Generalized Fractional Derivative, Fractional differential ring

There are many possible definitions of derivatives, here we present some and present one that we have called generalized that allows us to put some of the others as a particular case of this but, what interests us is to determine that there is an infinite number of possible definitions of fractional derivatives, all are correct as differential operators each of which must be properly defined in its algebra. We introduce a generalized version of the fractional derivative that extends the existing ones in the literature. To those extensions, it is associated with a differentiable operator and a differential ring and applications that show the advantages of the generalization. We also review the different definitions of fractional derivatives proposed by Michele Caputo in \cite{GJI:GJI529}, Khalil, Al Horani, Yousef, Sababheh in \cite{khalil2014new}, Anderson and Ulness in \cite{anderson2015newly}, Guebbai and Ghiat in \cite{guebbai2016new}, Udita N. Katugampola in \cite{katugampola2014new}, Camrud in \cite{camrud2016conformable} and it is shown how the generalized version contains the previous ones as a particular case.

existing in the literature and where the different algebras are unified under the notion of fractional differential ring.
The present paper is organized as follows: In the second section we give the previous definitions of fractional derivative and our generalized fractional derivative(GFD) definition, in the third section we introduce a fractional differential ring, in the forth section we give some result of GFD, in the fifth section we give a definition of partial fractional differential derivative, in the sixth section we give a definition of GFD when α ∈ (n, n + 1].

Fractional derivative
Let α ∈ (0, 1] be a fractional number, we want to give a definition of generalized fractional derivative of order α for a differentiable function f .We denote α−th derivative of f by D α (f ) and we denote the first derivative of f by D(f ).
We begin the present section listing previous definition of fractional derivative; later we present our proposal of generalized one showing how it contains the once already described.We finish the section providing some examples.
1.The Caputo fractional derivative was defined by Michele Caputo in [Cap69]: (1) 2. The conformable fractional derivative was defined by Khalil, Al Horani, Yousef and Sababheh in [KAHYS14] : (2) 3. The conformable fractional derivative was defined by Anderson and Ulness in [AU15a]: 4. The fractional derivative was defined by Udita N.Katugampola in [Kat14]: 5. The fractional derivative was defined by Guebbai and Ghiat in [GG16] for an increasing and positive function f : (5) 6.The conformable ratio derivative was defined by Camrud in [Cam16] for a function f (t) ≥ 0 with Df (t) ≥ 0: From all these definitions, we propose a definition that unifies almost all of them.
Definition 1.Given a differentiable function f : [0, ∞) → R, the generalized fractional derivative(GFD) for α ∈ (0, 1] at point t is defined by : , where w t,α is a function that may depend on α and t.
Remark 1.As a consequence of definition 1 we can see We denote C α [0, ∞) the set of α−generalized differentiable functions with real values in the interval [0, ∞) in variable t.The set (C α [0, ∞), +, .) is a ring.In the following we want to see the relation between GFD and the others definitions: 1.The fractional derivative of Khalil, Al Horani, Yousef and Sababheh in [KAHYS14] is a particular case of GFD where w t,α = 1.
2. The fractional derivative of Anderson and Ulness in [AU15a] is a particular case of GFD where In this fractional derivative w t,α depends on α and t.
We are particularly interested in discussing GFD where w t,α = g(t, α)τ α−1 such that g : [0, ∞) × (0, 1] → R is a function and τ is the characteristic of system with the properties If the system is periodic with period T , then we have τ = T .In the quantum systems τ is the Bohr radius and in astronomy τ is the light year.The characteristic of system τ depends on the systems and the derivative.If t is time, τ is time too.If t is space, τ is space too.In fact the unit of t is τ , i.e., t = cτ where c is a constant.In the general τ = 1. 2. If g(t, α) = α, we have the chain rule Example 4. We want to present the corresponding figure to the generalized fractional derivatives for α = 3 4 for a trigonometric, using all the fractional derivative definitions that we have already mentioned in this article.It can be seen from all the figures that in principle these definitions do not find a reason to discard them.That is, they have a fairly reasonable behavior.We consider f (t) = sin(2t), the figure of f can be seen in the following picture.

Generalized fractional differential ring
In this section we want to stress out that instead of defining a new derivative, we focus on the notion of differentiable operator and the ring that it carries with.Definition 5. Let R be a commutative ring with unity.A derivation on R is a map Proof.Since C α [0, ∞) is a commutative ring with unity f (t) = 1 and the derivation D α for α ∈ (0, 1] satisfies following properties from remark (1) Let α ∈ (0, 1] be a fractional number, let f 1 , f 2 ∈ C α [0, ∞) be two functions, GFD has the following properties: It is easy to see these properties from remark (1).If w t,α = t 1−α we have the equality wt,αw t,β = t we have the equality Parts 4 and 5 of the properties imply that we can create function spaces with different algebras using different expressions for w t,α .By considering previous properties of GFD we called C α [0, ∞) a w t,α −generalized fractional differential ring of functions and we denote it by By using the previous properties we can see the following result: Theorem 7. Let α ∈ (0, 1].Associated to any α and any w t,α there exists a fractional differential ring.

Some results of Generalized Fractional Derivative
Let α ∈ (0, 1] be a fractional number and t ∈ [0, ∞) then GFD has the following properties: ) ) = e But two limits have different signs.Then D α f (c) = 0.
Theorem 9. (Mean Value Theorem for α−Generalized Fractional Differentiable Functions) Let a > 0 and f : [a, b] → R be a function with the properties that Then, there exists c ∈ (a, b) such that Proof.Consider function Then, the function h satisfies the conditions of the fractional Rolle's theorem.Hence, there exists c ∈ (a, b) such that D α h(c) = 0. We have the result since

Generalized Partial Fractional Derivative
In this section we introduce a partial fractional derivative of first and second order, also we introduce a partial fractional differential ring.
R be a function with n variables such that ∀i, there exists the partial derivative of f respect to t i .Let α ∈ (0, 1] be a fractional number.We define α−generalized partial fractional derivative(GPFD) of f with respect to t i at point t = (t 1 , . . ., t n ) , where w t i ,α can be a function depend on α and t i .
By considering previous properties of GPFD we called the ring C α i [0, ∞) n a w t i ,α − generalized partial fractional differential ring.We denote it by ).We can see the following result by using the previous properties.
Theorem 12. Let α ∈ (0, 1] and 1 ≤ i ≤ n.Associated to any α and any w t i ,α there is a partial fractional differential ring.
Definition 14.Let α ∈ (0, 1] be a fractional number.We define α−generalized partial fractional derivative of second order with respect to t i and t j at point t Remark 3. As a consequence of definition (14) we can see for α ∈ (0, 1] and 1 ≤ i, j ≤ n; A partial differentiable function of second order f : [0, ∞) n → R is said to be a α−generalized fractional partial differentiable function of second order respect to t i and t ) n the set of α−generalized partial fractional differentiable functions of second order respect to t i and t j with real values in the interval [0, ∞) n in variable t = (t 1 , . . ., t n ).The set (C α 2 i,j [0, ∞) n , +, .) is a ring.
The generalized fractional derivative of order α is defined by where [α] is the smallest integer greater than or equal to α.

α-Fractional Taylor Series
There are some articles about fractional Taylor series see( [AU15b] , [Use08], [Mun04], [Yan11]).In this section we use GFD to define a fractional taylor series for a function f ∈ C r [0, ∞) for every fractional number r.
Let 0 < α < 1, we define the α-fractional taylor series of f at real number x 0 Let 1 < α ≤ 2, we define the α-fractional taylor series of f at real number x 0 Let n < α ≤ n + 1 such that α = n + A with 0 < A < 1 we define the α-fractional taylor series of f at real number x 0 ,

Application to differential equations
There are some articles for applications of fractional differential derivative such as [YZUHR13], [Aga12] ,[AHK14], [WZD17].In this section we solve some (partial)fractional differential equations by using our definitions.At the first we solve the fractional differential equations with the form aD α y + by = c, where y = f (t) be a differentiable function and 0 < α < 1.By substituting GFD in the equation ( 7) we have Example 17.We consider the partial fractional differential equation with boundary conditions where u(x, t) be a differentiable function respect to x and t, u(x, t) be a 1 3 −partial fractional differentiable function of first order respect to x , u t = ∂u ∂t and 3 3 by using remark (2) we can write We solve this equation by taking Laplace transform of equation respect to t, we denote by U (x, s) the Laplace of u(x, t) respect to t, we have the following equation .
Example 18.We consider the partial fractional differential equation of second order; where u(x, t) be a 1 5 −fractional partial differentiable function of second order respect to t, x.For w x, 1 5 = x 2 , w t,, 1 5 = 1 3 √ t by using remark (3) we have We consider a solution of this differential equation with the form u(x, t) = f (x)g(t) such that f a function depends on x and g a function depends on t.By substituting u(x, t) in the equation (11) we have We can write the equation (12) a form that divide the functions of t and x: two sides of the equality (13) is a constant k.We have ).

Conclusion
We define a generalized fractional derivative(GFD).We show that the previous derivatives are particular cases.We show how it is possible to have infinite fractional derivatives with their algebra.We present the fractional differential ring, the fractional partial derivatives and their applications.