Novel Definitions of α-Fractional Integral and Derivative of the Functions

An α-fractional integral and derivative of real function have been introduced in new definitions and then, they compared with the existing definitions. According to the properties of these definitions, the formulas demonstrate that they are most significant and suitable in fractional integrals and derivatives. The definitions of α-fractional derivative and integral coincide with the existing definitions for the polynomials for 0 ≤ α < 1. Furthermore, if α = 1, the proposed definitions and the usual definition of integer derivative and integral are identical. Some of the properties of the new definitions are discussed and proved, as well, we have introduced some applications in the α-fractional derivatives and integrals. Moreover, α-power series and α–rule of integration by parts have been proposed and implemented in this study.


Introduction
Fractional calculus was a hot topic in the 20 th and 21 th centuries.Evidently, over the past two decades, the use of fractional calculus has become increasingly prevalent in both pure and applied fields of science and engineering.Discrete versions of fractional calculus and their properties have been studied by many researchers.For example, some authors studied and introduced the properties and methods of solving fractional differential equations (FrDEs), as well as some concepts in FrDEs theory and applications [1][2][3][4].Some authors proposed new definitions of fractional derivatives.For example, R. Khalil et al. introduced a new definition of the fractional derivative and fractional integral and then, they show the new definition is the most fruitful and natural definition [5] while Zheng et al. proposed a new fractional derivative of the Caputo type and then, some of the basic properties this definition has been studied [6].In the review of the literature of the authors which are used the new definitions for solving FrDEs, namely Anderson and Avery have reformulated the second-order conjugate boundary value problem(BVP) using the new conformable fractional derivative [7], Cenesiz and Kurt discovered the precise answers to the time fractional heat differential equations (DEs) [8], Unal and Gan presented the conformable fractional differential transform method and its use with conformable FrDEs [9].Hammad and Khalil have studied the Legendre conformable FrDEs and the basic properties of such fractional polynomials [10], Abdel Hakim has proved the existence of the conformable fractional [11], Khalil and Abu-Hammad have gave the exact solution of the heat conformable FrDE.They also discussed some other differential equations [12], Khalil et al. studied the geometrical meaning of the conformable fractional derivative and the fractional orthogonal trajectories are also introduced [13].Unal et al. solved the variable coefficients, homogeneous sequential linear conformable FrDEs of order two using the power series around a regular point, also, they introduced the conformable fractional Hermite DEs [14], Abdel Jawad developed the definition of the fractional conformable derivative and he set the basic concepts in this fractional calculus [15].In addition, Ortega and Rosales introduced the properties of fractional conformable derivatives [16].Moreover, Qasim and Holel, studied the solution of some types of composition fractional order DEs corresponding to optimal control problems [17].Lastly, Mechee and Senu studied the numerical solutions of FrDEs of Lane-Emden type by the method of collocation, [18].On the other hand, many authors studied the properties and the applications of the definitions of solving fractional definitions derivative or integration [19]- [27].In this study, we introduced novel definitions of α-fractional integrals and derivatives.Indeed, their properties have been studied.Moreover, α-power series and α-rule of integration by parts have been proposed and implemented in this study.

Preliminary
The background related to this study has been introduced in this section.

Gamma Function
Albert Einstein created the non-integral factorial function known as the Gamma function (Gf).Gf is the most significant notation that is used in classical fractional calculus.

Definition 2.1 Gamma Function [28]
The function of Gamma Γ(x) is a function of a positive real number x, defined 1.
( ) ∫ ( ) In the following, some of the most important properties of the Gamma function are given: for x > 0; .

The Fractional Derivatives
Since the beginning of calculus, there have been fractional integrals and derivatives.
L'Hospital wondered what ( ) does it mean if .Many researchers have attempted to define the fractional derivative since then.For the fractional derivative, most of them are used in an integral form.Two of them are the most well-liked.Since that time, numerous researchers have attempted to define a fractional derivative.For the fractional derivative, the majority of them employed an integral form as follows.

Definition 2.2 Riemann-Liouville Derivative [5]
Let a function [ ) and Then, the Riemann-Liouville -derivative definition of the function f, for [ ) where n is an integer number n N, is defined as follows: Indeed, the Caputo derivative definition of f, for α [n -1, n), is defined as follows:

Definition 2.3 Caputo Derivative [5]
For α [n -1, n), the Caputo α-derivative of the function [ ) , and is defined as follows: ( in case that n is an integer number where n -1 < α < n, n N.

Definition 2.4 Conformable Fractional Derivative [5].
Given a function [ ) Then, the conformable fractional α -derivative of the function f of order α is defined by , for all x>0 and α (0, 1].

Definition 2.5 a-Conformable Fractional Derivative [27]
Sarikaya et al. have defined a-conformable fractional derivative for the function [ ) with 0 ≤ a < b of order α is as follows: [5] ( ( )) ( ) ( ) All of the above definitions, including Equations ( 2)-( 5), satisfy the fractional derivative's linear property.All of the definitions share this property with the first derivative.Some of the disadvantages of the other definitions are given in the following: 1.
The Riemann Liouville derivative does not meet the requirement for property ( ) . 2. The fractional Riemann Liouville derivative formula fails to meet the requirement for being the derivative of the product of two functions for integer derivatives.
( )( ) ( ) ( ( )) ( ) ( ( )) 3. The fractional Riemann Liouville derivative formula does not satisfy the property of the derivative of the quotient of two functions with integer order: ( ) 4. The fractional Riemann Liouville derivative formula does not satisfy the property of the derivative by the chain rule of two functions with integer order: ( )( ) ( )( ( )) ( ) 5. The property of composition of derivatives one function is not satisfied by any fractional derivative formula.Most fractional derivatives do not satisfy in general: ( ( )) ( ( )) 6.For all derivatives definitions assumes that the function f is differentiable.

Main Results
In this section, the main results of this article have been introduced

Proposed α-Fractional integral and Derivative of ( )
In this section, we have proposed the αfractional integral and derivative of the function f(τ) and introduced their properties.

Definition 3.1
The new definition of the α-fractional derivative of the real function ( ) [ ) is defined as follows: ( ( )) ( ) ( ) Accordingly, in Table 1: the α-fractional derivatives of some basic functions f(τ) of order α are given using the definition 3.1