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In this paper, we describe two approaches to the definition of fractional derivatives. We investigate the accuracy of the analysis method for solving the fractional order problem. We also give some improvements for the proof of the existence and uniqueness of the solution in fractional differential equations. Treatment of a fractional derivative operator has been made associated with the extended Appell hypergeometric functions of two or three variables and Lauricella hypergeometric function of three variables.

For compressible flows, in particular, there are additional degrees of fractional differential equations and treatment of a fractional derivative operator [

The fractional ordinary differential equations have the following general form

F ( x , y ( x ) , D α 1 y ( x ) , D α 2 y ( x ) , ⋯ , D α n y ( x ) ) = g ( x ) (1)

where F ( x , y 1 , y 2 , ⋯ , y n ) and g ( x ) are given function and D α n are the operators of fractional differentiation with real α n > 0 or complex α n , R e α n > 0 , n = 1 , 2 , 3 , ⋯ , k . for the nonlinear differential

D α y ( x ) = f ( x , y ( x ) ) (2)

With real α > 0 or complex α ( R e ( α ) > 0 ), and the linear differential equations

∑ n = 1 k c n ( x ) D α n y ( x ) + c 0 ( x ) y ( x ) = f ( x ) (3)

The fractional differentiation operators (1) and (3) can have different forms see [

( D α + α y ) ( x ) = ( d d x ) n ( I α + n − α y ) ( x ) , n = [ R e ( α ) ] + 1 , (4)

where ( I α + n − α y ) ( x ) is the fractional integral of order α .

( I α + n − α y ) ( x ) = 1 Γ ( x ) ∫ α x y ( t ) d t ( x − t ) 1 − α ( α ∈ ℂ , R e ( α ) > 0 ) (5)

Γ ( x ) is the Gamma function [

∫ α x d t ∫ α t d t 1 ⋯ ∫ α t n − 2 y ( t n − 1 ) d t n − 1 = 1 ( n − 1 ) ! ∫ α x ( x − t ) n − 1 y ( t ) d t (6)

(If we use the formula ( n − 1 ) ! = Γ ( n ) ) and replaced n by α ∈ ℂ ( R e ( α ) > 0 ), then (6) yields (5) and the fractional differentiation operator D α + α is inverse to the fractional integration one from the left:

( D α + α I α + α ) ( x ) = y ( x ) ( α ∈ ℂ , R e ( α ) > 0 ) (7)

For suitable function y ( x ) from (5). For real α > 0 , Equation (4) takes the form

( D α + α I α + α ) ( x ) = ( d d x ) n 1 Γ ( n − α ) ∫ α x y ( t ) d t ( x − t ) α − n + 1 , n = [ α ] + 1 , (8)

where [ α ] means the integral part of α in particular if 0 < α < 1 ,

( D α + α y ) ( x ) = d d x 1 Γ ( 1 − α ) ∫ α x y ( t ) d t ( x − t ) α , (9)

And if α = n ∈ ℕ = { 1 , 2 , ⋯ } , then ( D α + α y ) ( x ) ≡ ( D n y ) ( x ) , ( D = d d x ) is the usual derivative of order n [

In mathematics, many complex concepts developed from simple concepts. For example, we can refer to the extension of natural number to the real one in some mathematical formulae. Let’s give an example to clarify: the factorial of a non-negative integer n [

Γ ( x ) = ∫ 0 ∞ t x − 1 e − t d t (10)

One property of the function for n ∈ ℝ + is

Γ ( n + 1 ) = n Γ ( n ) (11)

Hence, this function is equal to factorial for the integer numbers. As a result, the gamma function could be considered as an extension of factorial function to real numbers. For instance, according to the above formalism, a factorial of 1/2 can be obtained as follows:

( 1 2 ) ! = Γ ( 3 2 ) = ( 3 2 ) Γ ( 1 2 ) = 3 π 2 (12)

According to Wikipedia, the gamma function can be seen as the solution to find a smooth curve that connects the points ( x , y ) given by y = ( x − 1 ) ! at the positive integer values for x.

Lemma 3.1. Let 1 < α ≤ 2 . The following relations hold:

1) if y ∈ L ∞ ( J ) , then D c α I α y ( t ) = y ( t ) ,

2) if y ∈ A C 1 ( J ) , then

I α c D α y ( t ) = y ( t ) − y ( 0 ) − y ′ ( 0 ) t .

then y ∈ A C 1 ( J ) is a solution of the boundary value problem

D c α y ( t ) + μ ( t ) = 0 , t ∈ J , y ( 0 ) = y 0 , y ( 1 ) = y 1 , (13)

if and only if y satisfies:

y ( t ) = ∫ 0 1 G ( t , z ) μ ( z ) d z + ( y 1 − y 0 ) t + y 0 , t ∈ J , (14)

where the Green’s function associated with (1) is defined by

G ( t , z ) = 1 Γ ( α ) { t ( 1 − z ) α − 1 − ( t − z ) α − 1 , 0 ≤ z ≤ t ≤ 1 t ( 1 − z ) α , 0 ≤ z ≤ t ≤ 1 (15)

Proof By Lemma 2.1 (2), we deduce from equation (13) that

0 = I α c D α y ( t ) + I α μ ( t ) = y ( t ) − y ′ ( 0 ) + ∫ 0 t ( t − z ) α − 1 Γ ( α ) μ ( z ) d z ,

and the boundary conditions give:

y ( 0 ) = y 0 , y ′ ( 0 ) = y 1 − y 0 + ∫ 0 t ( t − z ) α − 1 Γ ( α ) μ ( z ) d z ,

Thus, we obtain that:

y ( t ) = − ∫ 0 t ( t − z ) α − 1 Γ ( α ) μ ( z ) d z + t ∫ 0 t ( t − z ) α − 1 Γ ( α ) μ ( z ) d z + ( y 1 − y 0 ) t + y 0 = ∫ 0 t G ( t , z ) μ ( z ) d z + ( y 1 − y 0 ) t + y 0 , t ∈ J .

Inversely, if y ∈ A C 1 ( J ) satisfies (2), then we have:

y ( t ) = − I α μ ( t ) + ( ∫ 0 t ( t − z ) α − 1 Γ ( α ) μ ( z ) d z + y 1 − y 0 ) t + y 0 .

(which shows that D c α t = D c α 1 = 0 ) we have:

D c α y ( t ) = − μ ( t ) , t ∈ J ,

that is, y is a proof of (1) [

Definition 3.2 If α ∈ ℝ , then the fractional derivative α D exists almost everywhere on D . If α > 0 is not an integer, then it is expressed as follows:

( D c α μ ) ( t ) = 1 Γ ( n − α ) ∫ 0 t ( t − z ) n − α − 1 μ ( n ) ( z ) d z (16)

where n = [ α ] + 1 and [ α ] stands for the largest integer not greater than α [

Let’s use this approach to extend the concept of derivative to non-integer order; consider nth derivation of power function g ( x ) .

g ( x ) = x k , x ≥ 0 (17)

d n d x n g ( x ) = k ! ( k − n ) ! x k − n = Γ ( 1 + k ) Γ ( 1 + k − n ) x k − n (18)

where k and n are real integer number respectively, and k ≥ n . To generalize the above equation, it could be possible to extend the integer number n to a real value named α :

d α d x α g ( x ) = Γ ( 1 + k ) Γ ( 1 + k − α ) x k − α (19)

Then for fractional derivative of an arbitrary function, expand the function in a power series of x first, and then by using equation (18) derivate the expansion. For example, for derivative f ( x ) = e k x to α order, we rewrite f ( x ) function as follows:

f ( x ) = 1 + x + x 2 2 ! + x 3 3 ! + ⋯ (20)

Hence

d α d x α f ( x ) = 1 Γ ( 1 − α ) x − α + 1 Γ ( 2 − α ) x 1 − α + 1 Γ ( 3 − α ) x 3 − α + ⋯ = s i g n ( x ) ( s i g n ( x ) k ) α e k x ( 1 − Γ ( − α , k x ) Γ ( 2 α ) ) (21)

That Γ ( − α , k x ) is incomplete gamma function [

This is an arbitrary way to define fractional derivative and not the only way, for example, it is possible to use an exponential function f ( x ) = e k x instead of a power function, we can define:

D α f ( x ) = k α e k x (22)

The fractional derivative of the exponential function obtained by Liouville in 1832 [

In this section, we shall exploit the concept of Riemann-Liouville type fractional derivative operator [

D z v { f ( z ) } = ∫ 0 z ( z − t ) − v − 1 f ( t ) d t , R ( v ) < 0 (23)

where the integration path is a line from 0 to z in the complex t-plane [

The Riemann-Liouville fractional derivative of f ( z ) of order v is defined as:

D z v { f ( z ) } = d m d z m D z v − m { f ( z ) } = d m d z m { 1 Γ ( − v + m ) ∫ 0 z ( z − t ) − v + m − 1 f ( t ) d t } , R ( v ) < 0 (24)

When R ( v ) , let m ∈ ℕ be the smallest integer greater than R ( v ) and so m – 1 ≤ R ( v ) < m , then a new Riemann-Liouville fractional derivative of f ( z ) of order v can be defined as follows:

D z v , [ p ] q { f ( z ) } = d m d z m D z v − m , [ p ] q { f ( z ) } = d m d z m { 2 p π Γ ( − v + m ) ∫ 0 z f ( t ) ( z − t ) − v + m − 3 2 K q + 1 2 ( p z 2 t ( z − t ) ) d t } , ( ℜ ( p ) > 0 , ℜ ( q ) > 0 ) (25)

Remark On setting p = 0 , q = 0 in (18) and (19) we are left with the classical Riemann-Liouville fractional derivative. In the case q = 0 in Equations (24) and (25) reduces to the well-known fractional derivative operator given in [

Theorem 5.1

Let m − 1 ≤ ℜ ( v ) < m < R ( λ ) for some m ∈ ℕ . Suppose that a function f ( z ) is analytic at the origin with its Maclaurin expansion given by f ( z ) = ∑ n = 0 ∞ a n z n , ( | z | < ξ ) for some ξ ∈ ℝ + . Then we have

D z v , [ p ] q { z λ − 3 2 f ( z ) } = z λ − v − 2 Γ ( − v ) ∑ n = 0 ∞ a n B p , q ( λ + n , − v ) z n . (26)

Proof Now applying (24) in the definition (25) to the function z λ − 3 2 f ( z ) , and changing the order of integration and summation, we obtain

D z v , [ p ] q { z λ − 3 2 f ( z ) } = 2 p π Γ ( − v ) ∑ n = 0 ∞ a n ∫ 0 z t λ + n − 3 2 ( z − t ) − v − 3 2 K q + 1 2 ( p z 2 t ( z − t ) ) d t (27)

Putting t = ξ z in (27), we obtain

D z v , [ p ] q { z λ − 3 2 f ( z ) } = z λ − v − 2 2 p π Γ ( − v ) ∑ n = 0 ∞ a n z n ∫ 0 1 ξ λ + n − 3 2 ( 1 − ξ ) − v − 3 2 K q + 1 2 ( p ξ ( z − ξ ) ) d ξ . (28)

The applying definition of extended beta function, and after some simplification, we get the desired result as follows:

D z v , [ p ] q { z λ − 3 2 log z f ( z ) } = ∑ n = 0 ∞ z λ + n − v − 2 { a n log ( z ) B p , q ( λ + n , v ) + b n B p , q ( λ + n , − v + 1 ) } , (29)

Example 5.2 Let n – 1 ≤ ( v ) < n < ( λ ) for some m ∈ ℕ . Then we have

D z v , [ p ] q { z λ } = B p , q ( λ + 3 2 − v ) Γ ( − v ) z λ − v − 2 (30)

Solution We have applied the definition of the fractional derivative, we obtain

D z v , [ p ] q { z λ } = 2 p π Γ ( − v ) ∫ 0 z f ( t ) ( z − t ) − v + m − 3 2 K q + 1 2 ( p z 2 t ( z − t ) ) d t (31)

Putting t = μ z in (24), we obtain

D z v , [ p ] q { z λ } = z λ − v − 1 2 2 p π Γ ( − v ) ∫ 0 1 μ λ ( z − μ ) − v + − 3 2 K q + 1 2 ( p μ ( z − μ ) ) d μ (32)

We apply the definition of the extended beta function, we obtain the solution [

Example 5.3 Let n – 1 ≤ ( v ) < n < ( λ ) for some m ∈ ℕ . Then we have

D z λ − v , [ p ] q { z λ − 3 2 ( 1 − z ) − α } = Γ ( λ ) Γ ( v ) G p , q ( α , λ , v , z ) z v − 2 . (33)

Solution Applying the definition of the new fractional derivative operator, we obtain

D z λ − v , [ p ] q { z λ − 3 2 ( 1 − z ) − α } = 2 p π Γ ( − v ) ∫ 0 1 z λ − 3 2 ( 1 − t ) − α ( z − t ) v − λ − 3 2 K q + 1 2 ( p z 2 t ( z − t ) ) d t . (34)

Putting t = μ z in (27), we obtain

D z λ − v , [ p ] q { z λ − 3 2 ( 1 − z ) − α } = z v − 2 2 p π Γ ( v − λ ) ∫ 0 1 μ λ − 3 2 ( 1 − μ ) − α ( z − z μ ) − α K q + 1 2 ( p μ ( z − μ ) ) d μ . (35)

We apply the definition of the extended hypergeometric function, we get the solution [

In this paper, we have special issue to establish a collection of functions in the field of fractional calculus and explore the scope for applications in applied sciences. We have also defined fractional calculus, fractional differential, and an interesting Riemann-Liouville fractional derivative operator. In addition to that, we have important properties of the new fractional derivative operator. And we apply the definition of the new fractional derivative operator. We define the extended beta function that we obtained.

As an application of our new operator, we have established some interesting generating functions for the extended hypergeometric function F p , q , using the new operator.

I would like to thank my supervisor, Dr. Muhsin Hassan Abdallah who was a great help to me and also I thanks my husband Bashir Alfadol Albdawi without whose help, I could not have written this paper.

The author declares no conflicts of interest regarding the publication of this paper.

Mu’lla, M.A.M. (2020) Fractional Calculus, Fractional Differential Equations and Applications. Open Access Library Journal, 7: e6244. https://doi.org/10.4236/oalib.1106244