๐‘“(๐‘›)(๐‘ฅ) (1.99)

and lim๐‘Žโ†’๐‘›โˆ’1

๐ท๐‘ ๐‘Ž๐›ผ๐‘“(๐‘ฅ) = ๐‘“(๐‘›โˆ’1)(๐‘ฅ) โˆ’ ๐‘“(๐‘›โˆ’1)(0) (1.100)

where ๐›ผ โˆˆ [๐‘› โˆ’ 1, ๐‘›), ๐‘› โˆˆ โ„• and with the same condition of the preposition.

Preposition 1.5.4. [11, 19, 22]

The Caputo differential operator does not satisfy the following:

Page 40: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

31

1) ๐ท๐‘Ž๐›ผ(๐‘“โ„Ž)๐‘ = ๐‘“ ๐ท๐‘Ž

๐›ผ๐‘ (โ„Ž) + โ„Ž ๐ท๐‘Ž๐›ผ๐‘ (๐‘“)

2) ๐ท๐‘Ž๐›ผ๐‘“

โ„Ž๐‘ =

โ„Ž ๐ท๐‘Ž๐›ผ๐‘ (๐‘“) + ๐‘“ ๐ท๐‘Ž

๐›ผ๐‘ (โ„Ž)

โ„Ž2

3) ๐ท๐‘Ž๐›ผ(๐‘“ โˆ˜ โ„Ž)๐‘ = ๐‘“(๐›ผ)(โ„Ž(๐‘ฅ))โ„Ž(๐›ผ)(๐‘ฅ)

where ๐‘“(๐›ผ)(๐‘ฅ), โ„Ž(๐›ผ)(๐‘ฅ) are the Caputo ๐›ผ derivative.

Now, I will give counter example to show that the above rule does not satisfy for

Caputo Operator considers that:

๐ท๐›ผ๐‘ (๐‘ก) =1

ฮ“(2 โˆ’ ๐›ผ)๐‘ก1โˆ’๐›ผ โˆด ๐ท

13๐‘ (๐‘ก) = 1.1077๐‘ก2 3โ„

๐ท๐›ผ๐‘ (๐‘ก2) =2

ฮ“(3 โˆ’ ๐›ผ)๐‘ก2โˆ’๐›ผ โˆด ๐ท

13๐‘ (๐‘ก2) = 1.3293๐‘ก5 3โ„

๐ท๐›ผ๐‘ (๐‘ก3) =6

ฮ“(4 โˆ’ ๐›ผ)๐‘ก3โˆ’๐›ผ โˆด ๐ท

13๐‘ (๐‘ก3) = 1.4954๐‘ก8 3โ„

Let ๐‘“(๐‘ฅ) = ๐‘ก , ๐‘”(๐‘ฅ) = ๐‘ก2 , โ„Ž(๐‘ก) = ๐‘ก3

๐ท13๐‘ (๐‘“๐‘”) = ๐ท

13๐‘ (๐‘ก3) = 1.4954๐‘ก8 3โ„

๐‘“ ๐ท13๐‘ (๐‘”) + ๐‘” ๐ท

13๐‘ (๐‘“) = ๐‘ก(1.3293๐‘ก5 3โ„ ) + ๐‘ก2(1.1077๐‘ก2 3โ„ )

= 1.3293๐‘ก8 3โ„ + 1.1077๐‘ก8 3โ„ = 2.4370 ๐‘ก8 3โ„

Obviously

๐ท13(๐‘“๐‘”)๐‘ โ‰  ๐‘“ ๐ท

13๐‘ (๐‘”) + ๐‘” ๐ท

13๐‘ (๐‘“)

Also

๐ท13๐‘ (โ„Ž(๐‘ก)

๐‘“(๐‘ก)) = ๐ท

13๐‘ (๐‘ก3

๐‘ก) = ๐ท

13๐‘ (๐‘ก2) = 1.3293๐‘ก5 3โ„

But

Page 41: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

32

๐‘“(๐‘ก) ๐ท13๐‘ โ„Ž(๐‘ก) โˆ’ โ„Ž(๐‘ก) ๐ท

13๐‘ ๐‘“(๐‘ก)

๐‘“2(๐‘ก)=๐‘ก(1.4954๐‘ก5 3โ„ ) โˆ’ ๐‘ก3(1.1077๐‘ก2 3โ„ )

๐‘ก2

= 1.4954๐‘ก2 3โ„ โˆ’ 1.1077๐‘ก5 3โ„

Thus

๐ท13๐‘ (โ„Ž

๐‘“) โ‰ 

๐‘“ ๐ท13๐‘ (โ„Ž) โˆ’ โ„Ž ๐ท

13๐‘ (๐‘“)

โ„Ž2

It is easy to show that the composition Rule does not satisfy.

Preposition 1.5.5. [11, 19, 22] suppose that ๐›ผ โˆˆ [๐‘› โˆ’ 1, ๐‘›),๐‘š, ๐‘› โˆˆ โ„, and ๐ท๐‘Ž๐›ผ๐‘“(๐‘ฅ)๐‘

exist. Then

๐ท๐‘Ž๐›ผ๐‘ ๐ท๐‘š๐‘ ๐‘“(๐‘ฅ) = ๐ท๐›ผ+๐‘š๐‘ ๐‘“(๐‘ฅ) โ‰  ๐ท๐‘š๐‘ ๐ท๐‘Ž

๐›ผ๐‘ ๐‘“(๐‘ฅ) (1.101)

Now we give counter example to show that the Caputo derivative is not commute

Example:

๐ท๐‘Ž๐›ผ๐‘ ๐‘ฅ๐‘ = {

ฮ“(๐‘ + 1)

ฮ“(๐‘ โˆ’ ๐›ผ + 1)๐‘ฅ๐‘โˆ’๐›ผ ๐‘–๐‘“ ๐‘› โˆ’ 1 โ‰ค ๐›ผ < ๐‘› , ๐‘ > ๐‘› โˆ’ 1, ๐‘› โˆˆ โ„•

0 ๐‘–๐‘“ ๐‘› โˆ’ 1 โ‰ค ๐›ผ < ๐‘› , ๐‘ โ‰ค ๐‘› โˆ’ 1, ๐‘, ๐‘› โˆˆ โ„•

Now if we take ๐›ผ =1

2 , ๐‘š = 3 , ๐‘ = 2. Then

๐ท1 2โ„ ๐ท3๐‘๐‘ [๐‘ฅ2] = 0

But ๐ท3๐‘ ๐ท1 2โ„๐‘ [๐‘ฅ2] = ๐ท3๐‘ [ฮ“(3)

ฮ“(5

2)๐‘ฅ3

2]

= โˆ’38 ฮ“(3)ฮ“(52)๐‘ฅโˆ’

32

Page 42: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

33

Corollary 1.5.1. let ๐‘› โˆˆ โ„•,๐›ผ โˆˆ [๐‘› โˆ’ 1, ๐‘›), ๐œ‡ = ๐›ผ โˆ’ (๐‘› โˆ’ 1). Let ๐‘“(๐‘ฅ) be a function

such that ๐ท๐‘Ž๐›ผ๐‘“(๐‘ฅ)๐‘ exist, then

๐ท๐‘Ž๐›ผ๐‘ ๐‘“(๐‘ฅ) = ๐ท๐œ‡๐‘ ๐ท๐‘›โˆ’1๐‘ ๐‘“(๐‘ฅ) (1.102)

Theorem 1.5.2. [11, 19,22] (Some basic rules of Caputo fractional derivative):

Let ๐›ผ โˆˆ [๐‘› โˆ’ 1, ๐‘›), ๐‘› โˆˆ โ„•,

1) ๐ท๐‘Ž๐›ผ๐‘ ๐‘ = 0 , ๐‘ is constant (1.103)

2) ๐ท๐‘Ž๐›ผ๐‘ ๐‘ฅ๐œ‡ = {

ฮ“(๐œ‡ + 1)

ฮ“(๐œ‡ โˆ’ ๐›ผ + 1)๐‘ฅ๐œ‡โˆ’๐›ผ ๐‘–๐‘“ ๐œ‡ > ๐‘› โˆ’ 1, ๐‘ฅ > 0, ๐œ‡ โˆˆ โ„

0 ๐‘–๐‘“ ๐œ‡ โ‰ค ๐‘› โˆ’ 1, ๐‘ฅ > 0, ๐œ‡ โˆˆ โ„•

(1.104)

3) ๐ท๐‘Ž๐›ผ๐‘ ๐‘’๐‘๐‘ฅ = ๐‘๐‘›๐‘ฅ๐‘›โˆ’๐›ผ๐ธ1,๐‘›โˆ’๐›ผ+1(๐‘๐‘ฅ) (1.105)

4) ๐ท๐‘Ž๐›ผ๐‘ sin(๐‘๐‘ฅ) = โˆ’

1

2๐‘–(๐‘–๐‘)๐‘›๐‘ฅ๐‘›โˆ’๐›ผ[๐ธ1,๐‘›โˆ’๐›ผ+1(๐‘–๐‘๐‘ฅ) โˆ’ (โˆ’1)

๐‘›๐ธ1,๐‘›โˆ’๐›ผ+1(โˆ’๐‘–๐‘๐‘ฅ)] (1.106)

5) ๐ท๐‘Ž๐›ผ๐‘ cos(๐‘๐‘ฅ) =

1

2(๐‘–๐‘)๐‘›๐‘ฅ๐‘›โˆ’๐›ผ[๐ธ1,๐‘›โˆ’๐›ผ+1(๐‘–๐‘๐‘ฅ) + (โˆ’1)

๐‘›๐ธ1,๐‘›โˆ’๐›ผ+1(โˆ’๐‘–๐‘๐‘ฅ)] (1.107)

6) ๐ท๐‘Ž๐›ผ๐‘ cos(๐‘๐‘ฅ) =

1

2(๐‘–๐‘)๐‘›๐‘ฅ๐‘›โˆ’๐›ผ[๐ธ1,๐‘›โˆ’๐›ผ+1(๐‘–๐‘๐‘ฅ) + (โˆ’1)

๐‘›๐ธ1,๐‘›โˆ’๐›ผ+1(โˆ’๐‘–๐‘๐‘ฅ)] (1.108)

7) ๐ท๐‘Ž๐›ผ๐‘ sinh(๐‘๐‘ฅ) = โˆ’

1

2๐‘๐‘›๐‘ฅ๐‘›โˆ’๐›ผ[๐ธ1,๐‘›โˆ’๐›ผ+1(๐‘๐‘ฅ) โˆ’ (โˆ’1)

๐‘›๐ธ1,๐‘›โˆ’๐›ผ+1(โˆ’๐‘๐‘ฅ)] (1.109)

8) ๐ท๐‘Ž๐›ผ๐‘ cosh(๐‘๐‘ฅ) =

1

2๐‘๐‘›๐‘ฅ๐‘›โˆ’๐›ผ[๐ธ1,๐‘›โˆ’๐›ผ+1(๐‘๐‘ฅ) + (โˆ’1)

๐‘›๐ธ1,๐‘›โˆ’๐›ผ+1(โˆ’๐‘๐‘ฅ)] (1.110)

Proof:

Page 43: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

34

1) By applying the Caputo definition and because of the nโ€™th derivative ๐‘(๐‘›), (๐‘› โˆˆ

โ„• , ๐‘› โ‰ฅ 1) of constant equals 0, then

๐ท๐‘Ž๐›ผ๐‘ ๐‘ =

1

ฮ“(๐‘› โˆ’ ๐›ผ)โˆซ

๐‘(๐‘›)

(๐‘ฅ โˆ’ ๐‘)๐›ผโˆ’๐‘›+1

๐‘ฅ

๐‘Ž

๐‘‘๐‘ก = 0

2) The second case has an easy proof

( ๐ท๐‘Ž๐›ผ๐‘ ๐‘ก๐œ‡ = 0, ๐›ผ โˆˆ (๐‘› โˆ’ 1, ๐‘›), ๐œ‡ โ‰ค ๐‘› โˆ’ 1, ๐‘› โˆˆ โ„•)

It follows from the pattern of the proof of (1). But the first case is more

interesting. We can prove it by two ways. Directly by using Caputo definition.

Firstly, let ฮฑ โˆˆ (๐‘› โˆ’ 1, ๐‘›), ๐œ‡ > ๐‘› โˆ’ 1, ๐œ‡ โˆˆ โ„

๐ท๐‘Ž๐›ผ๐‘ฅ๐œ‡ =

1

ฮ“(๐‘› โˆ’ ๐›ผ)๐‘ โˆซ

๐ท๐‘›๐‘ก๐œ‡

(๐‘ฅ โˆ’ ๐‘ก)๐›ผ+1โˆ’๐‘›

๐‘ฅ

0

๐‘‘๐‘ก

=1

ฮ“(๐‘› โˆ’ ๐›ผ)โˆซ

ฮ“(๐œ‡ + 1) ๐‘ก๐œ‡โˆ’๐‘›

(๐‘ฅ โˆ’ ๐‘ก)๐›ผ+1โˆ’๐‘›ฮ“(๐œ‡ โˆ’ ๐‘› + 1)

๐‘ฅ

0

๐‘‘๐‘ก

Now by plugging t= ๐‘ฅ๐‘ข ; 0 โ‰ค ๐‘ข โ‰ค 1

=ฮ“(๐œ‡ + 1)

ฮ“(๐‘› โˆ’ ๐›ผ)ฮ“(๐œ‡ โˆ’ ๐‘› + 1)โˆซ(๐‘ฅ๐‘ข)๐œ‡โˆ’๐‘›((1 โˆ’ ๐‘ข)๐‘ฅ)๐‘›โˆ’๐›ผโˆ’11

0

๐‘ฅ๐‘‘๐‘ข

=ฮ“(๐œ‡ + 1)

ฮ“(๐‘› โˆ’ ๐›ผ)ฮ“(๐œ‡ โˆ’ ๐‘› + 1)๐‘ฅ๐œ‡โˆ’๐‘›๐›ฝ(๐œ‡ โˆ’ ๐‘› + 1, ๐‘› โˆ’ ๐›ผ)

=ฮ“(๐œ‡ + 1)

ฮ“(๐‘› โˆ’ ๐›ผ)ฮ“(๐œ‡ โˆ’ ๐‘› + 1)๐‘ฅ๐œ‡โˆ’๐‘›

ฮ“(๐œ‡ โˆ’ ๐‘› + 1)ฮ“(๐‘› โˆ’ ๐›ผ)

ฮ“(๐œ‡ โˆ’ ๐›ผ + 1)

=ฮ“(๐œ‡ + 1)

ฮ“(๐œ‡ โˆ’ ๐›ผ + 1)๐‘ฅ๐œ‡โˆ’๐‘›

Page 44: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

35

Secondly, we can prove it by the relation between the Caputo and

Riemann-Liouville derivatives:

๐ท๐‘Ž๐›ผ๐‘ ๐‘ฅ๐œ‡ = ๐ท๐›ผ๐‘ฅ๐œ‡ โˆ’โˆ‘

๐‘ฅ๐‘˜โˆ’๐›ผ

ฮ“(๐‘˜ + 1 โˆ’ ๐›ผ)

๐‘›โˆ’1

๐‘˜=0

๐ท๐‘˜[(๐‘ฅ)๐œ‡]๐‘ฅ=0

Now, the ๐ท๐‘˜[(๐‘ฅ)๐œ‡]๐‘ฅ=0 = 0, for ๐‘˜ โ‰ค ๐‘› โˆ’ 1 โ‰ค ๐œ‡

Then ๐ท๐‘Ž๐›ผ๐‘ ๐‘ฅ๐œ‡ =

ฮ“(๐œ‡+1)

ฮ“(ฮผโˆ’ฮฑ+1)๐‘ฅ๐œ‡โˆ’๐›ผ โˆŽ

3) To prove it, we need to use the relation between Caputo and Riemann-Liouville

fractional derivative as in (1.93) and use the exponential case of Riemann-

Liouville ๐›ผ โˆ’derivative in (1.80), then we have:

๐ท๐‘Ž๐›ผ๐‘’๐‘๐‘ฅ = ๐ท๐›ผ๐‘’๐‘๐‘ฅ โˆ’โˆ‘

๐‘ฅ๐‘˜โˆ’๐›ผ

ฮ“(๐‘˜ + 1 โˆ’ ๐›ผ)

๐‘›โˆ’1

๐‘˜=0

๐ท๐‘˜๐‘’๐‘๐‘ฅ|

๐‘ฅ=0

= ๐‘ฅโˆ’๐›ผ๐ธ1,1โˆ’๐›ผ(๐‘๐‘ฅ) โˆ’โˆ‘๐‘ฅ๐‘˜โˆ’๐›ผ๐‘๐‘˜

ฮ“(๐‘˜ + 1 โˆ’ ๐›ผ)

๐‘›โˆ’1

๐‘˜=0

= โˆ‘(๐‘๐‘ฅ)๐‘˜๐‘ฅโˆ’๐›ผ

ฮ“(k + 1 โˆ’ ฮฑ)

โˆž

๐‘˜=0

โˆ’โˆ‘๐‘ฅ๐‘˜โˆ’๐›ผ๐‘๐‘˜

ฮ“(๐‘˜ + 1 โˆ’ ๐›ผ)

๐‘›โˆ’1

๐‘˜=0

= โˆ‘๐‘ฅ๐‘˜โˆ’๐›ผ๐‘๐‘˜

ฮ“(๐‘˜ + 1 โˆ’ ๐›ผ)

โˆž

๐‘˜=๐‘›

=โˆ‘๐‘ฅ๐‘˜+๐‘›โˆ’๐›ผ๐‘๐‘˜+๐‘›

ฮ“(๐‘˜ + ๐‘› โˆ’ ๐›ผ + 1)

โˆž

๐‘˜=0

= ๐‘๐‘›๐‘ฅ๐‘›โˆ’๐›ผ๐ธ1,๐‘›โˆ’๐›ผ+1(๐‘๐‘ฅ) โˆŽ

4) Use (sin ๐‘ฅ =๐‘’๐‘–๐‘ฅโˆ’๐‘’โˆ’๐‘–๐‘ฅ

2๐‘–), then by using (1.105)

๐ท๐‘Ž๐›ผ๐‘ sin(๐‘๐‘ฅ) = ๐ท๐‘Ž

๐›ผ๐‘’๐‘–๐‘ฅ โˆ’ ๐‘’โˆ’๐‘–๐‘ฅ

2๐‘–๐‘ =

1

2๐‘–( ๐ท๐‘Ž

๐›ผ๐‘ ๐‘’๐‘–๐‘๐‘ฅ โˆ’ ๐ท๐‘Ž๐›ผ๐‘ ๐‘’โˆ’๐‘–๐‘๐‘ฅ)

Page 45: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

36

=1

2๐‘–[(๐‘–๐‘)๐‘›๐‘ฅ๐‘›โˆ’๐›ผ๐ธ1,๐‘›โˆ’๐›ผ+1(๐‘–๐‘๐‘ฅ) โˆ’ (โˆ’๐‘–๐‘)

๐‘›๐‘ฅ๐‘›โˆ’๐›ผ๐ธ1,๐‘›โˆ’๐›ผ+1(โˆ’๐‘–๐‘๐‘ฅ)]

= โˆ’1

2๐‘–(๐‘–๐‘)๐‘›๐‘ฅ๐‘›โˆ’๐›ผ[๐ธ1,๐‘›โˆ’๐›ผ+1(๐‘–๐‘๐‘ฅ) โˆ’ (โˆ’1)

๐‘›๐ธ1,๐‘›โˆ’๐›ผ+1(โˆ’๐‘–๐‘๐‘ฅ)] โˆŽ

5) Follows by using (cos๐‘ฅ =๐‘’๐‘–๐‘ฅ+๐‘’โˆ’๐‘–๐‘ฅ

2) and the same as (4).

6) Follows by using (sinh ๐‘ฅ =๐‘’๐‘ฅโˆ’๐‘’โˆ’๐‘ฅ

2) and the same as (4).

7) Follows by using (cosh๐‘ฅ =๐‘’๐‘ฅ+๐‘’โˆ’๐‘ฅ

2) and the same as (4).

1.6 Comparison between Riemann-Liouville and Caputo Fractional Derivative

Operators

Our goal in this section is to make a comparison between the definitions of

fractional derivative of Riemann-Liouville and Caputo, because the definition of

fractional integral is the same for both Riemann-Liouville and Caputo definitions

Remark 1.6.1. [11] If ๐‘“(๐‘) = ๐‘“โ€ฒ(๐‘) = โ‹ฏ = ๐‘“(๐‘›โˆ’1)(๐‘) = 0, then

๐ท๐‘Ž๐›ผ๐‘“(๐‘ฅ) = ๐ท๐‘Ž

๐›ผ๐‘ ๐‘“(๐‘ฅ)

Remark 1.6.2. [11] The difference between Caputo and Riemann-Liouville formulas

for the fractional derivatives leads to the following differences:

Caputo fractional derivative of a constant equals zero while (Riemann-Liouville)

fractional derivative of a constant does not equal zero.

The non-commutation, in Caputo fractional derivative we have:

Page 46: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

37

๐ท๐‘Ž๐›ผ๐‘ ( ๐ท๐‘Ž

๐‘š๐‘ ๐‘“(๐‘ฅ)) = ๐ท๐‘Ž๐›ผ+๐‘š๐‘ ๐‘“(๐‘ฅ) โ‰  ๐ท๐‘Ž

๐‘š ( ๐ท๐‘Ž๐›ผ๐‘ ๐‘“(๐‘ฅ)) ,๐‘ (1.111)

where ๐›ผ โˆˆ (๐‘› โˆ’ 1, ๐‘›), ๐‘› โˆˆ โ„•,๐‘š = 1,2,โ€ฆ

While for Riemann-Liouville derivative

๐ท๐‘Ž๐‘š(๐ท๐‘Ž

๐›ผ๐‘“(๐‘ฅ)) = ๐ท๐‘Ž๐›ผ+๐‘š๐‘“(๐‘ฅ) โ‰  ๐ท๐‘Ž

๐›ผ(๐ท๐‘Ž๐‘š๐‘“(๐‘ฅ)) , (1.112)

where ๐›ผ โˆˆ (๐‘› โˆ’ 1, ๐‘›), ๐‘› โˆˆ โ„•,๐‘š = 1,2,โ€ฆ

Note that the formulas as in (1.111) and (1.112) become equalities under the following

additional conditions:

๐‘“(๐‘ )(๐‘Ž) = 0 , ๐‘  = ๐‘›, ๐‘› + 1,โ€ฆ ,๐‘š โˆ’ 1 for ๐ท๐‘ ๐›ผ

๐‘“(๐‘ )(๐‘Ž) = 0 , ๐‘  = 0,1,2,โ€ฆ ,๐‘š โˆ’ 1 for ๐ท๐›ผ.

Page 47: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

38

Table 1: Comparison between Riemann-Liouville and Caputo [11]

๐‘“(๐‘ก )

=๐‘=

consta

nt

Non-c

om

muta

tion

Lin

earity

Inte

rpola

tion

Repre

senta

tion

Pro

perty

๐ท๐›ผ๐‘=

๐‘

ฮ“(1โˆ’๐›ผ)๐‘กโˆ’๐›ผโ‰ 0 ,๐‘=๐‘๐‘œ๐‘›๐‘ ๐‘ก

๐ท๐‘š๐ท๐›ผ๐‘“(๐‘ก )

=๐ท๐›ผ+๐‘š๐‘“(๐‘ก)

โ‰ ๐ท๐›ผ๐ท๐‘š๐‘“(๐‘ก)

๐ท๐›ผ(๐œ†๐‘“(๐‘ก )+๐‘”(๐‘ก) )

=๐œ†๐ท๐›ผ๐‘“(๐‘ก )

+๐ท๐›ผ๐‘”(๐‘ก)

lim๐›ผโ†’๐‘›๐ท๐›ผ๐‘“(๐‘ก )

=๐‘“(๐‘›)(๐‘ก)

lim๐›ผโ†’๐‘›โˆ’1๐ท๐›ผ๐‘“(๐‘ก)

=๐‘“(๐‘›โˆ’1)(๐‘ก)

๐ท๐›ผ๐‘“(๐‘ก )

=๐ท๐›ผ(๐ท

โˆ’(๐‘›โˆ’๐›ผ)๐‘“(๐‘ฅ))

Rie

ma

nn

-Lio

uv

ille

๐ท๐‘Ž ๐›ผ๐‘=0 ,๐‘=๐‘๐‘œ๐‘›๐‘ ๐‘ก

๐ท๐‘Ž ๐›ผ

๐‘๐ท๐‘š๐‘“(๐‘ก )

=๐ท๐‘Ž ๐›ผ+๐‘š๐‘“(๐‘ก)

โ‰ ๐ท๐‘š๐ท๐‘Ž ๐›ผ๐‘“(๐‘ก)

๐ท๐‘Ž ๐›ผ

๐‘(๐œ†๐‘“(๐‘ก )+๐‘”(๐‘ก) )

=๐œ†๐ท๐‘Ž ๐›ผ๐‘“(๐‘ก )

+๐ท๐‘Ž ๐›ผ๐‘”(๐‘ก)

lim๐›ผโ†’๐‘›๐ท๐‘Ž ๐›ผ

๐‘๐‘“(๐‘ก )

=๐‘“(๐‘›)(๐‘ก)

lim๐›ผโ†’๐‘›โˆ’1๐ท๐‘Ž ๐›ผ

๐‘๐‘“(๐‘ก )

=๐‘“(๐‘›โˆ’1)(๐‘ก )

โˆ’๐‘“(๐‘›โˆ’1)(0)

๐ท ๐‘๐‘Ž ๐›ผ๐‘“(๐‘ก )

=๐ทโˆ’(๐‘›โˆ’๐›ผ)๐ท๐‘Ž ๐‘›

๐‘๐‘“(๐‘ฅ)

Ca

pu

to

Page 48: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

39

1.7 Ordinary Differential Equations [2] :

This section shows some basic information about ordinary differential equation

which is needed in this thesis.

1.7.1. Bernoulli Differential Equation

Let us take a look at differential equation on the form

๐‘ฆโ€ฒ + ๐‘(๐‘ฅ)๐‘ฆ = ๐‘ž(๐‘ฅ)๐‘ฆ๐‘› (1.113)

where ๐‘(๐‘ฅ) and ๐‘ž(๐‘ฅ) both are continous, ๐‘› โˆˆ โ„.

Differential equation above is called Bernoulli equation

Now we solve (1.113) by dividing both sides by ๐‘ฆ๐‘›.

๐‘ฆโˆ’๐‘›๐‘ฆโ€ฒ + ๐‘(๐‘ฅ)๐‘ฆ1โˆ’๐‘› = ๐‘ž(๐‘ฅ) (1.114)

Let ๐‘ฃ = ๐‘ฆ1โˆ’๐‘›, then

๐‘ฃโ€ฒ = (1 โˆ’ ๐‘›)๐‘ฆโˆ’๐‘›๐‘ฆโ€ฒ

Multiply (1.113) by (1 โˆ’ ๐‘›)๐‘ฆโˆ’๐‘› , we get:

1

1 โˆ’ ๐‘›๐‘ฃโ€ฒ + ๐‘(๐‘ฅ)๐‘ฃ = ๐‘ž(๐‘ฅ) (1.115)

This is a linear differential equation.

1.7.2 Second-Order Linear Differential Equations [2]:

A second-order linear differential equation has the form

Page 49: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

40

๐ด(๐‘ฅ)๐‘ฆโ€ฒโ€ฒ + ๐‘ƒ(๐‘ฅ)๐‘ฆ + ๐‘„(๐‘ฅ) = ๐บ(๐‘ฅ( (1.116)

where A,P,Q and G are continuous functions, when ๐บ(๐‘ฅ) = 0, for all ๐‘ฅ, in equation

(1.116). Such equations are called homogenous linear equations. Thus, the form of a

second-order linear homogenous differential equation is:

๐ด(๐‘ฅ)๐‘‘2๐‘ฆ

๐‘‘๐‘ฅ2+ ๐‘ƒ(๐‘ฅ)

๐‘‘๐‘ฆ

๐‘‘๐‘ฅ+ ๐‘„(๐‘ฅ) = 0 (1.117)

( if ๐บ(๐‘ฅ) โ‰  0 for some ๐‘ฅ, equation (1.116) is called nonhomogeneous equation)

Page 50: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

41

Chapter Two: Conformable Fractional Definition

2.1 Conformable Fractional Derivative

When we study the previous definitions of derivative, we can illustrate that those

definitions have some inconveniences. The following are some of these shortcomings:

i) The Riemann-liouville derivative does not satisfy ๐ท๐‘Ž๐›ผ(1) = 0

(๐ท๐‘Ž๐›ผ(1) = 0 for the Caupto derivative) , if ฮฑ is not a natural number.

ii) All fractional derivatives do not satisfy the Known product rule:

๐ท๐‘Ž๐›ผ(๐‘“๐‘”) = ๐‘“๐ท๐‘Ž

๐›ผ(๐‘“) + ๐‘”๐ท๐‘Ž๐›ผ(๐‘“)

iii) All fractional derivatives do not satisfy the known quotient rule:

๐ท๐‘Ž๐›ผ(๐‘“ ๐‘”) =

๐‘”๐ท๐‘Ž๐›ผ(๐‘“)โˆ’๐‘“๐ท๐‘Ž

๐›ผ(๐‘”)

๐‘”2โ„

iv) All fractional derivatives do not satisfy the chain rule:

๐ท๐‘Ž๐›ผ(๐‘“ โˆ˜ ๐‘”)(๐‘ก) = ๐‘“(๐›ผ)(๐‘”(๐‘ก))๐‘”(๐›ผ)(๐‘ก)

v) All fractional derivatives don't satisfy: ๐ท๐›ผ๐ท๐›ฝ๐‘“ = ๐ท๐›ผ+๐›ฝ๐‘“ in general

vi) The Caputo definition assumes that the function f is differentiable.

Let us write Tฮฑ to denote the operator which is called the "Conformable

fractional derivative of order ฮฑ ".

Khalil, et al. [14] introduced a completely new definition of fractional calculus which

is more natural and effective than previous definitions of order ๐›ผ โˆˆ (0, 1]. Also, this

definition can be generalized to include any ฮฑ. However, the case ๐›ผ โˆˆ (0, 1] is the most

important one, and the other cases become easy when it is established..

Page 51: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

42

Definition 2.1.1. [14] Given a function โˆถ ๐‘“: [0,โˆž) โ†’ โ„ . Then the (conformable fractional

derivative) of ๐‘“ of order ๐›ผ is defined by

๐‘‡๐›ผ(๐‘“)(๐‘ก) = ๐‘™๐‘–๐‘š๐œ€โ†’0

๐‘“(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ) โˆ’ ๐‘“(๐‘ก)

๐œ€

For all ๐‘ก หƒ0, ๐›ผ โˆˆ (0.1), if ฦ’ is ฮฑ-differentiable in some (0, ๐›ผ). ๐›ผ หƒ 0 and, lim๐‘กโ†’0+ ๐‘“(๐›ผ)(๐‘ก)

exists, then define๐‘“(๐›ผ)(0) = lim๐‘กโ†’0+ ๐‘“(๐›ผ)(๐‘ก)

We sometimes, write ๐‘“(๐›ผ)(๐‘ก) for ๐‘‡๐›ผ(๐‘“ )(๐‘ก), to denote the conformable fractional

derivatives of ๐‘“ of order ๐›ผ. In addition, if the conformable fractional derivative of f

of order ฮฑ exists, then we say ๐‘“ is ฮฑ-differentiable.

We should take into consideration that ๐‘‡๐›ผ(๐‘ก๐‘) = ๐‘๐‘ก๐‘โˆ’๐›ผ. Further, this definition

coincides happen with the same of traditional definition of Riemannโ€“Liouville and

of Caputo on polynomials (up to a constant multiple).

Theorem 2.1.1. [14] if a function ๐‘“: [0,โˆž) โ†’ โ„ is ๐›ผ-differentiable at ๐‘ก0 หƒ 0. ๐›ผ โˆˆ

(0.1] then ๐‘“ is continuous at ๐‘ก0

Proof:

Because ๐‘“(๐›ผ)is differentiable at ๐‘ฅ = ๐‘ก0, we know that

๐‘“(๐›ผ)(๐‘ก0) = ๐‘™๐‘–๐‘š๐œ€โ†’0๐‘“(๐‘ก0+๐œ€๐‘ก0

1โˆ’๐›ผ)โˆ’๐‘“(๐‘ก0)

๐œ€ exists.

If we next assume that ๐‘ฅ โ‰  ๐‘ก0 we can write the following

๐‘“(๐‘ก0 + ๐œ€๐‘ก01โˆ’๐›ผ) โˆ’ ๐‘“(๐‘ก0) =

๐‘“(๐‘ก0 + ๐œ€๐‘ก01โˆ’๐›ผ) โˆ’ ๐‘“(๐‘ก0)

๐œ€๐œ€

Page 52: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

43

Then some basic properties of limits give us

๐‘™๐‘–๐‘š๐œ€โ†’0( ๐‘“(๐‘ก0 + ๐œ€๐‘ก0

1โˆ’๐›ผ) โˆ’ ๐‘“(๐‘ก0)) = ๐‘™๐‘–๐‘š๐œ€โ†’0

๐‘“(๐‘ก0 + ๐œ€๐‘ก01โˆ’๐›ผ) โˆ’ ๐‘“(๐‘ก0)

๐œ€. ๐‘™๐‘–๐‘š๐œ€โ†’0

๐œ€

๐‘™๐‘–๐‘š๐œ€โ†’0( ๐‘“(๐‘ก0 + ๐œ€๐‘ก0

1โˆ’๐›ผ) โˆ’ ๐‘“(๐‘ก0)) = ๐‘“โ€ฒ(๐‘ก0). 0

Let โ„Ž = ๐œ€๐‘ก01โˆ’๐›ผ. Then,

๐‘™๐‘–๐‘šโ„Žโ†’0 ๐‘“(๐‘ก0 + โ„Ž) = ๐‘“(๐‘ก0) . Hence, f is continuous at ๐‘ก0

It can be easily shown that ๐‘‡๐›ผ satisfies all properties in the following theorem

Theorem 2.1.2. [14] Let ๐›ผ โˆˆ (0.1] and ๐‘“, ๐‘” be ฮฑ-differentiable at a point ๐‘ก หƒ 0 .Then:

(1) ๐‘‡๐›ผ(๐‘“)(๐‘ก) = ๐‘ก1โˆ’๐›ผ ๐‘‘๐‘“

๐‘‘๐‘ก(๐‘ก), where f is differentiable

(2.1)

(2) ๐‘‡๐›ผ(af + bg) = a ๐‘‡๐›ผ (f ) + b ๐‘‡๐›ผ (g), for all ๐‘Ž, ๐‘ โˆˆ โ„ (2.2)

(3) ๐‘‡๐›ผ (๐‘ก๐‘) = ๐‘ ๐‘ก๐‘โˆ’๐›ผ for all ๐‘ โˆˆ โ„ (2.3)

(4) ๐‘‡๐›ผ (ฮป)=0 , for all constant functions ๐‘“ (๐‘ก) = ๐œ† (2.4)

(5) ๐‘‡๐›ผ (fg) = f ๐‘‡๐›ผ (g) + g ๐‘‡๐›ผ (f ) (2.5)

(6) ๐‘‡๐›ผ ( ฦ’

๐‘” ) =

๐‘” ๐‘‡๐›ผฦ’ โˆ’ ฦ’ ๐‘‡๐›ผ(๐‘”)

๐‘”2 (2.6)

Proof:

(1) Let โ„Ž = ๐œ€๐‘ก1โˆ’๐›ผ in definition (2.1.1). Then ๐œ€ = โ„Ž๐‘ก๐›ผโˆ’1

Page 53: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

44

๐‘‡๐›ผ(๐‘“)(๐‘ก) = lim๐œ€โ†’0

๐‘“(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ) โˆ’ ๐‘“(๐‘ก)

๐œ€= lim๐œ€โ†’0

๐‘“(๐‘ก + โ„Ž) โˆ’ ๐‘“(๐‘ก)

โ„Ž๐‘ก๐›ผโˆ’1

= ๐‘ก1โˆ’๐›ผ limโ„Žโ†’0

๐‘“(๐‘ก + โ„Ž) โˆ’ ๐‘“(๐‘ก)

โ„Ž= ๐‘ก1โˆ’๐›ผ๐‘“โ€ฒ(๐‘ก) โˆŽ

(2) ๐‘‡๐›ผ(๐‘Ž๐‘“ + ๐‘๐‘”) = lim๐œ€โ†’0(๐‘Ž๐‘“+๐‘๐‘”)(๐‘ก+๐œ€๐‘ก1โˆ’๐›ผ)โˆ’(๐‘Ž๐‘“+๐‘๐‘”)(๐‘ก)

๐œ€

= lim๐œ€โ†’0

๐‘Ž๐‘“(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ) + ๐‘๐‘”(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ) โˆ’ ๐‘Ž๐‘“(๐‘ก) โˆ’ ๐‘๐‘”(๐‘ก)

๐œ€

= lim๐œ€โ†’0

๐‘Ž (๐‘“(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ) โˆ’ ๐‘“(๐‘ก)

๐œ€) + lim

๐œ€โ†’0๐‘ (๐‘”(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ) โˆ’ ๐‘”(๐‘ก)

๐œ€)

= ๐‘Ž๐‘‡๐›ผ(๐‘“) + ๐‘๐‘‡๐›ผ(๐‘”) โˆŽ

(3) Recall (๐‘Ž + ๐‘)๐‘› = โˆ‘(๐‘›

๐‘˜) ๐‘Ž๐‘›โˆ’๐‘˜

๐‘›

๐‘˜=0

๐‘๐‘˜

Thus,

(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ)๐‘ =โˆ‘(๐‘

๐‘˜) ๐‘ก๐‘โˆ’๐‘˜

๐‘

๐‘˜=0

(๐œ€๐‘ก1โˆ’๐›ผ)๐‘˜

(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ)๐‘ = (๐‘

0) ๐‘ก๐‘ + (

๐‘

1) ๐‘ก๐‘โˆ’1(๐œ€๐‘ก1โˆ’๐›ผ)1 +โ‹ฏ+ (

๐‘

๐‘) ๐‘ก0(๐œ€๐‘ก1โˆ’๐›ผ)๐‘

To proof that ๐‘‡๐›ผ(๐‘ก๐‘) = ๐‘๐‘ก๐‘โˆ’๐›ผ

๐‘™๐‘–๐‘š๐œ€โ†’0

(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ)๐‘ โˆ’ ๐‘ก๐‘

๐œ€= ๐‘™๐‘–๐‘š

๐œ€โ†’0

๐‘ก๐‘ + (๐‘1)๐‘ก๐‘โˆ’1(๐œ€๐‘ก1โˆ’๐›ผ) + โ‹ฏ+ (๐‘

๐‘) ๐œ€๐‘(๐‘ก1โˆ’๐›ผ)๐‘ โˆ’ ๐‘ก๐‘

๐œ€

= ๐‘™๐‘–๐‘š ๐œ€โ†’0

๐œ€๐‘๐‘ก๐‘โˆ’1๐‘ก1โˆ’๐›ผ +โ‹ฏ+ (๐‘

๐‘) ๐œ€๐‘โˆ’1(๐‘ก1โˆ’๐›ผ)๐‘

๐œ€

= ๐‘๐‘ก๐‘โˆ’1๐‘ก1โˆ’๐›ผ = ๐‘๐‘ก๐‘โˆ’๐›ผ

Page 54: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

45

(4) ๐‘‡๐›ผ(๐œ†) = lim๐œ€โ†’0๐‘“(๐‘ก+๐œ€๐‘ก1โˆ’๐›ผ)โˆ’๐‘“(๐‘ก)

๐œ€

= lim๐œ€โ†’0

๐œ† โˆ’ ๐œ†

๐œ€= 0 โˆŽ

(5)

= lim๐œ€โ†’0

๐‘“(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ)๐‘”(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ) + ๐‘“(๐‘ก)๐‘”(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ) โˆ’ ๐‘“(๐‘ก)๐‘”(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ) โˆ’ ๐‘“(๐‘ก)๐‘”(๐‘ก)

๐œ€

= lim๐œ€โ†’0

๐‘“(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ) โˆ’ ๐‘“(๐‘ก)

๐œ€๐‘”(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ) + ๐‘“(๐‘ก) lim

๐œ€โ†’0

๐‘”(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ) โˆ’ ๐‘”(๐‘ก)

๐œ€

= ๐‘‡๐›ผ(๐‘“(๐‘ก)) lim๐œ€โ†’0

๐‘”(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ) + ๐‘“(๐‘ก)๐‘‡๐›ผ(๐‘”(๐‘ก))

= ๐‘”(๐‘ก)๐‘‡๐›ผ(๐‘“(๐‘ก)) + ๐‘“(๐‘ก)๐‘‡๐›ผ(๐‘”(๐‘ก)) โˆŽ

(6)

= lim๐œ€โ†’0

(๐‘“(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ)

๐‘”(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ)โˆ’

๐‘“(๐‘ก)

๐‘”(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ)+

๐‘“(๐‘ก)

๐‘”(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ)โˆ’๐‘“(๐‘ก)

๐‘”(๐‘ก)) .1

๐œ€

= lim๐œ€โ†’0

(๐‘“(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ) โˆ’ ๐‘“(๐‘ก)

๐œ€. ๐‘”(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ)) + ๐‘“(๐‘ก). lim

๐œ€โ†’0(

1

๐œ€๐‘”(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ)โˆ’

1

๐œ€๐‘”(๐‘ก))

= ๐‘‡๐›ผ(๐‘“) lim๐œ€โ†’0

1

๐‘”(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ)+ ๐‘“(๐‘ก) lim

๐œ€โ†’0(โˆ’(

๐‘”(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ) โˆ’ ๐‘”(๐‘ก)

๐œ€๐‘”(๐‘ก)๐‘”(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ)))

= ๐‘‡๐›ผ(๐‘“)1

๐‘”(๐‘ก)โˆ’ ๐‘“(๐‘ก) lim

๐œ€โ†’0(๐‘”(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ) โˆ’ ๐‘”(๐‘ก)

๐œ€) lim๐œ€โ†’0

(1

๐‘”(๐‘ก)๐‘”(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ))

๐‘‡๐›ผ(๐‘“๐‘”) = lim๐œ€โ†’0

๐‘“(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ)๐‘”(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ) โˆ’ ๐‘“(๐‘ก)๐‘”(๐‘ก)

๐œ€

๐‘‡๐›ผ (๐‘“

๐‘”) = lim

๐œ€โ†’0

๐‘“(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ)๐‘”(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ)

โˆ’๐‘“(๐‘ก)๐‘”(๐‘ก)

๐œ€

Page 55: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

46

=๐‘‡๐›ผ(๐‘“)

๐‘”(๐‘ก)โˆ’ ๐‘“(๐‘ก)๐‘‡๐›ผ(๐‘”(๐‘ก)).

1

๐‘”2(๐‘ก)

=๐‘”(๐‘ก)๐‘‡๐›ผ(๐‘“) โˆ’ ๐‘“(๐‘ก)๐‘‡๐›ผ(๐‘”(๐‘ก))

๐‘”2(๐‘ก) โˆŽ

Theorem 2.1.3. [14] (Conformable fractional derivative of Known functions)

1) ๐‘‡๐›ผ(๐‘’๐‘๐‘ก) = ๐‘๐‘ก1โˆ’๐›ผ๐‘’๐‘๐‘ก (2.7)

2) ๐‘‡๐›ผ(๐‘ ๐‘–๐‘› (๐‘Ž๐‘ก)) = ๐‘Ž๐‘ก1โˆ’๐›ผ ๐‘๐‘œ๐‘  (๐‘Ž๐‘ก) , ๐‘Ž โˆˆ โ„ (2.8)

3) ๐‘‡๐›ผ (๐‘๐‘œ๐‘  (๐‘Ž๐‘ก)) = โˆ’๐‘Ž๐‘ก1โˆ’๐›ผ ๐‘ ๐‘–๐‘› (๐‘Ž๐‘ก), ๐‘Ž โˆˆ โ„ (2.9)

4) ๐‘‡๐›ผ(tan(๐‘Ž๐‘ก)) = ๐‘Ž๐‘ก1โˆ’๐›ผ๐‘ ๐‘’๐‘2(๐‘Ž๐‘ก) , ๐‘Ž โˆˆ โ„ (2.10)

5) ๐‘‡๐›ผ(๐‘๐‘œ๐‘ก(๐‘Ž๐‘ก)) = โˆ’๐‘Ž๐‘ก1โˆ’๐›ผ๐‘๐‘ ๐‘2(๐‘Ž๐‘ก) , ๐‘Ž โˆˆ โ„ (2.11)

6) ๐‘‡๐›ผ(๐‘ ๐‘’๐‘(๐‘Ž๐‘ก)) = ๐‘Ž๐‘ก1โˆ’๐›ผ ๐‘ ๐‘’๐‘(๐‘Ž๐‘ก) ๐‘ก๐‘Ž๐‘›(๐‘Ž๐‘ก) , ๐‘Ž โˆˆ โ„ (2.12)

7) ๐‘‡๐›ผ(๐‘๐‘ ๐‘(๐‘Ž๐‘ก)) = โˆ’๐‘Ž๐‘ก1โˆ’๐›ผ ๐‘๐‘ ๐‘(๐‘Ž๐‘ก) ๐‘๐‘œ๐‘ก(๐‘Ž๐‘ก) , ๐‘Ž โˆˆ โ„ (2.13)

8) ๐‘‡๐›ผ (1

๐›ผ๐‘ก๐›ผ) = 1 (2.14)

Proof:

1. ๐‘‡๐›ผ(๐‘’๐‘๐‘ฅ) = ๐‘™๐‘–๐‘š๐œ€โ†’0

๐‘’๐‘(๐‘ก+๐œ€๐‘ก1โˆ’๐›ผ)โˆ’๐‘’๐‘๐‘ก

๐œ€= ๐‘’๐‘๐‘ก ๐‘™๐‘–๐‘š๐œ€โ†’0

๐‘’๐‘๐œ€๐‘ก1โˆ’๐›ผ

โˆ’1

๐œ€

= ๐‘’๐‘๐‘ก ๐‘™๐‘–๐‘š๐œ€โ†’0

๐‘ก1โˆ’๐›ผ๐‘’๐‘๐œ€๐‘ก1โˆ’๐›ผ

โˆ’ ๐‘ก1โˆ’๐›ผ

๐œ€๐‘ก1โˆ’๐›ผ= ๐‘’๐‘๐‘ก๐‘ก1โˆ’๐›ผ ๐‘™๐‘–๐‘š

๐œ€โ†’0

๐‘’๐‘๐œ€๐‘ก1โˆ’๐›ผ

โˆ’ 1

๐œ€๐‘ก1โˆ’๐›ผ

Let โ„Ž = ๐œ€๐‘ก1โˆ’๐›ผ . Then by using Lโ€™Hopitalโ€˜s rule, we get

Page 56: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

47

= ๐‘ก1โˆ’๐›ผ๐‘’๐‘๐‘ก ๐‘™๐‘–๐‘šโ„Žโ†’0

๐‘’๐‘โ„Ž โˆ’ 1

โ„Ž= ๐‘๐‘ก1โˆ’๐›ผ๐‘’๐‘๐‘ก ๐‘™๐‘–๐‘š

โ„Žโ†’0

๐‘’๐‘โ„Ž

1

= ๐‘๐‘ก1โˆ’๐›ผ๐‘’๐‘๐‘ก โˆŽ

(By Lโ€™Hopital Rule)

2. ๐‘‡๐›ผ(๐‘ ๐‘–๐‘›(๐‘Ž๐‘ก)) = ๐‘™๐‘–๐‘š๐œ€โ†’0

๐‘ ๐‘–๐‘› ๐‘Ž(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ) โˆ’ ๐‘ ๐‘–๐‘›(๐‘Ž๐‘ก)

๐œ€

= ๐‘™๐‘–๐‘š๐œ€โ†’0

๐‘ ๐‘–๐‘›(๐‘Ž๐‘ก) [๐‘๐‘œ๐‘ (๐‘Ž๐œ€๐‘ก1โˆ’๐›ผ) โˆ’ 1

๐œ€] + ๐‘™๐‘–๐‘š

๐œ€โ†’0

๐‘๐‘œ๐‘ (๐‘Ž๐‘ก) ๐‘ ๐‘–๐‘›(๐‘Ž๐œ€๐‘ก1โˆ’๐›ผ)

๐œ€

= ๐‘ก1โˆ’๐›ผ ๐‘ ๐‘–๐‘›(๐‘Ž๐‘ก) ๐‘™๐‘–๐‘š๐œ€โ†’0

[๐‘๐‘œ๐‘ (๐‘Ž๐œ€๐‘ก1โˆ’๐›ผ) โˆ’ 1

๐œ€๐‘ก1โˆ’๐›ผ]

+ ๐‘ก1โˆ’๐›ผ ๐‘๐‘œ๐‘ (๐‘Ž๐‘ก) ๐‘™๐‘–๐‘š๐œ€โ†’0

๐‘ ๐‘–๐‘›(๐‘Ž๐œ€๐‘ก1โˆ’๐›ผ)

๐œ€๐‘ก1โˆ’๐›ผ

Let โ„Ž = ๐œ€๐‘ก1โˆ’๐›ผ then we get

= ๐‘ก1โˆ’๐›ผ ๐‘ ๐‘–๐‘›(๐‘Ž๐‘ก) ๐‘™๐‘–๐‘šโ„Žโ†’0

[๐‘๐‘œ๐‘ (๐‘Žโ„Ž) โˆ’ 1

โ„Ž] + ๐‘ก1โˆ’๐›ผ ๐‘๐‘œ๐‘ (๐‘Ž๐‘ก) ๐‘™๐‘–๐‘š

โ„Žโ†’0

๐‘ ๐‘–๐‘›(๐‘Žโ„Ž)

โ„Ž

By using Lโ€™Hoputal Rule, we get

= ๐‘ก1โˆ’๐›ผsin (๐‘Ž๐‘ก) ๐‘™๐‘–๐‘šโ„Žโ†’0

โˆ’๐‘Ž ๐‘ ๐‘–๐‘›(๐‘Žโ„Ž)

1+ ๐‘ก1โˆ’๐›ผ ๐‘๐‘œ๐‘ (๐‘Ž๐‘ก) . ๐‘Ž

= ๐‘Ž๐‘ก1โˆ’๐›ผ ๐‘๐‘œ๐‘ (๐‘Ž๐‘ก) โˆŽ

3. Similar to (2)

= ๐‘™๐‘–๐‘š๐œ€โ†’0

๐‘ ๐‘–๐‘›(๐‘Ž๐‘ก) ๐‘๐‘œ๐‘ (๐‘Ž๐œ€๐‘ก1โˆ’๐›ผ) + ๐‘๐‘œ๐‘ (๐‘Ž๐‘ก) ๐‘ ๐‘–๐‘›(๐‘Ž๐œ€๐‘ก1โˆ’๐›ผ) โˆ’ ๐‘ ๐‘–๐‘›(๐‘Ž๐‘ก)

๐œ€

Page 57: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

48

4. ๐‘‡๐›ผ(๐‘ก๐‘Ž๐‘›(๐‘Ž๐‘ก)) = ๐‘‡๐›ผ (๐‘ ๐‘–๐‘›(๐‘Ž๐‘ก)

๐‘๐‘œ๐‘ (๐‘Ž๐‘ก))

=๐‘๐‘œ๐‘ (๐‘Ž๐‘ก) ๐‘‡๐›ผ(๐‘ ๐‘–๐‘›(๐‘Ž๐‘ก)) โˆ’ ๐‘ ๐‘–๐‘›(๐‘Ž๐‘ก) ๐‘‡๐›ผ(๐‘๐‘œ๐‘  ๐‘Ž๐‘ก)

๐‘๐‘œ๐‘ 2(๐‘Ž๐‘ก)

=๐‘๐‘œ๐‘ (๐‘Ž๐‘ก) (๐‘Ž๐‘ก1โˆ’๐›ผ ๐‘๐‘œ๐‘ (๐‘Ž๐‘ก)) โˆ’ ๐‘ ๐‘–๐‘›(๐‘Ž๐‘ก)(โˆ’๐‘Ž๐‘ก1โˆ’๐›ผ ๐‘ ๐‘–๐‘›(๐‘Ž๐‘ก))

๐‘๐‘œ๐‘ 2(๐‘Ž๐‘ก)

=๐‘Ž๐‘ก1โˆ’๐›ผ ๐‘๐‘œ๐‘ 2(๐‘Ž๐‘ก) + ๐‘Ž๐‘ก1โˆ’๐›ผ ๐‘ ๐‘–๐‘›2(๐‘Ž๐‘ก)

๐‘๐‘œ๐‘ 2(๐‘Ž๐‘ก)

= ๐‘Ž๐‘ก1โˆ’๐›ผ(1 + ๐‘ก๐‘Ž๐‘›2(๐‘Ž๐‘ก))

= ๐‘Ž๐‘ก1โˆ’๐›ผ ๐‘ ๐‘’๐‘2(๐‘Ž๐‘ก) โˆŽ

5. Similar to (4)

6. ๐‘‡๐›ผ(๐‘ ๐‘’๐‘(๐‘Ž๐‘ก)) = ๐‘‡๐›ผ (1

๐‘๐‘œ๐‘ (๐‘Ž๐‘ก)) =

(โˆ’1)(๐‘‡๐›ผ(๐‘๐‘œ๐‘ (๐‘Ž๐‘ก)))

๐‘๐‘œ๐‘ 2(๐‘Ž๐‘ก)

=(โˆ’1)(โˆ’๐‘Ž๐‘ก1โˆ’๐›ผ ๐‘ ๐‘–๐‘›(๐‘Ž๐‘ก))

๐‘๐‘œ๐‘ 2(๐‘Ž๐‘ก)= ๐‘Ž๐‘ก1โˆ’๐›ผ

๐‘ ๐‘–๐‘›(๐‘Ž๐‘ก)

๐‘๐‘œ๐‘ (๐‘Ž๐‘ก).

1

๐‘๐‘œ๐‘ (๐‘Ž๐‘ก)

= ๐‘Ž๐‘ก1โˆ’๐›ผ ๐‘ก๐‘Ž๐‘›(๐‘Ž๐‘ก) ๐‘ ๐‘’๐‘(๐‘Ž๐‘ก) โˆŽ

7. Similar to (7)

8. ๐‘‡๐›ผ (1

๐›ผ๐‘ก๐›ผ) = ๐‘™๐‘–๐‘š

๐œ€โ†’0

1๐›ผ(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ)๐›ผ โˆ’

1๐›ผ ๐‘ก

๐›ผ

๐œ€

=1

๐›ผ๐‘™๐‘–๐‘š๐œ€โ†’0

(๐‘ก + ๐œ€๐‘ก1โˆ’๐›ผ)๐›ผ โˆ’ ๐‘ก๐›ผ

๐œ€

=1

๐›ผ๐‘™๐‘–๐‘š๐œ€โ†’0

๐‘ก๐›ผ + (๐›ผ1) ๐‘ก๐›ผโˆ’1๐œ€๐‘ก1โˆ’๐›ผ +โ‹ฏ+ (

๐›ผ๐›ผ โˆ’ 1

)๐œ€๐›ผโˆ’1๐‘ก๐›ผโˆ’1 + (๐›ผ๐›ผ)๐œ€๐›ผ(๐‘ก1โˆ’๐›ผ)๐›ผ โˆ’ ๐‘ก๐›ผ

๐œ€

Page 58: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

49

=1

๐›ผ๐‘™๐‘–๐‘š๐œ€โ†’0

๐œ€ ((๐›ผ1) + (

๐›ผ๐›ผ โˆ’ 1

) ๐‘ก๐›ผ๐œ€๐›ผโˆ’2 +โ‹ฏ+ (๐›ผ๐›ผ)๐‘ก๐›ผโˆ’1(๐‘ก1โˆ’๐›ผ)๐›ผ)

๐œ€

=1

๐›ผ. ๐›ผ = 1 โˆŽ

Corollary 2.1.1. (Conformable fractional derivative of certain functions)

i) ๐‘‡๐›ผ (๐‘ ๐‘–๐‘›1

๐›ผ๐‘ก๐›ผ) = ๐‘๐‘œ๐‘ 

1

๐›ผ๐‘ก๐›ผ (2.15)

ii) ๐‘‡๐›ผ (๐‘ ๐‘–๐‘›1

๐›ผ๐‘ก๐›ผ) = ๐‘๐‘œ๐‘ 

1

๐›ผ๐‘ก๐›ผ (2.16)

iii) ๐‘‡๐›ผ (๐‘’1๐›ผ๐‘ก๐›ผ) = ๐‘’

1๐›ผ๐‘ก๐›ผ

(2.17)

Note: The function could be ฮฑ-differentiable at a point but not differentiable. For

example, let ๐‘“(๐‘ก) = 2โˆš๐‘ก.

Then, ๐‘‡12

(๐‘“)(0) = ๐‘™๐‘–๐‘š๐‘กโ†’0+ ๐‘‡12

(๐‘“)(๐‘ก) = 1 , when ๐‘‡12

(๐‘“)(๐‘ก) = 1 , for all t>0 , but

๐‘‡1(๐‘“)(0) does not exist.

The most important case for the range of ๐›ผ โˆˆ (0,1), when ๐›ผ โˆˆ (๐‘›, ๐‘› + 1] the

definition would be as the following

Definition 2.1.2. [14] Let ๐›ผ โˆˆ (๐‘›, ๐‘› + 1], and f be an n-differentiable at t ,

where t > 0, then the conformable fractional derivative of f of order ฮฑ is defined as:

๐‘‡๐›ผ(๐‘“)(๐‘ก) = limฮตโ†’0

ฦ’(โŒˆฮฑโŒ‰โˆ’1) ( ๐‘ก + ๐œ€๐‘ก(โŒˆฮฑโŒ‰โˆ’ฮฑ)) โˆ’ ฦ’(โŒˆฮฑโŒ‰โˆ’1)(t)

๐œ€

where [ฮฑ] is the smallest integer greater than or equal to ฮฑ.

Page 59: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

50

Remark 2.1.1. Let ๐›ผ โˆˆ (๐‘›, ๐‘› + 1], and f is (๐‘› + 1)-differentiable at ๐‘ก > 0. Then:

๐‘‡๐›ผ(๐‘“)(๐‘ก) = ๐‘ก(โŒˆฮฑโŒ‰โˆ’ฮฑ) ๐‘“

โŒˆฮฑโŒ‰(๐‘ก) (2.18)

Theorem 2.1.4 [14]

(Rolleโ€™s Theorem for Conformable Fractional Differentiable Functions).

Let ๐‘Ž > 0 and ๐‘“ โˆถ [๐‘Ž, ๐‘] โ†’ โ„ be a given function that satisfies

i. ๐‘“ is continuous on [๐‘Ž, ๐‘],

ii. ๐‘“ is ฮฑ-differentiable for some ๐›ผ โˆˆ (0,1),

iii. ๐‘“(๐‘Ž) = ๐‘“(๐‘).

Then, there exists ๐‘ โˆˆ (๐‘Ž, ๐‘), such that ๐‘“(๐›ผ)(๐‘) = 0.

Proof:

Since ๐‘“ is continuous on [๐‘Ž, ๐‘], and ๐‘“(๐‘Ž) = ๐‘“(๐‘), there is ๐‘ โˆˆ (๐‘Ž, ๐‘), which is

a point of local extrema. With no loss of generality, assume c is a point of local

minimum. So, ๐‘“(๐›ผ)(๐‘) = lim๐œ€โ†’ 0+๐‘“(๐‘+๐œ€๐‘1โˆ’๐›ผ)โˆ’๐‘“(๐‘)

๐œ€= lim๐œ€โ†’ 0โˆ’

๐‘“(๐‘+๐œ€๐‘1โˆ’๐›ผ)โˆ’๐‘“(๐‘)

๐œ€, but the

first limit is non โ€“ negative, and the second limit is non-positive. Hence, ๐‘“(๐›ผ)(๐‘) = 0.

Theorem 2.1.5. [14] (Mean Value Theorem for Conformable Fractional

Differentiable Functions). Let a > 0 and ๐‘“ : [๐‘Ž, ๐‘] โ†’ โ„ be a given function that

satisfies:

Page 60: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

51

i) ๐‘“ is continuous on [๐‘Ž, ๐‘].

ii) ๐‘“ is ฮฑ-differentiable for some ๐›ผ โˆˆ (0, 1).

Then, there exists ๐‘ โˆˆ (๐‘Ž, ๐‘), such that

Proof:

The equation of the secant through (๐‘Ž, ๐‘“(๐‘Ž)) and (๐‘, ๐‘“(๐‘)) is

๐‘ฆ โˆ’ ๐‘“(๐‘Ž) =๐‘“(๐‘) โˆ’ ๐‘“(๐‘Ž)

1๐›ผ ๐‘

๐›ผ โˆ’1๐›ผ ๐‘Ž

๐›ผ(1

๐›ผ๐‘ฅ๐›ผ โˆ’

1

๐›ผ๐‘Ž๐›ผ)

which we can write as

๐‘ฆ =๐‘“(๐‘) โˆ’ ๐‘“(๐‘Ž)

1๐›ผ ๐‘

๐›ผ โˆ’1๐›ผ ๐‘Ž

๐›ผ(1

๐›ผ๐‘ฅ๐›ผ โˆ’

1

๐›ผ๐‘Ž๐›ผ) + ๐‘“(๐‘Ž)

Let ๐‘”(๐‘ฅ) = ๐‘“(๐‘ฅ) โˆ’ [๐‘“(๐‘)โˆ’๐‘“(๐‘Ž)1

๐›ผ๐‘๐›ผโˆ’

1

๐›ผ๐‘Ž๐›ผ(1

๐›ผ๐‘ฅ๐›ผ โˆ’

1

๐›ผ๐‘Ž๐›ผ) + ๐‘“(๐‘Ž)].

Note that ๐‘”(๐‘Ž) = ๐‘”(๐‘) = 0 , ๐‘” is continuous on [๐‘Ž, ๐‘] and differentiable on (๐‘Ž, ๐‘). So

by Rollโ€™s theorem there are ๐‘ in (๐‘Ž, ๐‘) such that ๐‘”(๐›ผ)(๐‘) = 0.

But

๐‘“(๐›ผ)(๐‘) =ฦ’(๐‘) โˆ’ ฦ’(๐‘Ž)

1๐›ผ ๐‘

๐›ผ โˆ’ 1๐›ผ ๐‘Ž

๐›ผ

๐‘”(๐›ผ)(๐‘ฅ) = ๐‘“(๐›ผ)(๐‘ฅ) โˆ’ [๐‘“(๐‘) โˆ’ ๐‘“(๐‘Ž)

1๐›ผ ๐‘

๐›ผ โˆ’1๐›ผ ๐‘Ž

๐›ผ]

Page 61: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

52

So

๐‘“(๐›ผ)(๐‘) =๐‘“(๐‘) โˆ’ ๐‘“(๐‘Ž)

1๐›ผ ๐‘

๐›ผ โˆ’1๐›ผ ๐‘Ž

๐›ผ โˆŽ

2.2. Conformable Fractional Integrals

Suppose that the function is continuous

Let ๐›ผ โˆˆ (0,โˆž). Define ๐ฝ๐›ผ(๐‘ก๐‘) =

๐‘ก๐‘+๐›ผ

๐‘+๐›ผ , for any ๐‘ โˆˆ ๐‘… , ๐›ผ โ‰  โˆ’๐‘.

If ๐‘“(๐‘ก) = โˆ‘ ๐‘๐‘˜๐‘ก๐‘˜๐‘›

๐‘˜=0 , then we define ๐ฝ๐›ผ(๐‘“) = โˆ‘ ๐‘๐‘˜๐ฝ๐›ผ(๐‘ก๐‘˜)๐‘›

๐‘˜=0 = โˆ‘ ๐‘๐‘˜๐‘ก๐‘˜+๐›ผ

๐‘˜+๐›ผ

๐‘›๐‘˜=0

Cleary, ๐ฝ๐›ผis linear in its domain. Further, if ๐›ผ = 1, then ๐ฝ๐›ผ the usual integral.

Now according to conformable fractional definition, if ๐›ผ = 1 2โ„ ,then

sin ๐‘ก = โˆ‘(โˆ’1)๐‘›

(2๐‘˜+1)!โˆž๐‘›=0 ๐‘ก2๐‘›+1 then ๐ฝ๐›ผ(sin ๐‘ก) = โˆ‘

(โˆ’1)๐‘›๐‘ก2๐‘›+

32

(2๐‘›+3

2)(2๐‘›+1)!

โˆž๐‘›=0 .

Also, if ๐›ผ =1

2

cos(๐‘ก) = โˆ‘(โˆ’1)๐‘›๐‘ก2๐‘›

(2๐‘›)!โˆž๐‘›=0 then ๐ฝ๐›ผ(cos(๐‘ก)) = โˆ‘

(โˆ’1)๐‘›๐‘ก2๐‘›+

12

(2๐‘›+1

2)(2๐‘›)!

โˆž๐‘›=0

๐‘’๐‘ก = โˆ‘๐‘ก๐‘›

๐‘›!โˆž๐‘›=0 then ๐ฝ๐›ผ(๐‘’

๐‘ก) = โˆ‘๐‘ก๐‘›+

12

(๐‘›+1

2)(๐‘›)!

โˆž๐‘›=0

๐‘”(๐›ผ)(๐‘) = ๐‘“(๐›ผ)(๐‘) โˆ’ [๐‘“(๐‘) โˆ’ ๐‘“(๐‘Ž)

1๐›ผ ๐‘

๐›ผ โˆ’1๐›ผ ๐‘Ž

๐›ผ] = 0

Page 62: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

53

sinh(๐‘ก) = โˆ‘๐‘ก2๐‘›+1

(2๐‘›+1)!โˆž๐‘›=0 then ๐ฝ๐›ผ(sinh(๐‘ก)) = โˆ‘

๐‘ก2๐‘›+

32

(2๐‘›+3

2)(2๐‘›+1)!

โˆž๐‘›=0

cosh (๐‘ก) = โˆ‘๐‘ก2๐‘›

(2๐‘›)!โˆž๐‘›=0 then ๐ฝ๐›ผ(cosh(๐‘ก)) = โˆ‘

๐‘ก2๐‘›+

12

(2๐‘›+1

2)(2๐‘›)!

โˆž๐‘›=0 .

Definition 2.2.1 [14]

Let f be a continuous function. Then ๐›ผ-fractional integral of f is defined by:

๐ผ๐›ผ๐‘Ž๐‘“(๐‘ก) = ๐ผ1

๐‘Ž(๐‘ก๐›ผโˆ’1๐‘“(๐‘ก)) = โˆซ๐‘“(๐‘ฅ)

๐‘ฅ1โˆ’๐›ผ๐‘‘๐‘ฅ

๐‘ก

๐‘Ž

(2.19)

where ๐‘Ž > 0,๐›ผ โˆˆ (0,1) and the integral is the usual Riemann improper integral.

Examples:

1) ๐ผ12

0(โˆš๐‘ก cos(๐‘ก)) = โˆซcos(๐‘ฅ) . ๐‘‘๐‘ฅ

๐‘ก

0

= sin(๐‘ก)

2) ๐ผ12

0(cos(2โˆš๐‘ก)) = โˆซcos(2โˆš๐‘ฅ)

โˆš๐‘ฅ. ๐‘‘๐‘ฅ

๐‘ก

0

= sin(2โˆš๐‘ก)

Theorem 2.2.1 [14]

Let ๐‘“ be any continuous function in the domain of ๐ผ๐›ผ. Then

(๐‘‡๐›ผ๐ผ๐›ผ๐‘Ž(๐‘“(๐‘ก)) = ๐‘“(๐‘ก), for ๐‘ก โ‰ฅ ๐‘Ž) . (2.20)

Page 63: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

54

Proof: since f is continues, then ๐ผ๐›ผ๐‘Ž (๐‘“)(๐‘ก) is differentiable. So

๐‘‡๐›ผ (๐ผ๐›ผ๐‘Ž(๐‘“(๐‘ก))) = ๐‘ก1โˆ’๐›ผ

๐‘‘

๐‘‘๐‘ก๐ผ๐›ผ๐‘Ž๐‘“(๐‘ก)

= ๐‘ก1โˆ’๐›ผ๐‘‘

๐‘‘๐‘กโˆซ๐‘“(๐‘ฅ)

๐‘ฅ1โˆ’๐›ผ

๐‘ก

๐‘Ž

= ๐‘ก1โˆ’๐›ผ๐‘“(๐‘ก)

๐‘ก1โˆ’๐›ผ = ๐‘“(๐‘ก) โˆŽ

2.3 Applications [14]:

Now in this section we will solve fractional differential equations according to

conformable definitions:

Example (2.3.1):

๐‘ฆ(12โ„ ) + ๐‘ฆ = ๐‘ฅ3 + 3๐‘ฅ5 2โ„ , ๐‘ฆ(0) = 0 (2.21)

To find

๐‘ฆโ„Ž of ๐‘ฆ1 2โ„ + ๐‘ฆ = 0

we use

๐‘ฆโ„Ž = ๐‘’๐‘Ÿโˆš๐‘ฅ

Now

๐‘ฆ(1 2)โ„ + ๐‘ฆ = 0

Page 64: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

55

๐‘Ÿ

2๐‘’๐‘Ÿโˆš๐‘ฅ + ๐‘’๐‘Ÿโˆš๐‘ฅ = 0

๐‘’๐‘Ÿโˆš๐‘ฅ (๐‘Ÿ

2+ 1) = 0

๐‘Ÿ

2+ 1 = 0

๐‘Ÿ = โˆ’2

๐‘ฆโ„Ž = ๐‘’โˆ’2โˆš๐‘ฅ

And simply the particular solution is ๐‘ฆ๐‘ = ๐‘ฅ3

And by plugging the initial condition ๐‘ฆ๐‘ = ๐‘ฅ3 then A = 0

โˆด ๐‘ฆ = ๐‘ฆโ„Ž + ๐‘ฆ๐‘ = ๐‘’โˆ’2โˆš๐‘ฅ + ๐‘ฅ3

For more examples see [14].

2.4. Abelโ€™s Formula and Wronskain for Conformable Fractional Differential

Equation

In this section we will discuss the differential equation

๐‘ฆโ€ฒโ€ฒ + ๐‘ƒ(๐‘ฅ)๐‘ฆโ€ฒ +๐‘„(๐‘ฅ)๐‘ฆ = 0 (2.22)

In the sense of conformable fractional derivative, Abu Hammad, et al. [7] replaced

the derivative by conformable fractional derivative. They studied the form of

Wronskain for conformable fractional linear differential equation with variable

Page 65: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

56

coefficients. Finally, they study the Abel's formula. The result is similar to the case of

ordinary differential equation.

2.4.1. The Wronskain

For ๐›ผ โˆˆ (0,1], Abu Hammad, et al. discussed the equation [7].

๐‘‡๐›ผ๐‘‡๐›ผ๐‘ฆ + ๐‘ƒ(๐‘ฅ)๐‘‡๐›ผ๐‘ฆ + ๐‘„(๐‘ฅ)๐‘ฆ = 0 (2.23)

They discussed also the fractional Wronskain of two functions.

Definition 2.4.1. [7] For two functions ๐‘ฆ1 and ๐‘ฆ2 satisfying (2.24) and ๐›ผ โˆˆ (0,1] we set

๐‘Š๐›ผ[๐‘ฆ1, ๐‘ฆ2] = |๐‘ฆ1 ๐‘ฆ2๐‘‡๐›ผ๐‘ฆ1 ๐‘‡๐›ผ๐‘ฆ2

|

Theorem 2.4.1. [7] assume that ๐‘ฆ1, ๐‘ฆ2 satisfy equation (2.23), Then

๐‘Š๐›ผ[๐‘ฆ1, ๐‘ฆ2] = ๐‘’โˆ’๐ผ๐›ผ(๐‘ƒ)

Proof: applying the operator ๐‘‡๐›ผ on ๐‘Š๐›ผ[๐‘ฆ1, ๐‘ฆ2] to get

๐‘‡๐›ผ(๐‘Š๐›ผ[๐‘ฆ1, ๐‘ฆ2]) = ๐‘‡๐›ผ(๐‘ฆ1๐‘‡๐›ผ๐‘ฆ2 โˆ’ ๐‘ฆ2๐‘‡๐›ผ๐‘ฆ1)

= ๐‘‡๐›ผ๐‘ฆ1๐‘‡๐›ผ๐‘ฆ2 + ๐‘ฆ1๐‘‡๐›ผ๐‘‡๐›ผ๐‘ฆ2 โˆ’ ๐‘‡๐›ผ๐‘ฆ2๐‘‡๐›ผ๐‘ฆ1 โˆ’ ๐‘ฆ2๐‘‡๐›ผ๐‘‡๐›ผ๐‘ฆ1

But, ๐‘ฆ1 and ๐‘ฆ2 satisfy (2.24). So

๐‘‡๐›ผ๐‘‡๐›ผ๐‘ฆ1 = โˆ’๐‘ƒ(๐‘ฅ)๐‘‡๐›ผ๐‘ฆ1 โˆ’ ๐‘„(๐‘ฅ)๐‘ฆ1,

and

๐‘‡๐›ผ๐‘‡๐›ผ๐‘ฆ2 = โˆ’๐‘ƒ(๐‘ฅ)๐‘‡๐›ผ๐‘ฆ2 โˆ’ ๐‘„(๐‘ฅ)๐‘ฆ2,

Page 66: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

57

therefore,

๐‘‡๐›ผ(๐‘Š๐›ผ[๐‘ฆ1, ๐‘ฆ2]) = โˆ’๐‘ƒ(๐‘ฅ)(๐‘ฆ1๐‘‡๐›ผ๐‘ฆ2 โˆ’ ๐‘ฆ2๐‘‡๐›ผ๐‘ฆ1)

= โˆ’๐‘ƒ(๐‘ฅ)๐‘Š๐›ผ[๐‘ฆ1, ๐‘ฆ2] ,

thus,

๐‘‡๐›ผ(๐‘Š๐›ผ[๐‘ฆ1, ๐‘ฆ2])

๐‘Š๐›ผ[๐‘ฆ1, ๐‘ฆ2]= โˆ’๐‘ƒ(๐‘ฅ)

Consequently,

๐‘Š๐›ผ[๐‘ฆ1, ๐‘ฆ2] = ๐‘’โˆ’๐ผ๐›ผ(๐‘ƒ) (2.24)

2.4.2. Abelโ€™s Formula

First of all, it is important to discuss linear fractional differential equation

๐‘‡๐›ผ๐‘ฆ + ๐‘Ž(๐‘ฅ)๐‘ฆ = ๐‘(๐‘ฅ), ๐›ผ โˆˆ [0, 1] (2.25)

Multiply (2.26) by ๐‘’๐ผ๐›ผ(๐‘Ž(๐‘ฅ)) to get

๐‘’๐ผ๐›ผ(๐‘Ž(๐‘ฅ))๐‘‡๐›ผ๐‘ฆ + ๐‘’๐ผ๐›ผ(๐‘Ž(๐‘ฅ))๐‘Ž(๐‘ฅ)๐‘ฆ = ๐‘’๐ผ๐›ผ(๐‘Ž(๐‘ฅ))๐‘(๐‘ฅ)

๐‘‡๐›ผ(๐‘’๐ผ๐›ผ(๐‘Ž(๐‘ฅ))๐‘ฆ) = ๐‘’๐ผ๐›ผ(๐‘Ž(๐‘ฅ))๐‘(๐‘ฅ).

Hence

๐‘ฆ = ๐‘’โˆ’๐ผ๐›ผ(๐‘Ž(๐‘ฅ))๐ผ๐›ผ(๐‘’๐ผ๐›ผ(๐‘Ž(๐‘ฅ))๐‘(๐‘ฅ)) (2.26)

Is a solution of (2.26).

Now, let ๐‘ฆ1 be a solution of (2.24). To find a second solution ๐‘ฆ2 for equation (2.24).

Page 67: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

58

We have ๐‘Š๐›ผ[๐‘ฆ1, ๐‘ฆ2] = ๐‘’โˆ’๐ผ๐›ผ(๐‘ƒ), from which we get:

๐‘ฆ1๐‘‡๐›ผ๐‘ฆ2 โˆ’ ๐‘ฆ2๐‘‡๐›ผ๐‘ฆ1 = ๐‘’โˆ’๐ผ๐›ผ(๐‘ƒ),

And so

๐‘‡๐›ผ๐‘ฆ2 โˆ’ ๐‘ฆ2

๐‘‡๐›ผ๐‘ฆ1๐‘ฆ1

=๐‘’โˆ’๐ผ๐›ผ(๐‘ƒ)

๐‘ฆ1 (2.27)

Equation (2.28) is a fractional linear equation, with ๐‘Ž(๐‘ฅ) =๐‘‡๐›ผ๐‘ฆ1

๐‘ฆ1, and ๐‘(๐‘ฅ) =

๐ผ๐›ผ(โˆ’๐‘ƒ(๐‘ฅ))

๐‘ฆ1.

Hence, using the fact:

๐ผ๐›ผ (๐‘‡๐›ผ๐‘ฆ1๐‘ฆ1

) = ln๐‘ฆ1,

And formula (2.27) to get:

๐‘ฆ2 = ๐‘ฆ1๐ผ๐›ผ (

๐‘’โˆ’๐ผ๐›ผ(๐‘ƒ)

๐‘ฆ12 ). (2.28)

Page 68: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

59

Chapter 3: Exact Solution of Riccati Fractional Differential Equation

Riccati differential equation refers to the Italian Nobleman Count Jacopo Francesco

Riccati (1676-1754).

The fractional Riccati equation was studied by many researchers by using different

numerical methods [6, 9, 12, 13, 15, 20, 21, 24- 34]. Our interest in solving fractional

differential equations began when Prof. Khalil, et al.[14], presented the new and simple

conformable definition of fractional derivative.

In the rest of this chapter, we will find an exact solution to the fractional Riccati

differential equation (FRDE) precisely, we consider the following Problem:-

๐‘ฆ(๐›ผ) = ๐ด(๐‘ฅ)๐‘ฆ2 + ๐ต(๐‘ฅ)๐‘ฆ + ๐ถ(๐‘ฅ) (3.1)

๐‘ฆ(0) = ๐‘˜ , ๐‘˜: constant (3.2)

where ๐‘ฆ(๐›ผ) is the conformable fractional derivative of order ๐›ผ โˆˆ (0,1] , we should remark

that the method can be generalized to include any ๐›ผ .

3.1 Fractional Riccati Differential Equation (FRDE)

Riccati equation is studied by many researchers [8]. In this section, we found the exact

solution of fractional Riccati equation with known particular solution.

Theorem 3.1.1. (Reduction to second order equation)

The non-linear fractional Riccati equation can be reduced to a second order linear

ordinary differential equation of the form:

Page 69: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

60

๐‘ขโ€ฒโ€ฒ โˆ’ (

๐›ผ โˆ’ 1

๐‘ฅ+ ๐‘…(๐‘ฅ)) ๐‘ขโ€ฒ + ๐‘ฅ๐›ผโˆ’1๐‘†(๐‘ฅ)๐‘ข = 0 (3.3)

When ๐ด(๐‘ฅ) is non-zero and differentiable, such that ๐›ผ โˆˆ (0,1] ,also the solution of this

equation leads us to the solution.

๐‘ฆ =

โˆ’๐‘ˆโ€ฒ(๐‘ฅ) ๐‘ฅ1โˆ’๐›ผ

๐ด(๐‘ฅ)๐‘ˆ(๐‘ฅ) (3.4)

Proof:

Let ๐‘ฃ = ๐‘ฆ๐ด(๐‘ฅ)

๐‘ฃ(๐›ผ) = (๐‘ฆ๐ด(๐‘ฅ))(๐›ผ) = ๐‘ฆ(๐›ผ)๐ด(๐‘ฅ) + ๐‘ฆ๐‘ฅ1โˆ’๐›ผ๐ดโ€ฒ(๐‘ฅ)

๐‘ฆ(๐›ผ) satisfies the FRDE also by substituting ๐‘ฆ =๐‘ฃ

๐ด and some algebraic steps, then:

๐‘ฅ1โˆ’๐›ผ๐‘ฃโ€ฒ(๐‘ฅ) = ๐‘ฃ2 + ๐ต๐‘ฃ + ๐ถ๐ด + ๐‘ฃ๐‘ฅ1โˆ’๐›ผ๐ดโ€ฒ

๐ด

Divided both sides by ๐‘ฅ1โˆ’๐›ผ, then:

๐‘ฃโ€ฒ(๐‘ฅ) = ๐‘ฅ๐›ผโˆ’1๐‘ฃ2 + ๐‘ฅ๐›ผโˆ’1๐ต๐‘ฃ + ๐‘ฅ๐›ผโˆ’1๐ถ๐ด + ๐‘ฃ๐ดโ€ฒ

๐ด

Combining like terms, to get:

๐‘ฃโ€ฒ(๐‘ฅ) = ๐‘ฅ๐›ผโˆ’1๐‘ฃ2 + (๐‘ฅ๐›ผโˆ’1๐ต +

๐ดโ€ฒ

๐ด) ๐‘ฃ + ๐‘ฅ๐›ผโˆ’1๐ถ๐ด (3.5)

Assume: ๐‘…(๐‘ฅ) = ๐‘ฅ๐›ผโˆ’1๐ต +๐ดโ€ฒ

๐ด and ๐‘†(๐‘ฅ) = ๐‘ฅ๐›ผโˆ’1๐ถ๐ด , to get:

๐‘ฃโ€ฒ(๐‘ฅ) = ๐‘ฅ๐›ผโˆ’1๐‘ฃ2 + ๐‘…(๐‘ฅ)๐‘ฃ + ๐‘†(๐‘ฅ)

Page 70: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

61

Let ๐‘ฅ๐›ผโˆ’1๐‘ฃ = โˆ’

๐‘ขโ€ฒ

๐‘ข (3.6)

(๐›ผ โˆ’ 1)๐‘ฅ๐›ผโˆ’2๐‘ฃ + ๐‘ฅ๐›ผโˆ’1๐‘ฃโ€ฒ =โˆ’๐‘ข๐‘ขโ€ฒโ€ฒ + (๐‘ขโ€ฒ)2

๐‘ข2

(๐›ผ โˆ’ 1)๐‘ฅ๐›ผโˆ’2๐‘ฃ + ๐‘ฅ๐›ผโˆ’1๐‘ฃโ€ฒ =โˆ’๐‘ขโ€ฒโ€ฒ

๐‘ข+ ๐‘ฃ2(๐‘ฅ๐›ผโˆ’1)2

Divide both sides by ๐‘ฅ๐›ผโˆ’1

(๐›ผ โˆ’ 1)๐‘ฅโˆ’1๐‘ฃ + ๐‘ฃโ€ฒ = โˆ’๐‘ฅ1โˆ’๐›ผ๐‘ขโ€ฒโ€ฒ

๐‘ข+ ๐‘ฅ๐›ผโˆ’1๐‘ฃ2

๐›ผ โˆ’ 1

๐‘ฅ๐‘ฃ + ๐‘ฅ1โˆ’๐›ผ

๐‘ขโ€ฒโ€ฒ

๐‘ข= ๐‘ฅ๐›ผโˆ’1๐‘ฃ2 โˆ’ ๐‘ฃโ€ฒ

From equation (3.5)

๐›ผ โˆ’ 1

๐‘ฅ๐‘ฃ + ๐‘ฅ1โˆ’๐›ผ

๐‘ขโ€ฒโ€ฒ

๐‘ข= โˆ’(๐‘ฅ๐›ผโˆ’1๐ต +

๐ดโ€ฒ

๐ด) ๐‘ฃ โˆ’ ๐‘ฅ๐›ผโˆ’1๐ถ๐ด

๐›ผ โˆ’ 1

๐‘ฅ๐‘ฃ + ๐‘ฅ1โˆ’๐›ผ

๐‘ขโ€ฒโ€ฒ

๐‘ข= โˆ’๐‘…(๐‘ฅ)๐‘ฃ โˆ’ ๐‘†(๐‘ฅ)

combining like terms to get:

๐‘ฅ1โˆ’๐›ผ๐‘ขโ€ฒโ€ฒ

๐‘ข+ (๐›ผ โˆ’ 1

๐‘ฅ+ ๐‘…(๐‘ฅ)) ๐‘ฃ + ๐‘†(๐‘ฅ) = 0

divide both sides by ๐‘ฅ1โˆ’๐›ผ after substitute ๐‘ฃ = โˆ’๐‘ขโ€ฒ

๐‘ข๐‘ฅ1โˆ’๐›ผ

๐‘ฅ1โˆ’๐›ผ๐‘ขโ€ฒโ€ฒ

๐‘ข+ (๐›ผ โˆ’ 1

๐‘ฅ+ ๐‘…(๐‘ฅ)) (โˆ’

๐‘ขโ€ฒ

๐‘ข๐‘ฅ1โˆ’๐›ผ) + ๐‘†(๐‘ฅ) = 0

๐‘ขโ€ฒโ€ฒ

๐‘ขโˆ’ (๐›ผ โˆ’ 1

๐‘ฅ+ ๐‘…(๐‘ฅ))

๐‘ขโ€ฒ

๐‘ข+ ๐‘ฅ๐›ผโˆ’1๐‘†(๐‘ฅ) = 0

โˆด ๐‘ขโ€ฒโ€ฒ โˆ’ (๐›ผ โˆ’ 1

๐‘ฅ+ ๐‘…(๐‘ฅ))๐‘ขโ€ฒ + ๐‘ฅ๐›ผโˆ’1๐‘†(๐‘ฅ)๐‘ข = 0

Page 71: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

62

An answer of this equation will lead us to

๐‘ฆ =๐‘ฃ

๐ด=โˆ’๐‘ขโ€ฒ๐‘ฅ1โˆ’๐›ผ

๐‘ข๐ด โˆŽ

Theorem 3.1.2. (Transform FRDE to the Bernoulli equation)

For non-linear fractional Riccati equation the substitution ๐‘ฃ(๐‘ฅ) = ๐‘ฆ(๐‘ฅ) โˆ’ ๐‘ฆ1(๐‘ฅ) will

transform the (FRDE) into Bernoulli equation (ordinary differential equation of the first

order), when ๐‘ฆ1 is a known particular solution,

Proof:

Since ๐‘ฃ(๐‘ฅ) = ๐‘ฆ(๐‘ฅ) โˆ’ ๐‘ฆ1(๐‘ฅ)

โˆด ๐‘ฆ(๐‘ฅ) = ๐‘ฃ(๐‘ฅ) + ๐‘ฆ1(๐‘ฅ)

And ๐‘ฆ(๐›ผ)(๐‘ฅ) = ๐‘ฃ(๐›ผ)(๐‘ฅ) + ๐‘ฆ1(๐›ผ)(๐‘ฅ)

Since ๐‘ฆ1(๐‘ฅ) solves the (FRDE), it must be that

๐‘ฆ1(๐›ผ) = ๐ด(๐‘ฅ)๐‘ฆ1

2 + ๐ต(๐‘ฅ)๐‘ฆ1 + ๐ถ(๐‘ฅ)

Substitute in (3.1)

๐‘ฃ(๐›ผ)(๐‘ฅ) + ๐‘ฆ1(๐›ผ)(๐‘ฅ)โŸ

๐‘ฆ(๐›ผ)(๐‘ฅ)

= ๐ด(๐‘ฅ) [๐‘ฃ + ๐‘ฆ1]โŸ ๐‘ฆ(๐‘ฅ)

2+ ๐ต(๐‘ฅ) [๐‘ฃ + ๐‘ฆ1]โŸ

๐‘ฆ(๐‘ฅ)

+ ๐ถ(๐‘ฅ)

๐‘ฅ1โˆ’๐›ผ๐‘ฃ โ€ฒ(๐‘ฅ)โŸ ๐‘ฃ(๐›ผ)(๐‘ฅ)

+ ๐ด๐‘ฆ12 + ๐ต๐‘ฆ1 + ๐ถ = ๐ด๐‘ฃ

2 + 2๐ด๐‘ฃ๐‘ฆ1 + ๐ด๐‘ฆ12 + ๐ต๐‘ฃ + ๐ต๐‘ฆ1 + ๐ถ

๐‘ฅ1โˆ’๐›ผ๐‘ฃ โ€ฒ(๐‘ฅ) = ๐ด๐‘ฃ2(๐‘ฅ) + 2๐ด๐‘ฆ1๐‘ฃ(๐‘ฅ) + ๐ต๐‘ฃ(๐‘ฅ)

๐‘ฃ โ€ฒ(๐‘ฅ) = ๐ด๐‘ฅ๐›ผโˆ’1๐‘ฃ2(๐‘ฅ) + 2๐ด๐‘ฅ๐›ผโˆ’1๐‘ฆ1๐‘ฃ(๐‘ฅ) + ๐ต๐‘ฅ๐›ผโˆ’1๐‘ฃ(๐‘ฅ)

๐‘ฃ โ€ฒ(๐‘ฅ) + [โˆ’2๐‘ฅ๐›ผโˆ’1๐ด(๐‘ฅ)๐‘ฆ1 โˆ’ ๐‘ฅ๐›ผโˆ’1๐ต(๐‘ฅ)]โŸ

๐œ‘(๐‘ฅ)

๐‘ฃ = ๐ด๐‘ฅ๐›ผโˆ’1โŸ ๐‘ž(๐‘ฅ)

๐‘ฃ2(๐‘ฅ) (3.7)

This equation is of the form of Bernoulli equation with n=2 โˆŽ

Page 72: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

63

which could be transformed to first order linear differential equation.

Let ๐‘ข = ๐‘ฃโˆ’1(๐‘ฅ).

๐‘‘๐‘ข

๐‘‘๐‘ฅ= โˆ’๐‘ฃโˆ’2(๐‘ฅ)

๐‘‘๐‘ฃ

๐‘‘๐‘ฅ

Multiply (3.7) by โ€“ ๐‘ฃ(๐‘ฅ)โˆ’2

โˆ’๐‘ฃโˆ’2๐‘ฃ โ€ฒ + [2๐‘ฅ๐›ผโˆ’1๐ด๐‘ฆ1 + ๐‘ฅ๐›ผโˆ’1๐ต]๐‘ฃโˆ’2๐‘ฃ = โˆ’๐ด๐‘ฅ๐›ผโˆ’1

๐‘ฃโ€ฒ+ [2๐‘ฅ๐›ผโˆ’1๐ด๐‘ฆ1 + ๐‘ฅ๐›ผโˆ’1๐ต]๐‘ฃ = ๐ด ๐‘ฅ๐›ผโˆ’1โŸ

๐‘ž(๐‘ฅ)

(3.8)

The general solution is given by

๐‘ฃ =

โˆซ๐œ‡(๐‘ฅ)๐‘ž(๐‘ฅ). ๐‘‘๐‘ฅ + ๐‘(๐‘ฅ)

๐œ‡(๐‘ฅ) (3.9)

where ๐œ‡(๐‘ฅ) = ๐‘’(โˆซ[2๐‘ฅ๐›ผโˆ’1๐ด๐‘ฆ1+๐‘ฅ

๐›ผโˆ’1๐ต]๐‘‘๐‘ฅ) (3.10)

Theorem 3.1.3. (Obtaining solution of FRDE by Abelโ€™s formula)

Let ๐‘ฆ1 be a solution of (3.1), and assume that ๐‘ง = 1

๐‘ฆโˆ’ ๐‘ฆ1, then the solution of FRDE is

๐‘ง = ๐‘’โˆ’๐ผ(2๐ด๐‘ฆ1+๐ต)๐ผ๐›ผ(๐‘’๐ผ(2๐ด๐‘ฆ1+๐ต)(โˆ’๐ด(๐‘ฅ))) (3.11)

Proof: suppose that ๐‘ฆ1 is a solution of FRDE, and let =1

๐‘ฆโˆ’๐‘ฆ1 , then

๐‘ง(๐‘ฆ โˆ’ ๐‘ฆ1) = 1

๐‘ฆ =

1

๐‘ง+ ๐‘ฆ1 (3.12)

Page 73: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

64

Apply ๐›ผ-derivative definition to both sides of (3.12)

๐‘‡๐›ผ๐‘ฆ = ๐‘‡๐›ผ (1

๐‘ง) + ๐‘‡๐›ผ๐‘ฆ1

๐‘‡๐›ผ๐‘ฆ = โˆ’๐‘งโˆ’1โˆ’๐›ผ๐‘งโ€ฒ + ๐‘‡๐›ผ๐‘ฆ1

Substituting in the original FRDE

โˆ’๐‘งโˆ’1โˆ’๐›ผ๐‘งโ€ฒ + ๐‘‡๐›ผ๐‘ฆ1 = ๐ด [1

๐‘ง+ ๐‘ฆ1]

2

+ ๐ต [1

๐‘ง+ ๐‘ฆ1] + ๐ถ

โˆ’๐‘งโˆ’1โˆ’๐›ผ๐‘งโ€ฒ = ๐ด [1

๐‘ง2+2๐‘ฆ1๐‘ง+ ๐‘ฆ1

2] + ๐ต [1

๐‘ง+ ๐‘ฆ1] + ๐ถ โˆ’ ๐‘‡๐›ผ๐‘ฆ1

๐‘‡๐›ผ๐‘ฆ1 satisfies the FRDE

โˆ’๐‘งโˆ’1โˆ’๐›ผ๐‘งโ€ฒ =๐ด

๐‘ง2+2๐‘ฆ1๐ด

๐‘ง+ ๐ด๐‘ฆ1

2 +๐ต

๐‘ง+ ๐ต๐‘ฆ1 + ๐ถ โˆ’ ๐ด๐‘ฆ

2 โˆ’ ๐ต๐‘ฆ1 โˆ’ ๐ถ

Combining like terms and divide both sides by โˆ’๐‘งโˆ’1โˆ’๐›ผ

๐‘งโ€ฒ = โˆ’(2๐ด๐‘ฆ1 + ๐ต)๐‘ง๐›ผ โˆ’ ๐ด๐‘ง๐›ผโˆ’1, then

๐‘งโ€ฒ + (2๐ด๐‘ฆ1 + ๐ต)๐‘ง๐›ผ = โˆ’๐ด๐‘ง๐›ผโˆ’1 (3.13)

Multiply both sides of equation (3.13) by ๐‘ง1โˆ’๐›ผ

๐‘ง1โˆ’๐›ผ๐‘งโ€ฒ + (2๐ด๐‘ฆ1 + ๐ต)๐‘ง = โˆ’๐ด

๐‘ง(๐›ผ) + (2๐ด๐‘ฆ1 + ๐ต)๐‘ง = โˆ’๐ด (3.14)

which is Abelโ€™s formula as we mentioned in the previous chapter.

Thus, the solution is

Page 74: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

65

๐‘ง = ๐‘’โˆ’๐ผ(2๐ด๐‘ฆ1+๐ต)๐ผ๐›ผ(๐‘’๐ผ(2๐ด๐‘ฆ1+๐ต)(โˆ’๐ด(๐‘ฅ)))

Theorem 3.1.4. Assume that the coefficients ๐ถ(๐‘ฅ) + ๐ต(๐‘ฅ) + ๐ด(๐‘ฅ) = 0 of the fractional

Ricatii (3.1), if ๐ถ(๐‘ฅ) satisfies the integral condition, which is

๐ถ(๐‘ฅ) =๐‘“1(๐‘ฅ) โˆ’ {๐ต(๐‘ฅ) + ๐ด(๐‘ฅ) [โˆซ

๐‘“1(๐œ™) โˆ’ ๐ต2(๐œ™)

2๐ด(๐œ™)โˆ’ ๐ด1

๐‘ฅ]}2

4๐ด

(3.15)

where ๐ด1 is an arbitrary constant of integration.

and ๐‘“1 is the new generating function satisfying the differential condition (3.15) given by:

๐ต2(๐‘ฅ) + 4๐ด(๐‘ฅ)๐‘ฅ1โˆ’๐›ผ

๐‘‘๐‘ฆ๐‘

๐‘‘๐‘ฅ= ๐‘“1(๐‘ฅ) (3.16)

Then the general solution is given by:

๐‘ฆ(๐‘ฅ) =1

๐‘’โˆ’๐ผ(2๐ด๐‘ฆ1+๐ต)๐ผ๐›ผ(๐‘’๐ผ(2๐ด๐‘ฆ1+๐ต)(โˆ’๐ด(๐‘ฅ)))

+1

2[โˆซ

๐‘“1(๐œ™) โˆ’ ๐ต2(๐œ™)

2๐ด(๐œ™)๐‘‘๐œ™

๐‘ฅ

โˆ’ ๐ด1],

where ๐ด0 is an arbitrary constant of integration.

(3.17)

Proof.

Assume that the arbitrary function ๐ต(๐‘ฅ), ๐ด(๐‘ฅ) and ๐‘“1(๐‘ฅ) satisfying (3.15) then the

particular solution

๐‘ฆ๐‘ยฑ(๐‘ฅ) =โˆ’๐ต ยฑ โˆš๐‘“1 โˆ’ 4๐ด๐ถ

2๐ด

Page 75: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

66

=

โˆ’๐ต ยฑโˆš๐‘“1 โˆ’ 4๐ด

๐‘“1(๐‘ฅ) โˆ’ {๐ต(๐‘ฅ) + ๐ด(๐‘ฅ) [โˆซ๐‘“1(๐œ™) โˆ’ ๐ต2(๐œ™)

2๐ด(๐œ™)๐‘ฅ

โˆ’ ๐ด1]}2

4๐ด

2๐ด

=

โˆ’๐ต ยฑ โˆš๐‘“1 โˆ’ ๐‘“1(๐‘ฅ) โˆ’ {๐ต(๐‘ฅ) + ๐ด(๐‘ฅ) [โˆซ๐‘“1(๐œ™) โˆ’ ๐ต2(๐œ™)

2๐ด(๐œ™)โˆ’ ๐ด1

๐‘ฅ]}2

2๐ด

=โˆ’๐ต + ๐ต(๐‘ฅ) + ๐ด(๐‘ฅ) [โˆซ

๐‘“1(๐œ™) โˆ’ ๐ต2(๐œ™)

2๐ด(๐œ™)๐‘ฅ

โˆ’ ๐ด1]

2๐ด

=๐ด(๐‘ฅ) [โˆซ

๐‘“1(๐œ™) โˆ’ ๐ต2(๐œ™)

2๐ด(๐œ™)๐‘ฅ

โˆ’ ๐ด1]

2๐ด

=1

2[โˆซ๐‘“1(๐œ™) โˆ’ ๐ต

2(๐œ™)

2๐ด(๐œ™)

๐‘ฅ

โˆ’ ๐ด1]

Thus

๐‘ฆ๐‘ยฑ(๐‘ฅ) =โˆ’๐ต ยฑ โˆš๐‘“1 โˆ’ 4๐ด๐ถ

2๐ด=1

2[โˆซ๐‘“1(๐œ™) โˆ’ ๐ต

2(๐œ™)

2๐ด(๐œ™)

๐‘ฅ

โˆ’ ๐ด1] (3.18)

Differentiate equation (3.18)

๐‘‘๐‘ฆ

๐‘‘๐‘ฅ[โˆ’๐ต ยฑ โˆš๐‘“1 โˆ’ 4๐ด๐ถ

2๐ด] =

๐‘“1(๐œ™) โˆ’ ๐ต2(๐œ™)

2๐ด(๐œ™) (3.19)

Equation (3.19) can be integrated to get

โˆ’๐ต ยฑ โˆš๐‘“1 โˆ’ 4๐ด๐ถ

๐ด=โˆซ๐‘“1(๐œ™) โˆ’ ๐ต

2(๐œ™)2๐ด(๐œ™)

๐‘ฅ

1

Page 76: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

67

โˆ’๐ต ยฑ โˆš๐‘“1 โˆ’ 4๐ด๐ถ = ๐ด [โˆซ๐‘“1(๐œ™) โˆ’ ๐ต

2(๐œ™)

2๐ด(๐œ™)

๐‘ฅ

]

โˆš๐‘“1 โˆ’ 4๐ด๐ถ = ๐ต + ๐ด [โˆซ๐‘“1(๐œ™) โˆ’ ๐ต

2(๐œ™)

2๐ด(๐œ™)

๐‘ฅ

]

๐‘“1 โˆ’ 4๐ด๐ถ = {๐ต + ๐ด [โˆซ๐‘“1(๐œ™) โˆ’ ๐ต

2(๐œ™)

2๐ด(๐œ™)

๐‘ฅ

]}

2

โˆ’4๐ด๐ถ = โˆ’๐‘“1 + {๐ต + ๐ด [โˆซ๐‘“1(๐œ™) โˆ’ ๐ต

2(๐œ™)

2๐ด(๐œ™)

๐‘ฅ

]}

2

๐ถ(๐‘ฅ) =๐‘“1(๐‘ฅ) โˆ’ {๐ต(๐‘ฅ) + ๐ด(๐‘ฅ) [โˆซ

๐‘“1(๐œ™) โˆ’ ๐ต2(๐œ™)

2๐ด(๐œ™)โˆ’ ๐ด1

๐‘ฅ]}2

4๐ด

3.2 Applications:

Example: - find the solution of

๐‘ฆ(1

2) = (๐‘ฆ โˆ’ 2โˆš๐‘ฅ)

2+ 1 , ๐‘ฆ1(๐‘ฅ) = 2โˆš๐‘ฅ ; ๐‘ฆ(0) = 1 (3.20)

Solution: First we need to verify that ๐‘ฆ1 = 2โˆš๐‘ฅ is a solution to this equation by computing,

we find that ๐‘ฆ1 is a solution of (3.20).

Now we solve the equation.

Step1. Make the change of variables

Substituting ๐‘ฆ = ๐‘ฃ + 2โˆš๐‘ฅ and ๐‘ฆ(1

2) = ๐‘ฃ(

1

2) + 1 yields

Page 77: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

68

๐‘ฃ(12)+ 1 = (๐‘ฃ + 2โˆš๐‘ฅ โˆ’ 2โˆš๐‘ฅ)

2+ 1

Step 2. Simplify to a Bernoulli equation for ๐‘ฃ

๐‘ฅ12๐‘ฃโ€ฒ = ๐‘ฃ2

๐‘ฃ โ€ฒ = ๐‘ฅโˆ’

12๐‘ฃ2 (3.21)

This is a Bernoulli equation.

Step3. Solve the Bernoulli equation

Let ๐‘ข = ๐‘ฃโˆ’1

๐‘ขโ€ฒ = โˆ’๐‘ฃโˆ’2๐‘ฃโ€ฒ

Multiply equation (3.21) by โˆ’๐‘ฃโˆ’2

โˆ’๐‘ฃโˆ’2๐‘ฃ โ€ฒ = โˆ’๐‘ฅโˆ’12๐‘ฃโˆ’2๐‘ฃ2

๐‘ขโ€ฒ = โˆ’๐‘ฅโˆ’12 =

โˆ’1

โˆš๐‘ฅ

๐‘‘๐‘ข

๐‘‘๐‘ฅ=โˆ’1

โˆš๐‘ฅ โ†’ ๐‘‘๐‘ข =

โˆ’1

โˆš๐‘ฅ๐‘‘๐‘ฅ

๐‘ข = โˆซโˆ’1

โˆš๐‘ฅ. ๐‘‘๐‘ฅ = โˆ’2โˆš๐‘ฅ + ๐‘

1

๐‘ฃ= โˆ’2โˆš๐‘ฅ + ๐‘

๐‘ฃ =1

โˆ’2โˆš๐‘ฅ + ๐‘

Step 4. Reverse the substitution ๐‘ฆ = ๐‘ฃ + 2โˆš๐‘ฅ

Page 78: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

69

๐‘ฆ =1

โˆ’2โˆš๐‘ฅ + ๐‘โˆ’ 2โˆš๐‘ฅ

Finally, we use the initial condition ๐‘ฆ(0) = 1

โˆด ๐‘ = 1

โˆด The general solution is

๐‘ฆ =

3

โˆ’4โˆš๐‘ฅ3 + 1โˆ’๐‘ฅ2

2 (3.22)

Page 79: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

70

Future Work

The main aspect of the future work in the thesis is to take other conditions of fractional

Riccati Differential Equation (FRDE) and solve it.

Page 80: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

71

Conclusions

The objective of the present thesis is to use conformable fractional derivative which is

simpler and more efficient. The new definition reflects a natural extension of normal

derivative to solve fractional differential equation specifically fractional Riccati differential

equation.

In this thesis we found an exact solution of fractional Riccati differential equation and

introduced some theorems which lead us to find a second solution when we have a given

particular solution.

Page 81: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

72

REFERENCES

Page 82: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

73

[1] Abdeljawad, T. (2015), on Conformable Fractional Calculus, Journal of

Computational and Applied Mathematics, 279: 57-66.

[2] Boyce, William and Di/Prima, Richard (2010), Elementary Differential Equations

and Boundary Value Problems, (9th

ed), Asia: John Willey & Sons, Inc.

[3] Daftardar-Gejji, Varsha (2014), Fractional Calculus Theory and Applications,

India: Norosa Publishing House Pvt. Ltd.

[4] Das, Shantanu (2011), Functional Fractional Calculus, India: Scientific Publishing

Services Pvt. Ltd.

[5] Diethelm, Kai (2010), the Analysis of Fractional Differential Equations: An

Application โ€“Oriented Exposition Using Differential Operators of Caputo Type,

Germany: Springer Heidelberg Dordercht London New York.

[6] El-Sayed, A. M. A., El-Mesiry, A. E. M. and El-Saka, H. A. A. (2007), On the

Fractional-Order Logistic Equation, Applied Mathematics Letters, 20(7) : 817-823.

[7] Abu Hammad, M., and Khalil, R. (2014), Abel's formula and Wronskian for

Conformable Fractional Differential Equations, International Journal of

Differential Equations and Applications, 13(3).

[8] Harko, T., Lobo, F. and Mak, M. K. (2013), Analytical Solutions of the Riccati

Equation with Coefficients Satisfying Integral Or Differential Conditions With

Arbitrary Functions, Universal Journal of Applied Mathematics, 2: 109-118.

[9] He, J. H. (1999), Variational Iteration Method-a Kind Of Non-Linear Analytical

Technique: Some Examples, International Journal of Non-Linear Mechanics, 34:

699-708.

[10] Hilfer, R. (2000), Applications of Fractional Calculus in Physics, (2000), USA:

World Scientific Publishing Co. Pte. Ltd.

[11] Ishteva, M. (2005), Properties and Applications of the Caputo Fractional

Operator, Msc. Thesis, Universitรคt Karlsruhe (TH), Sofia, Bulgaria.

[12] Khader, M. M. (2011), On the Numerical Solutions for the Fractional Diffusion

Equation, Communications in Nonlinear Science and Numerical Simulation, 16:

2535-2542.

[13] Khader, M. M., EL Danaf, T. S. and Hendy, A. S. (2012), Efficient Spectral

Collocation Method for Solving Multi-Term Fractional Differential Equations Based

on the Generalized Laguerre Polynomials, Journal of Fractional Calculus and

Applications, 3(13), 1-14.

Page 83: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

74

[14] Khalil, Roshdi, Horani, M., Yousef, Abdelrahman and Sababheh, M. (2014), A New

Definition of Fractional Derivative, Computational and Applied Mathematics,

246: 65-70.

[15] Khan, Najeeb, Ara, Asmat and Jamil, Muhammad (2011), An Efficient Approach for

Solving the Riccatii Equation with Fractional Orders, Computers and Mathematics

with Applications, 61(9): 2683-2689.

[16] Leszczynski, Jacek (2011), An Introduction of Fractional Mechanics,

Czestochowa: The Publishing Office of Czestochowa University of Technology.

[17] Mathai A. and Haubold H. (2008), Special Functions for Applied Scientists, New

York: Springer Science and Business Media LLC .

[18] Merdan, Mehmet (2012), On the Solutions Fractional Riccatii Differential Equation

with Modified Riemann-Liouville Derivative, International Journal of Differential

Equations, 2012.

[19] Miller, Kenneths and Ross, Bertram (1993), An Introduction to the Fractional

Calculus and Fractional Differential Equations, Canada: John Willey & Sons, Inc.

[20] Momani, Shaher and Shawagfeh, Nabil (2006), Decomposition Method for Solving

Fractional Riccati Differential Equations, Applied Mathematics and Computation,

182(2): 1083-1092.

[21] Odibat, Zaid and Momani, Shaher (2008), Modified Homotopy Perturbation Method:

Application to Quadratic Riccati Differential Equation of Fractional Order, Chaos,

Solutions & Fractals, 36(1), 167-174.

[22] Podlubny, I. (1998), Fractional Differential Equations: An Introduction to

Fractional Derivatives, Fractional Differential Equations, to the Methods of

Their Solutions and Some of Their Applications, USA: Academic Press.

[23] Sebah, Pascal and Gourdon, Xavier (2002), Introduction to Gamma Function,

Numbers. Computation. Free. Fr/Constant/constants.html.

[24] Sweilam, N. H., Khader, M. M. and Al-Bar, R. F. (2007), Numerical Studies for a

Multi-Order Fractional Differential Equation, Physics Letters A, 371: 26-33.

[25] Sweilam, N. H., Khader, M. M. and Nagy, A. M. (2011), Numerical Solution of two-

Sided Spacefractional Wave Equation Using Finite Difference Method, Journal of

Computional and Applied Mathematics, 235: 2832-2841.

Page 84: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

75

[26] Sweilam, N. H., Khader M. M. and Adel, M. (2012), On the Stability Analysis of

Weighted Average Finite Difference Methods for Fractional Wave Equation,

Accepted in Fractional Differential Calculus, 2(1): 17-29.

[27] Sweilam, N. H., Khader, M. M. and Mahdy, A. M. S. (2012), Crank-Nicolson Finite

Difference Method for Solving Time-Fractional Diffusion Equation, Journal of

Fractional Calculus and Applications, 2(2): 1-9.

[28] Sweilam, N. H., Khader, M. M. and Mahdy, A. M. S. (2012), Numerical Studies for

Solving Fractional-Order Logistic Equation, International Journal of Pure and

Applied Mathematics, 78(8): 1199-1210.

[29] Sweilam, N. H., Khader, M. M. and Mahdy, A. M. S. (2012), Numerical Studies for

Fractional Order Logistic Differential Equation with Two Different Delays,

Accepted in Journal of Applied Mathematics.

[30] Sweilam, N.H., Khader, M.M. and Mahdy, M.S. (2012), Numerical Studies for

Solving Fractional Riccati Differential Equation, Applications and Applied

Mathematics (AAM), 7(2): 595-608.

[31] Tan, Y. and Abbasbandy, S. (2008), Homotopy Analysis Method for Quadratic

Riccati Differential Equation, Communications in Nonlinear Science and

Numerical Simulation, 13(3).

[32] Wazwaz, A. M. (1998), A Comparison between Adomian Decomposition Method

and Taylor Series Method in the Series Solution, Applied Mathematics and

Computation, 97, 37- 44.

[33] Weilbeer, Marc (2005), Efficient Numerical Methods for Fractional Differential

Equations and their Analytical Background, Ph. D. Thesis, Technische University

Sitat Braunshweig.

[34] Wu, Fei and Huang, Lan.Lan (2014), Approximate Solutions of Fractional Riccati

Equations using the Adomain decomposition Method, Abstract and Applied

Analysis, 2014.

Page 85: SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING ...SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS BY USING CONFORMABLE FRACTIONAL DERIVATIVES DEFINITION By Shadi Ahmad Al-Tarawneh Supervisor

76

ุญู„ ู…ุนุงุฏุงู„ุช ุชูุงุถู„ูŠุฉ ูƒุณุฑูŠุฉ ุจุงุณุชุฎุฏุงู… ุชุนุฑูŠู ุงู„ู…ุทุงุจู‚ ู„ู„ู…ุดุชู‚ุงุช ุงู„ูƒุณุฑูŠุฉ

ุฅุนุฏุงุฏ

ุดุงุฏูŠ ุฃุญู…ุฏ ุงู„ุทุฑุงูˆู†ุฉ

ุงู„ู…ุดุฑู

ุฏ. ุฎุงู„ุฏ ุฌุงุจุฑ

ุงู„ู…ู„ุฎุต

ุน ุงู„ู…ูƒุงู†ูŠูƒูŠุฉุŒ ูˆุงุฃู„ุญูŠุงุก ุŒ ูˆุงู„ููŠุฒูŠุงุก ุŒ ุฆุงู„ู…ุนุงุฏุงู„ุช ุงู„ุชูุงุถู„ูŠุฉ ุงู„ุนุงุฏูŠุฉ ูˆุงู„ุฌุฒุฆูŠุฉ ู…ู‡ู…ุฉ ุฌุฏุงู‹ ููŠ ู…ุฌุงุงู„ุช ุนุฏูŠุฏุฉุ› ู…ุซู„ ุงู„ู…ูˆุง

ูˆู†ุธุฑูŠุฉ ุงู„ุชุญูƒู… ููŠ ุงุฃู„ู†ุธู…ุฉ ูˆุงู„ุจุตุฑูŠุงุชุŒ ูˆุงู„ูƒู‡ุฑูˆูƒูŠู…ูŠุงุฆูŠุฉ ุŒ ูˆุงู„ู‡ู†ุฏุณุฉุŒ ูˆุงู„ู„ุฒูˆุฌุฉ ุงู„ู…ุทุงุทูŠุฉ ุŒ ูˆุงู„ุดุจูƒุงุช ุงู„ูƒู‡ุฑุจุงุฆูŠุฉ ุŒ

ุงู„ุฏูŠู†ุงู…ูŠูƒูŠุฉ.

ู…ู† ู‚ุจู„ ุจุงุญุซูŠู† ุนุฏุฉุŒ ุจุงุณุชุฎุฏุงู… ุทุฑู‚ ุนุฏูŠุฏุฉ ู…ุฎุชู„ูุฉ. ุจุฏุฃ ู…ูˆุถูˆุน ุงู‡ุชู…ุงู…ู†ุง ูˆู‡ูˆ ุญู„ ุฑูŠูƒุงุชูŠุชู‘ู… ุฏุฑุงุณุฉ ู…ุนุงุฏู„ุฉ

ูˆู…ุฌู…ูˆุนุฉ ุจุงุญุซูŠู† ุขุฎุฑูŠู† ู„ุชู‚ุฏูŠู… ุชุนุฑูŠู ุฌุฏูŠุฏ ูˆุจุณูŠุท ูˆุฃูƒุซุฑ ุฎู„ูŠู„ ุฑุดุฏูŠู…ุนุงุฏุงู„ุช ุชูุงุถู„ูŠุฉ ูƒุณุฑูŠุฉุŒ ุนู†ุฏู…ุง ู‚ุงู… ุงู„ุฏูƒุชูˆุฑ

ุฑูŠุฉ. ู‡ุฐุง ุงู„ุชุนุฑูŠู ุงู„ุฌุฏูŠุฏ ู‡ูˆ ุงู…ุชุฏุงุฏ ู„ู„ู…ุดุชู‚ุงุช ุงู„ุนุงุฏูŠุฉ ูˆุงู„ุฐูŠ ูŠุณู‘ู…ู‰ "ุชุนุฑูŠู ุงู„ู…ุทุงุจู‚ ู„ู„ู…ุดุชู‚ุงุช ูƒูุงุกุฉู‹ ู„ู„ู…ุดุชู‚ุงุช ุงู„ูƒุณ

ุงู„ูƒุณุฑูŠุฉ".

ุงู„ุชูุงุถู„ูŠุฉ ุงู„ูƒุณุฑูŠุฉุŒ ูˆู‚ุฏู‘ู…ู†ุง ุจุนุถ ุงู„ู†ุธุฑูŠุงุช ุงู„ุฐูŠ ุชุณุงุนุฏู†ุง ููŠ ุฑูŠูƒุงุชูŠููŠ ู‡ุฐุง ุงู„ุจุญุซุŒ ุฃูˆุฌุฏู†ุง ุญู„ ุฏู‚ูŠู‚ ู„ู…ุนุงุฏู„ุฉ

ุฉ.ุงู„ุชูุงุถู„ูŠุฉ ุงู„ูƒุณุฑูŠ ุฏ ุญู„ ุซุงู†ูŠ ุนู†ุฏู…ุง ูŠุนุทูŠ ุญู„ ู„ู…ุนุงุฏู„ุฉ ุฑูŠูƒุงุชูŠุฅูŠุฌุง