Fractional Integration and Differentiation by new Transformation

: In opposite to differentiation and integration of integer order, an important type of differentiation and integration is the so - called Fractional Calculus (FC) in which the differentiation and integration is of non-integer order. The idea of this work is to use a transformation known as the Extension AL-Zughair Transform (EZT) for fractional calculus, so it reviewed some basic properties and definitions of (FC) such as differentiation and integration with Riemann-Liouvial operator. This transformation for fractional differentiation and integration reinforced with some application examples at the end of the article for simplicity the Fractional Integrals and Fractional Derivatives


Introduction
Fractional calculus [1] has an important role in many applied sciences, especially applied mathematics [2]- [4]. It is known that calculus means integration and differentiation. Fractional calculus, as it is name suggests, refer to fractional integration and fractional differentiation [5].
For ≤ ≤ , is called the Riemann-liouville fractional integral operator of order .

Theorem (2.1.2):
Let , ≥ 0, and ∈ 1 [ , ] then, which is the defining property for a non-local operator of fractional type.

The power function (by Riemann-liouville fractional integral)
The function fractional degree which considers in this subsection is the important function = , where is initially arbitrary. It shall see, however, that must exceed -1 for integration to have the properties it demands of the operator.
From classical calculus, the first encounter with non-integer will be restricted to negative so that it may exploit the Riemann-Liouvial definition, thus: Therefore,

Fractional Derivative
Definition (2.3.1): Let ∈ and ∈ , the Riemann-Liouville fractional differential operator defined as following that [10]  Lead to an alternative decomposition of fractional derivative into an ordinary standard derivative followed by a fractional integral.

Expansion of Al-zughair transform
Definition (3.1): the al-zughair transform of a given function ( ) is defined as [7] [ Where is positive constant

Definition (3.2):
From definition (3.1) and by transforming the limits of integration, will have This is called expansion of al-zughair transform (EZT).

property (3.3): EZT is distinguished by the linear property, which is
Where are constants, 0 < < 1.

EZT for some selected function:
Expansion of Al-Zughair transformation for Some basic functions are given by [9]:

EZT for fractional calculus
This paper present two important properties that will be useful in obtaining the EZT of fractional integral and derivative operators. Property

Property (4.3)
: this property states that the EZT of ( ( ) ( )) is given by: Where

EZT for fractional integration
Let's start with Expansion Al-zughair transform (EZT) of fractional integral. According to the following proposition.

EZT for fractional differentiation
In this subsection, let's turn to Expansion Al-zughair transform (EZT) of Riemannliouville fractional derivative operator with , according to the following proposition.

Examples 5.1 The Unit Function of Fractional Order:
Considering first the integration to order of the function ≡ 1, for which it is convenient to reserve the special notion, It shall refer to this function as the unit function.
In order to find a Compute [ ( )], where ( ) is unit function and 0 < < 1, using Proposition

The Constant Function:
From a function = , where k is any constant and 0 < < 1 , one has

The power function :
The function fractional degree that considering in this subsection is the important function = , where q is initially arbitrary. Then to find EZT for fractional integration for this function, by using Proposition

Power Natural logarithm function ( ) :
With k arbitrary constant, the EZT of fractional integration for the power natural logarithm function ( ) = ( ) by using Proposition (4.1.1), as following:

Conclusion
The subject of fractional calculus is extremely challenging and has many illdefined concepts to deal with. In any case, the requirement for new methods of solving differential equations when fractional order is necessary in physical and engineering applications. Therefore, it used a new transformation which is the Expansion Al-Zughair transformation of fractional calculus, to be able to get the fractional derivatives and fractional integrals that are an extension of solving fractional differential equations.