Al – Neelain University

Faculty of Mathematical Sciences & Statistics

Fractional Adomian Decomposition Method Compared by

Jumarie Fractional Derivative

A Thesis Submitted in partial Fulfillment of the Academic Requirements for

the Degree of M.Sc. in Mathematics

By :

Mohammed Ali Mohammed Shaeldin

Supervisor :

Dr . Hayam Mohammed Elbarbary

February 2021

I

الآية كاج فا قال تعانى : ) شأ ك ض يثم ر رأ الأ اخ ا ر انسه الله

ي قذ كة در أ ا ك جاجح كأه ثاح ف سجاجح انش صأ ثاح انأ ي يصأ

أ ن أتا ضء تهح كاد س ل غزأ قهح أتح له شزأ ثاركح س شجزج ي

زب ضأ ي شاء نر أذي الله ار ر عهى ر سسأ أ نىأ ت

ء عه أ تكم ش الله ثال نههاص يأ الأ .ى (الله

صذق الل انعظى

53سرج انراح

II

Dedication

To my parents, with love and respect, for everyone who loves

mathematics, and for all my loved ones and my

companionship.

III

Acknowledgements

firstly, I am deeply thankful to allah for his almighty.....

also I am grateful to Dr. Hayam Mohammed Elbarbary

for her advices and help until I finished this study.

IV

Abstract

This research aims to study the Fractional derivatives by

means of Jumarie and we found the solution to the equations

for wave and heat (linear and nonlinear) respectively, using

two methods: the Jumarie Fractinal Derivative Via Tanh-

Method.

V

الخلاصةاجذا انحم نعادنت انحزارج يزياسح انشتقاخ انكسزح ع طزق جذف ذا انثحث ان در

اديا انكسزح انجح انخطح غز انخطح عه انتان تاستخذاو طزقتا : انطزقح النى

يع انشتقح انكشزح نجيزي تاسطح طزقح انظم انشائذح .يقارتا

TABLE OF CONT ENTS

Subject Bage I الآيةDedication II Acknowledgements III Abstract IV Abstract (Arabic) V Table of Contents VI Chapter 1: introduction 1 1.1 The Origin of Fractional Calculus 1 1.2 Definition of Fractional Calculus 1 1.3 Useful Mathematical Functions 1 1.4 Definition of the Riemann Liouville Fractional Integral 3 1.5 Derivatives of the Fractional Integral and the Fractional Integral of Derivatives

4

1.6 Fractional differential equations 5 Chapter 2: Solve The fractional differential equations using by

Jumarie’s 8

2.0 Definition of fractional derivatives 8 2.1 The Mittag-Leffler Function 9 2.2 Non-Homogeneous Fractional Differential Equations and Some Basic Solutions

10

2.3 α−order non-homogeneous fractional differential equations 12 2.4 Particular Integral for g(tα) = Eα(ct

α) 13 2.5 Particular Integral for g(tα) = tα 14 2.6 Evaluation 15 2.7 2α− order non-homogeneous fractional differential equations 18 2.8 Use of method of un-determinant coefficient method to calculate the particular integrals for different functional forms of g(tα)

18

2.9 Use of direct method to calculate the Particular integrals for different functional format of g(tα)

21

2.10 Some examples of the fractional differential-using by jumarie’s 23 Chapter 3: Adomian decomposition method and jumari’s Fractional Derivative

26

3.1 The Adomian decomposition method 26 3.2 The jumari’s Fractional Derivative 28 Chapter 4: Compared between Jumerie’s-method and Tanh –mothed

32

4.1 Solve the linear\ non linear using by Tanh -mothed 32 CONCLUSION 38 References 39

1

Chapter 1

INTRODUCTION 1.1 The Origin of Fractional Calculus :- Fractional calculus owes its origin to a question of whether the meaning of a derivative to an integer order could be extended to still be valid when is not an integer. This question was first raised by L’Hopital on September , 1695. On that day, in a letter to Leibniz, he posed a question about , Leibniz’s notation for the derivative of

the linear function ( ) . L’Hopital curiously asked what the result would be if

Leibniz responded that it would be “an apparent paradox, from which one day useful consequences will be drawn.

Starting with where m is a positive integer, Lacroix found the derivative,

( ) (1.1.1)

And using Legendre’s symbol , for the generalized factorial, he wrote

( )

( ) (1.1.2)

Finally by letting and

, he obtained

√

√ (1.1.3)

1.2 Definition of Fractional Calculus :- Over the years, many mathematicians, using their own notation and approach, have found various definitions that fit the idea of a non-integer order integral or derivative. One version that has been popularized in the world of fractional calculus is the Riemann- Liouville definition. It is interesting to note that the Riemann-Liouville definition of a fractional derivative gives the same result as that obtained by Lacroix in equation (1.1.3). Since most of the other definitions of fractional calculus are largely variations of the Riemann-Liuoville version, it is this version that will be mostly addressed.

1.3 Useful Mathematical Functions Before looking at the definition of the Riemann-Liouville Fractional Integral or Derivative, we will first discuss some useful mathematical definitions that are inherently tied to fractional calculus and will commonly be encountered. These include the Gamma function, the Beta function, the Error function, the Mittag-Leffler function, and the Mellin-Ross function.

1.3.1 The Gamma Function The most basic interpretation of the Gamma function is simply the generalization of the factorial for all real numbers. Its definition is given by

2

( ) ∫

( )

The Gamma function has some unique properties. By using its recursion relations we can obtain formulas

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

From equation (1.3.1.2b) we note that ( ) . We now show that (

) √ .

By definition (1.3.1.1) we have (

) ∫

1.3.2 The Beta Function:- The Beta function is defined by a definite integral. Its given by

( ) ∫ ( )

(1.3.2.1a)

The Beta function can also be defined in terms of the Gamma function:

( ) ( ) ( )

( ) (1.3.2.1b)

1.3.3 The Error Function:- The definition of the error function is given by

( )

√ ∫

(1.3.3.1)

The complementary error function ( ) is a closely related function that can be written in terms of the error function as

( ) (1.3.3.2) As a result of (1.3.3.1) we note that ( ) and ( )

1.3.4 The Mittag-Leffler Function The Mittag-Leffler function is named after a Swedish mathematician who defined and studied it in 1903 . The function is a direct generalization of the exponential function, , and it plays a major role in fractional calculus. The one and two-parameter representations of the Mittag-Leffler function can be defined in terms of a power series as

( ) ∑

( )

(1.3.4.1)

( ) ∑

( )

(1.3.4.2)

The exponential series defined by (1.3.4.2) gives a generalization of (1.3.4.1). This more generalized form was introduced by R.P. Agarwal in 1953 . As a result of the definition given in (1.3.4.2), the following relations hold:

( ) ( ) (1.3.4.3)

and

3

( ) ( )

( ) (1.3.4.4)

Observe that (1.3.4.4) implies that

( )

( ) ( )

So

( )

( ) ( ) ( ) (1.3.4.5)

Now prove (1.3.4.3).

By definition(1.3.4.2), ( ) ∑

( )

∑

( ( ) )

∑

( ( )

∑

( ( )

( ) ∑

( ( )

( ) ( )

Note that ( ) Also, for some specific values of and , the Mittag-Leffler

function reduces to some familiar functions. For example,

( ) ∑

( ) ∑

(1.3.4.6a)

( ) ∑

(

)

( )

(1.3.4.6b)

( ) ∑

( )

∑

( )

(1.3.4.6c)

1.4 Definition of the Riemann Liouville Fractional Integral :- Let be a real nonnegative number. Let f be piecewise continuous on ( ) and integrable on any finite subinterval of Then for

( )

( )∫ ( ) ( )

(1.4.a)

we call (1.4.a) the Riemann-Liouville fractional integral of ( ) of order . Definition (1.4.a) can be obtained in several ways. We shall consider one approach that uses the theory of linear differential equations.

4

1.4.1 Examples of Fractional Integrals: Let

evaluate where

By definition

( )∫( ) ( )

=

( )∫ (

)

( )∫( ) ( )

( ) ∫( )

( ) ( )

( )

( )

In the above example, we have established that

( )

( ) (1.4.1.1)

We refer to (1.4.1.1) as the Power Rule . The power rule tells us that the fractional integral of a constant of order is

( ) ( )

( )

And in particular, if

( ) ( )

( )

√

( ) ( )

( )

√

The above examples may give the reader the notion that fractional integrals are generally easy to evaluate. This notion is false. In fact, some fractional integrals, even of such elementary functions as exponentials, sines and cosines, lead to higher transcendental functions. We now demonstrate that fact by looking at some more examples. Suppose ( ) , where is a constant. Then by definition (1.4.a) we have

( )∫ ( )

(1.4.1.3)

If we make the substitution , then (1.4.1.3) becomes

( )∫

(1.4.1.4)

5

Clearly, (1.4.1.4) is not an elementary function. If we refer back to (1.3.4.2) we observe that (1.4.1.4) can be written as

( ) ( ) (1.4.1.5)

1. 5 Derivatives of the Fractional Integral :- For fractional integral

( ) ( ) We now develop a similar relation involving derivatives. However, generally

( ) ( ) Theorem (1) :- Let be continuous on and let If is continuous, then for all

( ) ( ) ( )

( ) (1.5.1)

Proof By definition

( )

( )∫( ) ( )

If we make the substitution , where

, we obtain

( )

( )∫( )

( )( )

which simplifies to

( )

( )∫ ( )

Using the Leibniz’s Integral Rule which states that

∫ ( ) ( ( )) ( ) ∫

( )

( )

( )

(1.5.2)

we then have

( )

( ) * ( ) ∫

( )

+

Now, if we reverse our substitution i.e. let , we obtain

( ) ( )

( )

( ) ∫(

)

( )

Finally, since

and ( )

the preceding equation simplifies to

( ) ( )

( )

( ) ∫( )

( )

6

which implies that

( ) ( ) ( )

( )

1.6 FRACTIONAL DIFFERENTIAL EQUATIONS : We will apply the Laplace transform to solve some fractional order differential equations. The procedure will be simple: We will find the Laplace transform of the equation, solve for the transform of the unknown function, and finally find the inverse Laplace to obtain our desired solution. To start off, let’s briefly look at the definition of the Laplace transform, and particularly the Laplace transform of the fractional integral and derivative.

1.6.1 The Laplace Transform We recall that a function ( ) defined on some domain is said to be of exponential order if there exist constants such that | ( )| for all . If ( )

is of exponential order , then ∫ ( )

exists for all . The Laplace

transform of ( ) is then defined as

( ) ( ) ∫ ( )

(1.6.1.1)

1.6.2 Laplace Transform of the Fractional Integral :- The fractional integral of ( ) of order is

( )

( ) ∫ ( ) ( )

(1.6.2.1)

Equation (1.6.2.1) is actually a convolution integral. we find that

( )

( ) ( ) ( ) (1.6.2.2)

Equation (1.6.2.2) is the Laplace transform of the fractional integral. As examples, we see For that

( )

and

( ) (1.6.2.3)

1.6.3 Laplace Transform of the Fractional Derivative :- We recall that in the integer order operations, the Laplace transform of is given by

( ) ( ) ( )

( ) ∑ ( ) (1.6.3.1)

the fractional derivative of ( ) of order is

( ) ( ) (1.6.3.2a)

where, is the smallest integer greater than , and Observe that we can write equation (1.6.3.2a) as

7

( ) ( ) ( ) (1.6.3.2b) Now, if we assume that the Laplace transform of ( ) exists, then by the use of (1.6.3.1) we have

( ) ( ) ( ) [ ( ) ( )] ∑ [ ( ) ( )]

[ ( ) ( )] ∑ ( )

( ) ∑ ( )

(1.6.3.3)

In particular, if and , we respectively have

( ) ( ) ( ) ( ) (1.6.3.4) And

( ) ( ) ( ) ( ) ( ) ( ) (1.6.3.5) Table (1) gives a brief summary of some useful Laplace transform pairs. We will frequently refer to this Table. Notice that the Mittag-Leffler function is very prominent. As will become more evident later on, this function plays an important role when solving fractional differential equations.

Table (1). Laplace transform pairs

In this table, is real constants; are arbitrary.

8

Chapter 2

Solve the Fractional Differential Equations Using by Jumarie’s

The fractional differential equations and its solutions arises in different branches of applied

science, engineering, applied mathematics and biology [1-9]. The solutions of fractional

difference equations are obtained by different methods which includes Exponential-Function

Method [10], Homotopy Perturbation Method [11], Variation Iteration Method [12], Differential

transform Method [13] and Fractional Sub-equation Method [14], Analytical Solutions in terms

of Mittag-Leffler function [15]. In developing those Methods the usually used fractional

derivative is Riemann-Liouvellie (R-L) [6], Caputo derivative [6], Jumarie’s left handed

modification of R-L fractional derivative [16-17]. In [15] we have developed an algorithm to

solve the homogeneous fractional order differential equations in terms of Mittag-Leffler function

and fractional sine and cosine functions. However, there are no standard methods to find

solutions of non-homogeneous fractional differential equations. In this research we describe a

method to solve the fractional order non-homogeneous differential equations. Organizations of

the research are as follows; we describe the different definitions of fractional derivatives and

properties of Mittag-Leffler function. In section-3.0 we describe the solutions of α−order

fractional differential equations. In section 4.0 the solutions of 2α−order fractional differential

equations is described, with several types of forcing functions. this methods has been applied to

solve both un-damped and damped fractional order forced oscillator equations. In this research

the fractional derivative operator will be of Jumarie type fractional derivative.

2.0 Definition of Fractional Derivatives :-

The useful definitions of the fractional derivatives are the Grunwald-Letinikov (G-L) definition

and Riemann-Liouville(R-L) definition [6] and Modified R-L-definitions [16-17].

2.0.1 Grunwald-Letinikov definition

Let ( ) be any function then the α-th order derivative of ( ) is defined by

( )

∑( )

( )

∑

( ) ( )

( ) ∫ ( ) ( )

( ) ∫

( )

( )

Where α is any arbitrary number real or complex; and the generalized binomial coefficients are

described as follows [1], [16-17]

( )

( )

( )

( ) ( )

The above formula becomes fractional order integration if we replace by which is

Riemann fractional integration formula

9

∫ ( )

( )

( )

( ) ∫ ( ) ( )

In above we have noted several notations used for fractional integration.

2.0.2 Riemann-Liouville fractional derivative definition

Let the function ( ) is one time integrable then the integro-differential expression as following

defines Riemann-Liouvelli fractional derivative [1], [6]

( )

( )(

)

∫ ( )( )

( )

Here the is a positive integer number just greater than real number The above expression is

known as the Riemann-Liouville definition of fractional derivative with( ) In the above definition fractional derivative of a constant is non-zero. [6]

2.0.3 Modified Riemann-Liouville definition : To overcome the short coming fractional derivative of a constant, as non-zero, another

modification of the definition of left R-L type fractional derivative of the function ( ),in the

interval was proposed by Jumarie in the form described below [16]

( )

{

( )∫ ( )

( )

( )

∫ ( )

( ) ( )

( ( ))

Here we state that for ( ) , However in we will this left-Jumarie

fractional derivative that is ( )

for with condition ( ) for all will

simplify the symbol and drop differentiationg variable and simply write ( )

Using the above definition Jumarie [16] proved

( ( ) ( ))

( ) ( ) ( ) ( )

We have recently modified the right R-L definition of fractional derivative of the

function ( ),in the interval in the following form [17]

( )

{

( )∫ ( )

( )

( )

∫ ( )

( ) ( )

( ( ))

Using both the modified definition we investigate the characteristics of the non-differentiable

points of some continuous functions in [17]. The above defined all the derivatives are non-local

type, and obtained solution to homogeneous FDE, with Jumarie derivative [15]. Subsequently we

will be using as fractional derivative operator of Jumarie type, with start point , and

stating the function ( ) for all in following sections.

10

2.1 The Mittag-Leffler Function

The Mittag-Leffler function was introduced by Gösta Mittag-Leffler [18] in 1903. The one-

parameter Mittag-Leffler function is denoted by ( ) and defined by following series

( ) ∑

( )

( )

Again from the Jumarie definition of fractional derivative we have

we apply this

property to get α order Jumarie Derivative of the Mittag-Leffler function ( ) as follows

( )

( )

( )

( )

( )

( )

( )

Therefore the fractional differential equation

has solution in the form

( )

where is an arbitrary constant.

2.2 Non-Homogeneous Fractional Differential Equations and Some Basic

Solutions The general format of the fractional linear differential equation is

( ) ( ) (2.2.1)

Where ( ) is a linear differential operator . The above differential equation is

said to be linear non-homogeneous fractional differential equation when ( ) , otherwise it

is homogeneous. Solution of the linear fractional differential equations (composed via Jumarie

Derivative) can be easily obtained in terms of Mittag-Leffler function and fractional sine and

cosine functions [15].

The function ( ) is forcing function. We have written this as function of purposely for

ease. For example we will use ( ) ( ) are taken as forcing functions.

There will be other functions in the derivations like ( ) ( )all functions described with

scaled variable that is . Nevertheless the forcing functions can be written as simple ( )

though.

In that research [15] we found the following (theorems) which we will be using :

(i) The fractional differential equation ( )

( ) ( )

. Has solution of the

form ( ) ( ) where and are constants.

(ii) The fractional differential equation

. Has solution of

the form ( ) ( )where and are constants.

11

(iii) Solution of the fractional differential

equation

( )

is of the form

( ) ( ) ( ) where and are constants .

From now we indicate Jumarie fractional derivative with start point of differentiation as as

instead

Theorem (2.1) :

If and are two solutions of the fractional differential equation ( )

then is also a solution where and are arbitrary constants.

Proof:

Since ( ) has solutions and

( ) and ( )

( )( ) ( ) ( )

Hence is also a solution of the given fractional differential equation.

Hence the theorem is proved.

Similarly we can prove if are solutions of the fractional differential equation

( ) then is also a solution of it.

Theorem (2.2): If ( ) ( )( ) ( ) , then solution of

the homogeneous equation ( ) is ∑ ( )

where ’s are arbitrary

constants and all are distinct.

Proof:

Since Jumarie type fractional derivative of Mittag-Leffler function ( ) is with

constant ( ) ( ) Thus solution of the differential

equation ( ) is ( ) where is a constant [15].

Let ( ) be a non-trivial trial solution of the differential equation ( )

then ( ) or we write the following after subtracting from both the sides as

demonstrated below

( ) ( ) ( ) ( )

( ) ( )

We apply the above result sequentially as demonstrated below

( ) ( ) ( )

( )( ) ( ) ( )

12

( )(

) ( ) ( ) ( )

( )( )( ) ( ) ( )

( )( ) ( ) ( ) ∏( )

Since ( ) we get

∏ ( ) (1.2.2)

Implying that

Hence the general solution is

( ) (

) ( ) ∑

( )

Hence the theorem is proved.

The above theorem implies principal of superposition holds for the linear fractional differential

equations (composed via Jumarie fractional derivative) also.

Note: In the above theorem if two or more roots of the equation (1.2.2) are equal roots or

complex then the solution [15] form is given below.

For and then solution of the is

(

) ( ) ( )

For and

then the solution is

( ) (

) ( )

where are arbitrary constants.

For and

then the solution is

( ) ( ) ( ) ( ) ( )

Thus solutions of linear homogeneous fractional differential equation with Jumarie fractional

derivative is express in terms of Mittag-Leffler functions and fractional type sine and cosine

series.

Now the question arises what will be solution of linear non-homogeneous fractional differential

equations. The solution corresponding to the homogeneous equation will be called as the

complementary function, it contains the arbitrary constants and this solution will be denoted

by the other part, that is a solution which is free from integral constant, and depending on the

forcing function will be called as Particular Integral (PI) and will be denoted by . Thus the

general solution will be . We will develop simple method to evaluate Particular

Integral.

13

2.3 α−order non-homogeneous fractional differential equations Consider the linear α− order non-homogeneous fractional differential equation with

for for of the following form,

( ) ( ) (2.3.1)

The solution of the corresponding homogeneous part is [15]

( ) is arbitrary constant

Multiply both side of equation (2.3.1) by ( ) as demonstrated below

[ ( )][ ) ] = ( ) ( )

[ ( )][ )] = ( ) ( )

[ ( )][

( )] = ( ) ( )

[ ( )][( ) ( ( ( )))] = ( ) ( )

( ) ( ) ( )

In the above steps we have used ( ) ( )

. Now operating

on

both The sides are ( ) ( ) ( ) Also we add a constant since

Jumarie type derivative of a constant is zero and from here we get the following

( ) ( ) ( ) where is a constant

( ) [ ( ) ( ) ] (2.3.2)

Or

( ) [ ( ) ( ) ] ( )

the first part corresponds to solution of homogeneous equation, that is

( ) and the other part ( ) [ ( ) ( ) ] corresponds to

the effect of non-homogeneous part and free from integral constant, but depending on the nature

of forcing function, this part is named as Particular Integral (PI) as in case of classical

differential equations. Now we take several forms of forcing function.

2.4 Particular Integral for ( ) ( )

We consider the linear first order non-homogeneous fractional differential equation of order

with with for

( ) ( ) ( )

( )

then the Particular Integral (PI) described in the previous section is

( ) [ ( ) ( ) ]

Putting ( ) ( ) in above we get the following

( ) [ ( ) ( ) ]

= ( ) [ (( ) ) ]

= ( ) (

) (( ) )

14

=

( )

For , P.I is

( ) [ ( ) ( ) ]

( ) [ ]

( )

=

( ) ( )

2.4.1 Short Procedure for Calculating Particular Integral for ( ) ( )

This procedure is similar and in conjugation with classical integer order calculus. In classical

order calculus .Hence the forced function reduce to ( ) ( ) Therefore the

particular integral will be

for

( ) [ ( ) ( ) ] ( ) [

( ) ]

( ) [

( ) ]

( )

And for is

( ) [ ( ) ( ) ] ( ) [

] ( )

Here we observe that the derivative operator is replaced by in the first case, i.e. for ( ).

In the second case the derivative operator is replaced by . We can replace the fractional

Jumarie derivative operator by for the first case ( ) and by ( ) for second

case ( ) then short procedure as follows for Particular Integral, that is,

for ( ) is

[

] ( replace

by

=

( )

And for is

[

] ( replace

( ) *

+ ( ) *

+ ( )[ ]

( ) ( )

Hence the general solution of equation (2.3.1) is

{

( )

( )

( )

( ) ( )

}

15

2.5 Particular Integral for ( )

Again when ( ) then the differential equation (2.3.1) becomes

( ) ( )

The solution of the homogeneous part [15] that is ( ) is ( )

Let ( ) ( ) the solution of the corresponding non-homogeneous equation

where ( ) is an unknown function of Then using the definition by Jumarie [16] that is

( ( ) ( ))

( ) ( ) ( ) ( )

We get the following

( ) ( )

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

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

putting this in (2.5.1) we get

( )

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

( ) ( ( ) )

Therefore we get

( )

( ) ( )

We now apply fractional integration by parts by Jumarie formula [16] as depicted below

∫ ( )

( ) ( ) ( ) ( ) ∫ ( ) ( ) ( )

Here we mention that the symbol ∫ ( )( ) ( ) implies Jumarie fractional

integration We will use also derived expression that is

( ) ( ) ( ) the following derivation. [15]

( ) ( )

∫ ( ) ( )

∫ ( ) ( * ( )

+)

( )

( )

|

∫ ( )

( )( )

( )

( )

|

∫ ( )

( ( ))( )

( )

|

( )

∫ ( )

( )

16

( )

|

( )

* ( )

+

( )

( )

( )

Then is constant

Hence the general solution is

( ) ( )

*

( )

+ (

( )

)

( )

( ) the first part in above expression is solution of homogeneous equation and

the second part of the above that is

*

( )

+ is particular integral.

2.4.2 Short Procedure for Calculating Particular Integral for ( )

This procedure is similar and in conjugation with classical integer order calculus. Here for

( ) , and the corresponding particular integral is

(

)

*

+

[

]

In the same way we can have a short procedure as follows for Particular Integral that is,

( )

*

+

[

]

*

( )

+

In the above derivation ( ) ( ) is used.

Thus all the Jumarie derivatives for , where is Natural number. Therefore

we have discussed the solutions of non-homogeneous order differential equations for different

forcing functions ( ).

2.6 Evaluation of

( ) ( ) ( ) where

can be factorized as

(

) (

)

[

], and we

use this in following derivation.

As here we replace by and by for the operations

( ) and

( ) respectively , as is demonstrated below.

( )

17

[

] ( )

[

( )

( )]

( ) ⇒

( ) ( )

( )

[

( )

( )]

*

( )

( )+ ( )

( )

[

( )

( )]

*

( )

( )+ ( )

Therefore (a-b) we obtains

( )

[

( )

( )

( )

( )]

[(

) ( ) (

) ( )]

[

( )

( )]

[

( ) ( ) ]

( )

Similarly we get by following above procedure

( )

( )

Thus to find the particular integral

( ) replace by .

This procedure is similar and in conjugation with classical integer order calculus. In classical

order calculus hence the forced function reduce to ( ) ( ) Therefore the particular

integral will be

18

( )

[

] ( )

[

( )

( )]

( ) ⇒

( ) ( )

( )

[

( )

( )]

[

( )

( )]

( )

[

( )

( )]

[

( )

( )]

Therefore

( )

[

( )

( )

( )

( )]

[(

) ( ) (

) ( )]

[

( )

( )]

[

( ) ( ) ]

( )

2.7 2α− Order Non-Homogeneous Fractional Differential Equations

General formulation of non-homogeneous fractional differential equation of 2α-order is

( ) ( ) .

where and are constant here. Consider the 2α− order non-homogeneous fractional differential

equation ( ) ( ) where ( ) ( )( ) then solution of the

non-homogeneous part that is ( ) given by ( ) ( )

19

2.8 Use of Method of un-Determinant Coefficient Method to Calculate the

Particular Integrals for Different Functional Forms of ( )

For ( ) ( ) we have the given equation is

( ) ( )( )

( ( ) )( ) ( )

( )

Here let the particular integral be ( ) where is constant.

Then

[ ] ( ) [ ] ( )

and putting in the given equation (2.8.1) we get the following

( ( ) )( ) ( )

( ) ( )

( ) ( ) ( ) ( ) ( )

( )

( )

( )( )

Therefore

( )( )

and consequently the Particular integral is

( )( ) ( )

Hence the general solution is

( ) ( )

( )( ) ( ) ( )

20

For implying ( )( ) then the solution (2.8.2) does not exists. In this case

the fractional differential equation

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

If we consider ( ) in this case also then putting in (2.8.3) we get ( )

which is free from P i.e. P is non-determinable. This form of PI is not suitable here; consider

the modified form as following

( )

Then

[ ] ( ) ( ) ( )

[ ] ( ) ( ) ( )

and putting in (2.8.3) we get the following

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

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( )

In this case the general solution of the fractional differential equation is of following form

( ) ( )

( ) ( ) ( )

When then

( ) ( ) ( ) ( )

and take the particular integral in the form

( )

Then

21

[ ] ( ) ( ( )

( )) ( )

[ ] ( ) ( ( )

( )) ( ) ( ) ( )

Putting this in (2.8.4) and after simplification we get

( )

( ) ( )

In this case the general solution will be of following form

( ) ( )

( ) ( )

Thus we can summarize the result as a theorem in the following form

Theorem:(2.3)

The differential equation the ( ) ( ) has particular integral

(

( )) ( ) for ( )

(i) When ( ) ( )( ) , for then solution of the fractional

differential equation will be

( ) ( )

( )( ) ( )

(ii) When ( ) ( )( ) for then solution of the fractional

differential equation will be

( ) ( )

( ) ( ) ( )

(iii) When ( ) ( )( ) for then solution of the fractional

differential equation will be

( ) ( )

( ) ( )

22

2.9 Use of Direct Method to Calculate the Particular Integrals for Different

Functional Format of ( ) Using the direct method we can easily calculate the Particular integrals for different functional

format of ( )

• For ( ) ( ) we have

(

( )) ( )

(

(( )( ))) ( )

[

( )

( )] ( )

[

] ( )

( )( ) ( )

In this case the general solution is

( ) ( )

( )( ) ( )

And for is

(

( )) ( )

[

( )

( )] ( )

* ( )

( )

+ ( )

In this case the second part will be adjusted in the complementary function and hence the

general solution is

( ) ( )

( ) ( ) ( )

And for is

23

(

( )) ( )

( ) ( )

*

( ) + ( )

( ) *

( ) + ( )

( )

( )

In this case the general solution will be

( ) ( )

( ) ( )

In generalized case for any polynomial type function ( ) the particular integral is

(

( )) ( )

( ) ( ) ( )

For ( ) ( ) must contain a factor of the form ( ) positive integer i.e.

( ) ( ) ( ) with ( ) then

(

( )

(

)) ( )

( )

( )

( )

( )

( )

( )

( )

24

2.10 Some Examples of the Fractional Differential-Using by Jumarie’s :-

Example :(3.1)

We take the following fractional differential equation ( ) for

(

)

Solution:

Solution of the corresponding homogeneous equation is [15]

(

)

(

)

The particular integral calculation is done in following steps

(

) (

)

*

+

[

(

)

(

)

]

[

(

)

(

)

]

(

)

(

)

( )

( )

( )

( )

Hence the general solution is

25

(

)

(

)

( )

( )

( )

( )

Where and are arbitrary constants.

Example : (3.2)

Consider the fractional order forced differential equation ( ) , for

( )

Solution:

Here solution of the corresponding homogeneous equation

is

( ) ( )

The particular integral is

( )

( )

Hence the general solution is

( ) ( )

( )

For particular integral we replaced ( )

Example : (3.3)

Take the fractional order damped-forced differential equation ( ) , for

( ) ( )

Solution:

Here solution of the corresponding homogeneous equation [15]

( )

Is

( ) ( ) ( )

26

The particular integral is

( ) ( )

( ) ( )

( ) ( ) here replace

multiply the numerator and denominator by ( )

( )

( ) ( )

( ) ( ) ( )

( )

Hence the general solution is

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

( )

Example : (3.4)

We solve the following homogeneous FDE using Fractional Laplace

Transform * (

)

+ ( ) where ( )

( )

Solution:

The equation can be written in the form

( )

Applying fractional Laplace transforms to both sides, we have

[ (

)]

[

]

( ) ( )

( ) [

( ) ( )] ( )

[

]

( )

27

( ) (

)

[

( )

*(( )

+]

( )

[

( )

*(( )

+]

(

)

(

)

28

Chapter 3

Adomian Decomposition Method and Jumari’s

Fractional Derivative

3.1 The Adomian Decomposition Method:

The decomposition method was recently introduced by Adomian [1] . The method has much

similarity with the Neumann series as has been discussed in the previous. The decomposition

method has been proved to be reliable and efficient for a wide class of differential and integral

equations of linear and nonlinear models. Like Neumann series method, the method provides the

solution in a series form and the method can be applied to ordinary and partial differential

equations and recently its use to the integral equations was found in the literature (see Ref. [9]).

The concept of uniform convergence of the infinite series was addressed by Adomian ([2], [3])

and Adomian and Rach [4] for linear problems and extended to nonlinear problems by

Cherruault et al [5] and Cherruault and Adomian [6]. we do not want to repeat the convergence

problems.

In the decomposition method, we usually express the solution of the linear integral equation

( ) ( ) ∫ ( ) ( )

(3.1.1)

in a series form like regular perturbation series (see Van Dyke [8]) defined by

( ) ∑ ( ) (3.1.2)

Substituting the decomposition equation (3.1.2) into both sides of equation (3.1.1) gives

∑ ( ) ( ) ∫ ( ) ∑ ( )

(3.1.3)

The components ( ) ( ) ( ) ( ) of the unknown function ( ) are completely

determined in a recurrence manner if we set

( ) ( )

( ) ∫ ( ) ( )

( ) ∫ ( ) ( )

( ) ∫ ( ) ( )

( ) ∫ ( ) ( )

(3.1.4)

29

and so on. The main idea here like perturbation technique is to determine the zeroth

decomposition ( ) by the known function ( ). Once ( ) is known, then successively

determine ( ) ( ) ( ) and so on.

A compact recurrence scheme is then given by

( ) ( )

( ) ∫ ( ) ( )

In view of equations (3.1.2) and (3.1.3), the components ( ) ( ) ( ) ( ) follow

immediately. Once these components are determined, the solution u(x) can be obtained using the

series (3.1.3). It may be noted that for some problems,the series gives the closed-form solution;

however, for other problems, we have to

determine a few terms in the series such as ( ) ∑ ( ) by truncating the series at

certain term. Because of the uniformly convergence property of the infinite series a few terms

will attain the maximum accuracy

Example :(3.1)

Solve the D’Alembert’s wave equation with the given initial conditions by the decomposition

method

( ) ( )

( ) ( )

Solution :

The D’Alembert’s wave equation can be transformed into the integral equation by using the

given initial conditions as follows:

( ) ( ) ∫

( ) ( ) ( ) ∫ ∫

(3.1.1)

Consider the infinite series

( ) ∑ ( ) (3.1.2)

which is known as the decomposition series. Using this series solution into the equation (3.1.1)

we have

∑ ( ) ( ) ( ) ∫ ∫ ∑ ( )

(3.1.3)

Now the various iterates are obtained as

( ) ( ) ( )

( ) ∫ ∫ ∑( )

( ) ∫ ∫ ∑( )

( ) ∫ ∫ ∑( )

30

( ) ∫ ∫ ∑( )

Performing the indicated integrations, we can write the solutions of each iterate as follows:

*

( )

( )+

( ) *

( )( )

( )( )+

( ) *

( )

( )

( ) ( )+

Hence the solution can be written as

( )

[∑( )

( )

( )] [∑( )

( )

( )]

( ) ( )

∫ ( )

(3.1.4)

3.2 The jumari’s Fractional Derivative :

In order to elucidate the solution procedure of the Adomian’s Decomposition Algorithm, we

consider the following fractional differential equation:

( ) ( ) ( ) (3.2.1)

( ) ( )

where is the differential operator in , ( ) and ( ) are continuous function . According

to the ADM, we can construct recurrence relation for Eq. (3.2.1) as follows

( ) ( ) [( ( ) ( ))] (3.2.2)

( ) ( )

( )∫ ( )

[( ( ) ( ))]

Then

( ) ( )

( )

( )∫

[( ( ) ( ))] ( ) (3.2.3)

It is obvious that the successive approximations for can be obtained. The initial values are

usually used for selecting the zeros approximation . Consequently, the exact solution may be

obtained by using

( ) ∑ ( ) (3.2.4)

Example : (3.2)

Consider the nonlinear time-fractional advection partial differential equation solve

D’Alembert’s wave equation :

( )

( ) (3.2.1)

31

( ) ( )

( ) ( )

Solution :

Applying Adomian’s Decomposition Method, we get

( ) ( )

( )∫

( ) ( )

( ) ( ) ( )

( )∫ ∫

( ) ( )

( ) (3.2.2)

Substitutions the initial conditions from eq(3.2.2) we obtain

( ) ( ) ( ( )) ( )

( )∫ ∫

( ) ( )

( ) (3.2.3)

Hence

( ) ( ) ( ) ( )

( )

( )∫ ∫

( ) ( )

( )

When

( )

( )∫ ∫

( ) ( )

( )

( )

( )∫ ∫

( ) ( ) ( ) ( )

( )

( ) ( ) ( ( ) ( ) )

( )

( )∫ ∫ ( ) ( ) ( ) ( )

( )

( )∫ ( ) ( ) ( ) ( ) ( )

( )* ( ) ( )

( )

( ) ( ) ( )

( )+

( )

( )* ( ) ( )

( )

( ) ( ) ( ) ( )

(( ) )+

Hence the solution can be written as

( )

32

Chapter 4

Compared Between Jumerie’s-Method and Tanh -mothed

4.1 Solve the linear and non linear Using by Tanh -Mothed :

The solution of linear and non linear differential equation , partial differential equation

and non linear fractional differential equation is current research in Applied science. Here

tanh-method and fraction sub equation methods and used to solve three non-linear

differential equation and the corresponding fraction differential equation .

Let the linear partial differential equation is of the form :

( ) (4.1.1)

Where ( ) and H is functions linear operator of the u and its derivative . is

order of fractions derivatives of Jumerie type .

From equation (4.1.1)

Using the travelling wave transformation the eq(4.1.1) reduce to

( ) (4.1.2)

Where ( ) is function of one variable using the same transformation , the eq(4.1.2)

reduce to

Thus the operator

, in terms of with

[

]

[

]

[

]

And

[

]

[

]

[

]

33

Hence

(4.1.3)

The solution of eq(4.1.2) is taken in the following series form

( ) ∑

Where ( ) satisfy the fraction Riccati equation

( )

Where is real constant . say for

Integrating both sides once we get

∫

[

]

Similar way we write in terms of fractional integration (Jumarie type) the following

[

]

Thus :

( )

{

√ (√ )

√ (√ ) }

√ ( )

√ ( )}

( )

}

Here we consider the solution in form

( )

Assume that ( ) ( ) Then

34

[

]

[( )

]

( )

( )

Thus :

( )

( )

( )

( )

(4.2.4)

When we get

( )

Putting the value of in (4.2.4) we get

Hence the solution given by :

( ) ( )

Example (4.1) :

Let the non- linear partial differential equation is of the form :

( ) (4.1.1)

Where ( ) and H is functions linear operator of the u and its derivative . is

order of fractions derivatives of Jumerie type .

From equation (4.1.1)

Using the travelling wave transformation the eq(4.1.1) reduce to

( ) (4.1.2)

35

Where ( ) is function of one variable using the same transformation , the eq(4.1.2)

reduce to

Thus the operator

, in terms of with

And

[

]

[

]

[

]

Hence

(4.1.3)

The solution of eq(4.1.2) is taken in the following series form

( ) ∑

Assume that ( ) ( ) Then

[

]

[( )

]

( )

( )

Thus :

( )

( )

( )

(4.1.4)

When we get

36

( )

Putting the value of in (4.1.4) we get

Hence the solution given by :

( )

( )

Example (4.2) : Consider the nonlinear wave equation

subject to initial conditions

( ) ( )

Applying Adomian’s Decomposition Method, we get

( ) ( )

( ) ( )

( )∫

( )

Where

, hence

( )

( )

( ) ∫

when

( )

( ) ∫

( ) ∫( )( )

( )

( ) ∫

( )

( ) ( )

( )

37

( )

Since

( ) ( )

( ) (

( )

( ) )

( ) ( )

38

Conclusion

We use these solutions in several ways, but the best

way to arrive at’s the solution of the fractional

Adomian method because it is easier to solve the

problem. If Tanh-method is used, the solution cannot

be easily found because it ignores the conditions.

39

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